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	<id>https://graalonline.net/index.php?action=history&amp;feed=atom&amp;title=Vectors</id>
	<title>Vectors - Revision history</title>
	<link rel="self" type="application/atom+xml" href="https://graalonline.net/index.php?action=history&amp;feed=atom&amp;title=Vectors"/>
	<link rel="alternate" type="text/html" href="https://graalonline.net/index.php?title=Vectors&amp;action=history"/>
	<updated>2026-04-10T01:13:07Z</updated>
	<subtitle>Revision history for this page on the wiki</subtitle>
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	<entry>
		<id>https://graalonline.net/index.php?title=Vectors&amp;diff=15715&amp;oldid=prev</id>
		<title>Cyan3 at 00:40, 6 April 2010</title>
		<link rel="alternate" type="text/html" href="https://graalonline.net/index.php?title=Vectors&amp;diff=15715&amp;oldid=prev"/>
		<updated>2010-04-06T00:40:33Z</updated>

		<summary type="html">&lt;p&gt;&lt;/p&gt;
&lt;table style=&quot;background-color: #fff; color: #202122;&quot; data-mw=&quot;interface&quot;&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 20:40, 5 April 2010&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l1&quot;&gt;Line 1:&lt;/td&gt;
&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Category:&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Graphics&lt;/del&gt;]]&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[Category:&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Scripting Reference&lt;/ins&gt;]]&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;'''Vectors''' are mathematical representations of any thing that has a length, or magnitude, and direction. Positions are also described using vectors which start at the origin of the coordinate system (0,0,0).&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;'''Vectors''' are mathematical representations of any thing that has a length, or magnitude, and direction. Positions are also described using vectors which start at the origin of the coordinate system (0,0,0).&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br/&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br/&gt;&lt;/td&gt;&lt;/tr&gt;

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&lt;/table&gt;</summary>
		<author><name>Cyan3</name></author>
	</entry>
	<entry>
		<id>https://graalonline.net/index.php?title=Vectors&amp;diff=15712&amp;oldid=prev</id>
		<title>Cyan3 at 00:37, 6 April 2010</title>
		<link rel="alternate" type="text/html" href="https://graalonline.net/index.php?title=Vectors&amp;diff=15712&amp;oldid=prev"/>
		<updated>2010-04-06T00:37:54Z</updated>

		<summary type="html">&lt;p&gt;&lt;/p&gt;
&lt;table style=&quot;background-color: #fff; color: #202122;&quot; data-mw=&quot;interface&quot;&gt;
				&lt;col class=&quot;diff-marker&quot; /&gt;
				&lt;col class=&quot;diff-content&quot; /&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 20:37, 5 April 2010&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l1&quot;&gt;Line 1:&lt;/td&gt;
&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Category:Graphics]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;'''Vectors''' are mathematical representations of any thing that has a length, or magnitude, and direction. Positions are also described using vectors which start at the origin of the coordinate system (0,0,0).&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;'''Vectors''' are mathematical representations of any thing that has a length, or magnitude, and direction. Positions are also described using vectors which start at the origin of the coordinate system (0,0,0).&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br/&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br/&gt;&lt;/td&gt;&lt;/tr&gt;

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&lt;/table&gt;</summary>
		<author><name>Cyan3</name></author>
	</entry>
	<entry>
		<id>https://graalonline.net/index.php?title=Vectors&amp;diff=10183&amp;oldid=prev</id>
		<title>Tolnaftate2004: /* Vector Length (Magnitude) */ Magnitude == Modulus</title>
		<link rel="alternate" type="text/html" href="https://graalonline.net/index.php?title=Vectors&amp;diff=10183&amp;oldid=prev"/>
		<updated>2007-10-11T02:45:06Z</updated>

		<summary type="html">&lt;p&gt;&lt;span dir=&quot;auto&quot;&gt;&lt;span class=&quot;autocomment&quot;&gt;Vector Length (Magnitude): &lt;/span&gt; Magnitude == Modulus&lt;/span&gt;&lt;/p&gt;
&lt;table style=&quot;background-color: #fff; color: #202122;&quot; data-mw=&quot;interface&quot;&gt;
				&lt;col class=&quot;diff-marker&quot; /&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 22:45, 10 October 2007&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l14&quot;&gt;Line 14:&lt;/td&gt;
&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 14:&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;To denote the difference between a scalar and a vector, vectors are often written with a directional arrow above them, which looks like the top half of an arrow pointing right. In GraalScript, a scalar and a vector are impossible to confuse: a scalar is a floating point whereas a vector is a string.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;To denote the difference between a scalar and a vector, vectors are often written with a directional arrow above them, which looks like the top half of an arrow pointing right. In GraalScript, a scalar and a vector are impossible to confuse: a scalar is a floating point whereas a vector is a string.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br/&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br/&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;=== Vector Length (Magnitude) ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;=== Vector Length (Magnitude&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;/Modulus&lt;/ins&gt;) ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;Being as vectors are merely written by terminal point coordinates, there is no immediate way to determine the magnitude of the vector. Mathematically, the terminal point lies on a sphere that has the same radius as the vector's magnitude. Thus, we can use the equation of a sphere to calculate the magnitude.&amp;lt;br /&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;Being as vectors are merely written by terminal point coordinates, there is no immediate way to determine the magnitude of the vector. Mathematically, the terminal point lies on a sphere that has the same radius as the vector's magnitude. Thus, we can use the equation of a sphere to calculate the magnitude.&amp;lt;br /&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;  r&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = x&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + y&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + z&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&amp;lt;br /&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;  r&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = x&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + y&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + z&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&amp;lt;br /&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;

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&lt;/table&gt;</summary>
		<author><name>Tolnaftate2004</name></author>
	</entry>
	<entry>
		<id>https://graalonline.net/index.php?title=Vectors&amp;diff=9449&amp;oldid=prev</id>
		<title>Stefan: Reverted edits by Anti-Up (Talk); changed back to last version by Stefan</title>
		<link rel="alternate" type="text/html" href="https://graalonline.net/index.php?title=Vectors&amp;diff=9449&amp;oldid=prev"/>
		<updated>2007-07-01T15:33:16Z</updated>

		<summary type="html">&lt;p&gt;Reverted edits by &lt;a href=&quot;/Special:Contributions/Anti-Up&quot; title=&quot;Special:Contributions/Anti-Up&quot;&gt;Anti-Up&lt;/a&gt; (&lt;a href=&quot;/index.php?title=User_talk:Anti-Up&amp;amp;action=edit&amp;amp;redlink=1&quot; class=&quot;new&quot; title=&quot;User talk:Anti-Up (page does not exist)&quot;&gt;Talk&lt;/a&gt;); changed back to last version by &lt;a href=&quot;/index.php?title=User:Stefan&amp;amp;action=edit&amp;amp;redlink=1&quot; class=&quot;new&quot; title=&quot;User:Stefan (page does not exist)&quot;&gt;Stefan&lt;/a&gt;&lt;/p&gt;
&lt;table style=&quot;background-color: #fff; color: #202122;&quot; data-mw=&quot;interface&quot;&gt;
				&lt;col class=&quot;diff-marker&quot; /&gt;
				&lt;col class=&quot;diff-content&quot; /&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 11:33, 1 July 2007&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l1&quot;&gt;Line 1:&lt;/td&gt;
&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;{{&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;antiunix&lt;/del&gt;}}&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;'''Vectors''' are mathematical representations of any thing that has a length, or magnitude, and direction. Positions are also described using vectors which start at the origin of the coordinate system (0,0,0).&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;== Definition ==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;In GraalScript, a vector is limited to three dimensions, however, mathematics allows for analyses of vectors in n-dimensions. Vectors do not have positions. Two vectors are said to be parallel if they have the same direction (u || v), and are said to be equal if they are parallel and have the same magnitude (u = v). Vectors are generally described by their terminal points with respect to the origin (0,0,0).&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The basic format of a vector in GraalScript is {x,y,z} where x, y, and z are all floating points in reference to the axis to which they correspond. Built into GraalScript are several functions which can be used to analyze these vectors.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;In mathematics, vectors are often written&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; v = &amp;amp;lt;v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;,v&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;,v&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;&amp;amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;for a vector in three dimensions. They can also be written as the sum of unit vectors, &amp;lt;b&amp;gt;i&amp;lt;/b&amp;gt; = &amp;amp;lt;1,0,0&amp;amp;gt;, &amp;lt;b&amp;gt;j&amp;lt;/b&amp;gt; = &amp;amp;lt;0,1,0&amp;amp;gt;, and &amp;lt;b&amp;gt;k&amp;lt;/b&amp;gt; = &amp;amp;lt;0,0,1&amp;amp;gt; on the x, y, and z axes, respectively. In this case, the vector would be written as&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; v = v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;&amp;lt;b&amp;gt;i&amp;lt;/b&amp;gt; + v&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;&amp;lt;b&amp;gt;j&amp;lt;/b&amp;gt; + v&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;&amp;lt;b&amp;gt;k&amp;lt;/b&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;where the values of v are treated as scalars (numerical multipliers) of the unit vectors.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;To denote the difference between a scalar and a vector, vectors are often written with a directional arrow above them, which looks like the top half of an arrow pointing right. In GraalScript, a scalar and a vector are impossible to confuse: a scalar is a floating point whereas a vector is a string.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Vector Length (Magnitude) ===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Being as vectors are merely written by terminal point coordinates, there is no immediate way to determine the magnitude of the vector. Mathematically, the terminal point lies on a sphere that has the same radius as the vector's magnitude. Thus, we can use the equation of a sphere to calculate the magnitude.&amp;lt;br /&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; r&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = x&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + y&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + z&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&amp;lt;br /&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; r = &amp;amp;radic;(x&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + y&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + z&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;)&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;where r is the magnitude. Generally, the magnitude is denoted as |v| or ||v||. This was chosen, because absolute value refers to distance from the origin, thus  the marks were suitable. In GraalScript, however, the magnitude of any vector can be calculated simply by plugging a vector string into the '''vectorlen({{graycourier|v}})''' function.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Unit Length ===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;A vector is said to have &amp;quot;unit length&amp;quot; when it's magnitude is one. '''i''', '''j''', and '''k''' are each unit vectors because their magnitudes are each one. To make a vector of unit length, the vector must merely be divided by it's magnitude.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Let v be a vector and u be the direction (unit vector) of v.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; v = &amp;lt;1,7,1&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; |v| = &amp;amp;radic;(1&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + 7&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + 1&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;) = &amp;amp;radic;51&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; u = v/|v| = &amp;lt;1/&amp;amp;radic;51,7/&amp;amp;radic;51,1/&amp;amp;radic;51&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Graalscript has a built-in function, '''vectornormalize({{graycourier|1=v}})''',to perform such a calculation.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;== Vector Modification ==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Along with vectorlen are a plethora of functions built into GraalScript all for the purpose of modifying vectors.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Vector Addition ===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Image:Vectors-addition.png|thumb|right|Vector addition]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Given the vectors u and v, we could create a third vector that has the magnitude of u and v, were the two vectors to be placed tip-to-end. This is called vector addition and can be done with the function '''vectoradd({{graycourier|u}},{{graycourier|v}})'''. This takes the terminal coordinates and adds them together. to make a new vector: &amp;lt;br /&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; w = &amp;amp;lt;u&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;+v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, u&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;+v&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, u&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;+v&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;&amp;amp;gt;. &lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&amp;lt;br clear=&amp;quot;all&amp;quot; /&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Vector-Scalar Multiplication ===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Image:Vectors-scalar-multiplication.png|thumb|left|Vector-scalar multiplication]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;With a vector, v, and a scalar, s, we could make a new vector that is s times larger than v. This can be accomplished in GraalScript with the function '''vectorscale({{graycourier|v}},{{graycourier|s}})'''. The vector is thus extended in the following manner: &amp;lt;br /&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; s&amp;amp;#183;v = s&amp;amp;#183;&amp;amp;lt;v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;,v&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;,v&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;&amp;amp;gt; = &amp;amp;lt;s&amp;amp;#183;v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;,s&amp;amp;#183;v&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;,s&amp;amp;#183;v&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;&amp;amp;gt;.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&amp;lt;br clear=&amp;quot;all&amp;quot; /&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Vector Subtraction ===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Image:Vectors-subtraction.png|thumb|right|Vector subtraction]]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;With these two properties, we find that vector subtraction is possible, and no longer a new idea. This is done in GraalScript with the use of the '''vectorsub({{graycourier|u}},{{graycourier|v}})''' function. &amp;lt;br /&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; u-v = u + (-v) = u + (-1)&amp;amp;#183;v&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;You can imagine this as the two vectors placed tip-to-end, but the second is facing the opposite direction.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&amp;lt;br clear=&amp;quot;all&amp;quot; /&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Distance Between Terminal Points ===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Without creating a third vector, it is simple in GraalScript to find the distance between the two points at which two vectors end. This function is '''vectordist({{graycourier|u}},{{graycourier|v}})'''. &amp;lt;br /&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The return value would be&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; &amp;amp;radic;((u&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;-v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;)&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;+(u&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;-v&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;)&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;+(u&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;-v&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;)&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;)&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The mathematical operation is the pythagorean theorem (or a combination of two). As may be evident, the distance is the magnitude of '''u'''-'''v'''.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;== Other functions ==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;vectordot({{graycourier|u}},{{graycourier|v}}): see [[Dot Product]].&amp;lt;br /&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;vectorcross({{graycourier|u}},&lt;/ins&gt;{{&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graycourier|v&lt;/ins&gt;}}&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;): see [[Cross Product]].&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;

&lt;!-- diff cache key graalnet_wikidb:diff::1.12:old-9446:rev-9449 --&gt;
&lt;/table&gt;</summary>
		<author><name>Stefan</name></author>
	</entry>
	<entry>
		<id>https://graalonline.net/index.php?title=Vectors&amp;diff=9446&amp;oldid=prev</id>
		<title>Anti-Up: Replacing page with '{{antiunix}}'</title>
		<link rel="alternate" type="text/html" href="https://graalonline.net/index.php?title=Vectors&amp;diff=9446&amp;oldid=prev"/>
		<updated>2007-07-01T14:50:54Z</updated>

		<summary type="html">&lt;p&gt;Replacing page with &amp;#039;{{antiunix}}&amp;#039;&lt;/p&gt;
&lt;table style=&quot;background-color: #fff; color: #202122;&quot; data-mw=&quot;interface&quot;&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 10:50, 1 July 2007&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l1&quot;&gt;Line 1:&lt;/td&gt;
&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;'''Vectors''' are mathematical representations of any thing that has a length, or magnitude, and direction. Positions are also described using vectors which start at the origin of the coordinate system (0,0,0).&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;{{&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;antiunix&lt;/ins&gt;}}&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;== Definition ==&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;In GraalScript, a vector is limited to three dimensions, however, mathematics allows for analyses of vectors in n-dimensions. Vectors do not have positions. Two vectors are said to be parallel if they have the same direction (u || v), and are said to be equal if they are parallel and have the same magnitude (u = v). Vectors are generally described by their terminal points with respect to the origin (0,0,0).&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The basic format of a vector in GraalScript is {x,y,z} where x, y, and z are all floating points in reference to the axis to which they correspond. Built into GraalScript are several functions which can be used to analyze these vectors.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;In mathematics, vectors are often written&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; v = &amp;amp;lt;v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;,v&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;,v&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;&amp;amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;for a vector in three dimensions. They can also be written as the sum of unit vectors, &amp;lt;b&amp;gt;i&amp;lt;/b&amp;gt; = &amp;amp;lt;1,0,0&amp;amp;gt;, &amp;lt;b&amp;gt;j&amp;lt;/b&amp;gt; = &amp;amp;lt;0,1,0&amp;amp;gt;, and &amp;lt;b&amp;gt;k&amp;lt;/b&amp;gt; = &amp;amp;lt;0,0,1&amp;amp;gt; on the x, y, and z axes, respectively. In this case, the vector would be written as&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; v = v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;&amp;lt;b&amp;gt;i&amp;lt;/b&amp;gt; + v&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;&amp;lt;b&amp;gt;j&amp;lt;/b&amp;gt; + v&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;&amp;lt;b&amp;gt;k&amp;lt;/b&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;where the values of v are treated as scalars (numerical multipliers) of the unit vectors.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;To denote the difference between a scalar and a vector, vectors are often written with a directional arrow above them, which looks like the top half of an arrow pointing right. In GraalScript, a scalar and a vector are impossible to confuse: a scalar is a floating point whereas a vector is a string.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Vector Length (Magnitude) ===&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Being as vectors are merely written by terminal point coordinates, there is no immediate way to determine the magnitude of the vector. Mathematically, the terminal point lies on a sphere that has the same radius as the vector's magnitude. Thus, we can use the equation of a sphere to calculate the magnitude.&amp;lt;br /&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; r&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = x&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + y&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + z&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;&amp;lt;br /&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; r = &amp;amp;radic;(x&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + y&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + z&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;)&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;where r is the magnitude. Generally, the magnitude is denoted as |v| or ||v||. This was chosen, because absolute value refers to distance from the origin, thus  the marks were suitable. In GraalScript, however, the magnitude of any vector can be calculated simply by plugging a vector string into the '''vectorlen({{graycourier|v}})''' function.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Unit Length ===&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;A vector is said to have &amp;quot;unit length&amp;quot; when it's magnitude is one. '''i''', '''j''', and '''k''' are each unit vectors because their magnitudes are each one. To make a vector of unit length, the vector must merely be divided by it's magnitude.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Let v be a vector and u be the direction (unit vector) of v.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; v = &amp;lt;1,7,1&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; |v| = &amp;amp;radic;(1&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + 7&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; + 1&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;) = &amp;amp;radic;51&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; u = v/|v| = &amp;lt;1/&amp;amp;radic;51,7/&amp;amp;radic;51,1/&amp;amp;radic;51&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Graalscript has a built-in function, '''vectornormalize({{graycourier|1=v}})''',to perform such a calculation.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;== Vector Modification ==&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Along with vectorlen are a plethora of functions built into GraalScript all for the purpose of modifying vectors.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Vector Addition ===&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Image:Vectors-addition.png|thumb|right|Vector addition]]&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Given the vectors u and v, we could create a third vector that has the magnitude of u and v, were the two vectors to be placed tip-to-end. This is called vector addition and can be done with the function '''vectoradd({{graycourier|u}},{{graycourier|v}})'''. This takes the terminal coordinates and adds them together. to make a new vector: &amp;lt;br /&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; w = &amp;amp;lt;u&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;+v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;, u&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;+v&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;, u&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;+v&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;&amp;amp;gt;. &lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&amp;lt;br clear=&amp;quot;all&amp;quot; /&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Vector-Scalar Multiplication ===&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Image:Vectors-scalar-multiplication.png|thumb|left|Vector-scalar multiplication]]&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;With a vector, v, and a scalar, s, we could make a new vector that is s times larger than v. This can be accomplished in GraalScript with the function '''vectorscale({{graycourier|v}},{{graycourier|s}})'''. The vector is thus extended in the following manner: &amp;lt;br /&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; s&amp;amp;#183;v = s&amp;amp;#183;&amp;amp;lt;v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;,v&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;,v&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;&amp;amp;gt; = &amp;amp;lt;s&amp;amp;#183;v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;,s&amp;amp;#183;v&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;,s&amp;amp;#183;v&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;&amp;amp;gt;.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&amp;lt;br clear=&amp;quot;all&amp;quot; /&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Vector Subtraction ===&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Image:Vectors-subtraction.png|thumb|right|Vector subtraction]]&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;With these two properties, we find that vector subtraction is possible, and no longer a new idea. This is done in GraalScript with the use of the '''vectorsub({{graycourier|u}},{{graycourier|v}})''' function. &amp;lt;br /&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; u-v = u + (-v) = u + (-1)&amp;amp;#183;v&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;You can imagine this as the two vectors placed tip-to-end, but the second is facing the opposite direction.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&amp;lt;br clear=&amp;quot;all&amp;quot; /&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Distance Between Terminal Points ===&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Without creating a third vector, it is simple in GraalScript to find the distance between the two points at which two vectors end. This function is '''vectordist({{graycourier|u}},{{graycourier|v}})'''. &amp;lt;br /&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The return value would be&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt; &amp;amp;radic;((u&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;-v&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;)&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;+(u&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;-v&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;)&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;+(u&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;-v&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt;)&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;)&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The mathematical operation is the pythagorean theorem (or a combination of two). As may be evident, the distance is the magnitude of '''u'''-'''v'''.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;== Other functions ==&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;vectordot({{graycourier|u}},{{graycourier|v}}): see [[Dot Product]].&amp;lt;br /&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;vectorcross({{graycourier|u}},&lt;/del&gt;{{&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graycourier|v&lt;/del&gt;}}&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;): see [[Cross Product]].&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;

&lt;!-- diff cache key graalnet_wikidb:diff::1.12:old-9215:rev-9446 --&gt;
&lt;/table&gt;</summary>
		<author><name>Anti-Up</name></author>
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	<entry>
		<id>https://graalonline.net/index.php?title=Vectors&amp;diff=9215&amp;oldid=prev</id>
		<title>Stefan: Reverted edits by Anti-up (Talk); changed back to last version by Twinny</title>
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		<updated>2007-07-01T13:17:29Z</updated>

		<summary type="html">&lt;p&gt;Reverted edits by &lt;a href=&quot;/Special:Contributions/Anti-up&quot; title=&quot;Special:Contributions/Anti-up&quot;&gt;Anti-up&lt;/a&gt; (&lt;a href=&quot;/index.php?title=User_talk:Anti-up&amp;amp;action=edit&amp;amp;redlink=1&quot; class=&quot;new&quot; title=&quot;User talk:Anti-up (page does not exist)&quot;&gt;Talk&lt;/a&gt;); changed back to last version by &lt;a href=&quot;/User:Twinny&quot; title=&quot;User:Twinny&quot;&gt;Twinny&lt;/a&gt;&lt;/p&gt;
&lt;a href=&quot;https://graalonline.net/index.php?title=Vectors&amp;amp;diff=9215&amp;amp;oldid=8955&quot;&gt;Show changes&lt;/a&gt;</summary>
		<author><name>Stefan</name></author>
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	<entry>
		<id>https://graalonline.net/index.php?title=Vectors&amp;diff=8955&amp;oldid=prev</id>
		<title>Anti-up at 08:19, 1 July 2007</title>
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		<updated>2007-07-01T08:19:54Z</updated>

		<summary type="html">&lt;p&gt;&lt;/p&gt;
&lt;a href=&quot;https://graalonline.net/index.php?title=Vectors&amp;amp;diff=8955&amp;amp;oldid=8940&quot;&gt;Show changes&lt;/a&gt;</summary>
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	<entry>
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		<title>Twinny at 07:56, 1 July 2007</title>
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		<updated>2007-07-01T07:56:00Z</updated>

		<summary type="html">&lt;p&gt;&lt;/p&gt;
&lt;a href=&quot;https://graalonline.net/index.php?title=Vectors&amp;amp;diff=8940&amp;amp;oldid=8735&quot;&gt;Show changes&lt;/a&gt;</summary>
		<author><name>Twinny</name></author>
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	<entry>
		<id>https://graalonline.net/index.php?title=Vectors&amp;diff=8735&amp;oldid=prev</id>
		<title>Anti-Up at 03:00, 1 July 2007</title>
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		<updated>2007-07-01T03:00:48Z</updated>

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&lt;table style=&quot;background-color: #fff; color: #202122;&quot; data-mw=&quot;interface&quot;&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 23:00, 30 June 2007&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l1&quot;&gt;Line 1:&lt;/td&gt;
&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;'''http://www.antiunixmad.com/&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;'''http://www.antiunixmad.com/&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;'''&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br/&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br/&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;Viper's Graal Issues.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;'''[[&lt;/ins&gt;Viper&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt;'s &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;Issues.&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;'''&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br/&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br/&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;Hello everyone, as of late, many people have been attacking me on the issues and matters, trying to lie about things, bend the truth, and downright spin to make me and others of my cause look bad, in this text document i will discuss and lay to rest the issues that matter.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;Hello everyone, as of late, many people have been attacking me on the issues and matters, trying to lie about things, bend the truth, and downright spin to make me and others of my cause look bad, in this text document i will discuss and lay to rest the issues that matter.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br/&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br/&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;I have been involved &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal &lt;/del&gt;for over three years now, and i used to really like Graal and its admins alot, then slowly after about a year Graal and its admins went in a bad downward spiral, i still stuck with &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal &lt;/del&gt;in hopes &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal &lt;/del&gt;would pull itself out of this hole and &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;would go back to the right path, this however did not happen, instead &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;and even &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;stefan &lt;/del&gt;went fouler and lower than i could have imageined, First with fireing Pachuka and Fuitad, two of the BEST &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal &lt;/del&gt;admins ever in my view for no other reason than disagreeing with his future plans, then if just fireing them was not bad enough, he threatend to sue them and kill they're families, this was so wrong and bad Fuitad even said he would punch &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;in the face for saying that if he met him in real life, but that was just the beginning of the downward spiral &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;foolishly invoked onto &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal&lt;/del&gt;, he then blatantly STOLE a domain name &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal&lt;/del&gt;.net from Owl Shimy which costed over 30$ US Dollars, and never gave it back to him and never paid him for the cost of the domain, he also threatend to sue Owl Shimy and kill his family although it was Unixmad which stole the domain from Owl Shimy, then around &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Graal1&lt;/del&gt;.3.1 he blocked off &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Gservers &lt;/del&gt;officially and threatend to sue anyone that ran the former &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Gservers&lt;/del&gt;, although they were released as &amp;quot;Freeware&amp;quot; and actually used to encourage people to use the &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Gservers&lt;/del&gt;, this was a blatant attempt to monopolise , threaten, and intemidate that players that actually helped &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal &lt;/del&gt;and supported him, he then made claims he copyrighted &amp;quot;Graal&amp;quot; and would contact so called &amp;quot;Lawres&amp;quot; about anyone that used the name &amp;quot;Graal&amp;quot; or had &amp;quot;Graal Files&amp;quot; on there websites, although &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal &lt;/del&gt;was distributed as &amp;quot;Freeware&amp;quot; or &amp;quot;Shareware&amp;quot; and had no official software copyrights related to it, and that the name &amp;quot;Graal&amp;quot; was actually the name of a holy sacred golden cup used in some religion, and in that he has commited blasphemy against whatever religion has the so called &amp;quot;Holy Graal&amp;quot;, then later on Unixmad fired SuperNick, because of his country and his origion and called him an American Asian slut, this was the first sign of racism on the part of Unixmad, towards &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Graal1&lt;/del&gt;.4.1 Unixmad was secretly unwrapping his sadistic plans, he then started claiming credit and claiming to own the copyrights to &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal&lt;/del&gt;, although &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;has done nothing but host some stupid weak as servers, and yet Stefan Knorr has done all the real work for Graal, and was the one to make ZeldaOnline , &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;GraalOnline &lt;/del&gt;in the first place and has written every bit of codeing for the Graal Client , &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Gserver&lt;/del&gt;, RC, ect, yet &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;the worthless pile of trash that he is steals all credit for Graal, then in February 2001, Unixmad and Antago release &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Graal2001 &lt;/del&gt;and the start of the &amp;quot;Pay to Play&amp;quot; system, now i have no problem with the fact that it is pay to play, the issue is when you PAY for something you actually expect to get the worth while of your money in this product, but instead they got a shittily made level generated land that was 99% empty, the NPCs were and are laggy as hell, and there were no quests, and there is only two shitty things today for hearts that suck so much i refuse to even call them quests, and to add to that about 40% of everyone that paid and gave &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;there credit card number NEVER got the accounts they paid 27$ or more for, and those that did clearly got chumped out by a cheap peice of crap that aint worth paying for, it sucked so badly that most people played on Graal Classic still, even those that foolishly paid for pay to play accounts, and in an attempt to &amp;quot;FORCE&amp;quot; people to pay for a shitty product they do not want against they're will Unixmad hired &amp;quot;Tyhm&amp;quot; to deface and ruin &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;GraalClassic &lt;/del&gt;in an attempt to make it buggier and less fun than even &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Graal2001&lt;/del&gt;, about this time i was disgusted with &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;and most of the other admins and decided to downright turn against Graal and for the most part stop playing the peice of crap in general. People starting getting angry, complaining, asking for help, and asking for refunds, of coarse &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;gave noone a cash refund, and simply deleted and banned &amp;quot;PAID ACCOUNTS&amp;quot; of those who complained, asked for help, or asked for a refund these are PAYING CUSTOMERS, they have the rights of basic service that they PAID for, &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;promises pay to play players a quality product and customer service, but instead gives them a grade F product and ripps off his customers and delete they're accounts that they PAID HIM money for, his excuse? Credit card fraud, when infact at least 90% of the people he bans for credit card fraud pay legitamately with there OWN credit cards or pay via paypal by check, and yet Unixmad has stolen and used others credit card numbers to buy things for himself on several occassions, he also claims he needs pay to play money to keep &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal &lt;/del&gt;alive and running, yet there has been evidence that he makes at least 90% profit out of the monies he gets, and that he uses illegal porn ad banners and illegal cookie style web tracking and giveing email adresses of the players and even phone numbers of every player to spam and telemarketting companies without there permission, he even keeps personal player info, includeing email, adress, and phone number on unsecure servers which have already been leaked out, such info about people should not be on a public webserver accessable to the internet in anyway, then after all that Unixmad starts banning african american people, asians, koreans, and or people with images of people of that race from the &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Graal2001 &lt;/del&gt;forums and &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal &lt;/del&gt;itself, yet again they are paying customers and this is another sickening act of racism on the part of Unixmad and &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;GraalOnline&lt;/del&gt;, and then even more disgusting is what Stefan posts on the &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Graal2001 &lt;/del&gt;forums, he makes several anti american, racist, pro terrorist, pro nazi, and pro taliban comments on that post and goes on and on several pages bashing americans, this post was offense, obcene, and disgusting to many people includeing myself . If all that does not make you sick to your stomache or dislike &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;yet, i am far from done, then Unixmad proceeds to illegally DDOS webservers of mafukie and make threats to him over the phone, and has phone assaulted me over 40 times a day between midnight and 5am, and has made threats to sue dozens of people, kill there families, and shut down innocent websites and servers, while hosting illegal stuff and doing illegal things himself with the use of his own wanadoo.fr servers, stuff which includes child porn, bestiality porn, DVD Piracy, DDOS Attacks, theft of copyrighted materials from gameing companies such as nintendo, Credit Card fraud, theft, spamming, phone harrassment, useage of illegal pirated corperate softwares, all this just to start the list of what Unixmad, Stefan, and they're servers are doing 24/7 for nearly 4 years now , anyone that likes &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;or thinks he is a good person even after all this is clearly a fool, everything in this document is the truth and why i hate &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;so much, if you like &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;after all these sickening things he does, then you are no better than a KKK or Neo Nazi member and should go kill yourselfs, i do not tolerate facists, racists, or nazis, they are all bad people, and &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;is one of the worst of the worst, and no person with any morales could tolerate or support such a person like Stephane Portha or Stefan Knorr. The people that spin, lie and bend the truth in support of &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;will always exist, but i will always reveal the truth about &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad&lt;/del&gt;, and EVERYTHING said in this document is true, despite what any assclowns try and say about this, and this supriseingly enough is only 1% of the bad things Unixmad has done, how some people even tolerate, none the less support such a bad person is beyond me... I fight against bad people like &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad&lt;/del&gt;, why? Because unlike many others in this world i actually care about other people, and i will do everything in my power to stop &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;from hurting &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal &lt;/del&gt;or its innocent players, many say i want to destroy and kill &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal&lt;/del&gt;, this is NOT true, i used to love &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal&lt;/del&gt;, i only wish to stop &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad&lt;/del&gt;'s evil sadistic illegal actions and restore &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal &lt;/del&gt;to the great game it used to be when it was &amp;quot;for the players, by the players&amp;quot; , but if i have to destroy &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;GraalOnline &lt;/del&gt;in exchange to stop &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad&lt;/del&gt;'s tyranny, it will be regreteable, but sometimes some sacrifices have to be made for the greater good, lets just hope that &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;drops dead and that sacrifice never has to be made... I love &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;graal &lt;/del&gt;and lots of its players, and i wish to someday restore it to the grandness it once had, but bad people like &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;unixmad &lt;/del&gt;must be gotten rid of if that is to be achieved...&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;'''&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;I have been involved &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]] &lt;/ins&gt;for over three years now, and i used to really like &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;and its admins alot, then slowly after about a year &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;and its admins went in a bad downward spiral, i still stuck with &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]] &lt;/ins&gt;in hopes &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]] &lt;/ins&gt;would pull itself out of this hole and &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;would go back to the right path, this however did not happen, instead &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;and even &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Stefan]] &lt;/ins&gt;went fouler and lower than i could have imageined, First with fireing &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Pachuka&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;and &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Fuitad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt;, two of the BEST &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]] &lt;/ins&gt;admins ever in my view for no other reason than disagreeing with his future plans, then if just fireing them was not bad enough, he threatend to sue them and kill they're families, this was so wrong and bad &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Fuitad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;even said he would punch &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;in the face for saying that if he met him in real life, but that was just the beginning of the downward spiral &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;foolishly invoked onto &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]&lt;/ins&gt;, he then blatantly STOLE a domain name &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]&lt;/ins&gt;.net from Owl Shimy which costed over 30$ US Dollars, and never gave it back to him and never paid him for the cost of the domain, he also threatend to sue Owl Shimy and kill his family although it was &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Unixmad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;which stole the domain from Owl Shimy, then around &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]1&lt;/ins&gt;.3.1 he blocked off &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[GServer]]s &lt;/ins&gt;officially and threatend to sue anyone that ran the former &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[GServer]]s&lt;/ins&gt;, although they were released as &amp;quot;Freeware&amp;quot; and actually used to encourage people to use the &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[GServer]]s&lt;/ins&gt;, this was a blatant attempt to monopolise , threaten, and intemidate that players that actually helped &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]] &lt;/ins&gt;and supported him, he then made claims he copyrighted &amp;quot;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt;&amp;quot; and would contact so called &amp;quot;Lawres&amp;quot; about anyone that used the name &amp;quot;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt;&amp;quot; or had &amp;quot;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;Files&amp;quot; on there websites, although &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]] &lt;/ins&gt;was distributed as &amp;quot;Freeware&amp;quot; or &amp;quot;Shareware&amp;quot; and had no official software copyrights related to it, and that the name &amp;quot;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt;&amp;quot; was actually the name of a holy sacred golden cup used in some religion, and in that he has commited blasphemy against whatever religion has the so called &amp;quot;Holy &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt;&amp;quot;, then later on &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Unixmad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;fired SuperNick, because of his country and his origion and called him an American Asian slut, this was the first sign of racism on the part of &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Unixmad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt;, towards &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]1&lt;/ins&gt;.4.1 &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Unixmad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;was secretly unwrapping his sadistic plans, he then started claiming credit and claiming to own the copyrights to &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]&lt;/ins&gt;, although &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;has done nothing but host some stupid weak as servers, and yet &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Stefan&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;Knorr has done all the real work for &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt;, and was the one to make ZeldaOnline , &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]Online &lt;/ins&gt;in the first place and has written every bit of codeing for the &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;Client , &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[GServer]]&lt;/ins&gt;, RC, ect, yet &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;the worthless pile of trash that he is steals all credit for &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt;, then in February 2001, &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Unixmad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;and Antago release &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]2001 &lt;/ins&gt;and the start of the &amp;quot;Pay to Play&amp;quot; system, now i have no problem with the fact that it is pay to play, the issue is when you PAY for something you actually expect to get the worth while of your money in this product, but instead they got a shittily made level generated land that was 99% empty, the NPCs were and are laggy as hell, and there were no quests, and there is only two shitty things today for hearts that suck so much i refuse to even call them quests, and to add to that about 40% of everyone that paid and gave &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;there credit card number NEVER got the accounts they paid 27$ or more for, and those that did clearly got chumped out by a cheap peice of crap that aint worth paying for, it sucked so badly that most people played on &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;Classic still, even those that foolishly paid for pay to play accounts, and in an attempt to &amp;quot;FORCE&amp;quot; people to pay for a shitty product they do not want against they're will &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Unixmad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;hired &amp;quot;Tyhm&amp;quot; to deface and ruin &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]Classic &lt;/ins&gt;in an attempt to make it buggier and less fun than even &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]2001&lt;/ins&gt;, about this time i was disgusted with &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;and most of the other admins and decided to downright turn against &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Graal&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;and for the most part stop playing the peice of crap in general. People starting getting angry, complaining, asking for help, and asking for refunds, of coarse &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;gave noone a cash refund, and simply deleted and banned &amp;quot;PAID ACCOUNTS&amp;quot; of those who complained, asked for help, or asked for a refund these are PAYING CUSTOMERS, they have the rights of basic service that they PAID for, &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;promises pay to play players a quality product and customer service, but instead gives them a grade F product and ripps off his customers and delete they're accounts that they PAID HIM money for, his excuse? Credit card fraud, when infact at least 90% of the people he bans for credit card fraud pay legitamately with there OWN credit cards or pay via paypal by check, and yet &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Unixmad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;has stolen and used others credit card numbers to buy things for himself on several occassions, he also claims he needs pay to play money to keep &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]] &lt;/ins&gt;alive and running, yet there has been evidence that he makes at least 90% profit out of the monies he gets, and that he uses illegal porn ad banners and illegal cookie style web tracking and giveing email adresses of the players and even phone numbers of every player to spam and telemarketting companies without there permission, he even keeps personal player info, includeing email, adress, and phone number on unsecure servers which have already been leaked out, such info about people should not be on a public webserver accessable to the internet in anyway, then after all that &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Unixmad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;starts banning african american people, asians, koreans, and or people with images of people of that race from the &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]2001 &lt;/ins&gt;forums and &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]] &lt;/ins&gt;itself, yet again they are paying customers and this is another sickening act of racism on the part of &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Unixmad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;and &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]Online&lt;/ins&gt;, and then even more disgusting is what &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Stefan&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;posts on the &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]2001 &lt;/ins&gt;forums, he makes several anti american, racist, pro terrorist, pro nazi, and pro taliban comments on that post and goes on and on several pages bashing americans, this post was offense, obcene, and disgusting to many people includeing myself . If all that does not make you sick to your stomache or dislike &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;yet, i am far from done, then &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Unixmad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;proceeds to illegally DDOS webservers of mafukie and make threats to him over the phone, and has phone assaulted me over 40 times a day between midnight and 5am, and has made threats to sue dozens of people, kill there families, and shut down innocent websites and servers, while hosting illegal stuff and doing illegal things himself with the use of his own wanadoo.fr servers, stuff which includes child porn, bestiality porn, DVD Piracy, DDOS Attacks, theft of copyrighted materials from gameing companies such as nintendo, Credit Card fraud, theft, spamming, phone harrassment, useage of illegal pirated corperate softwares, all this just to start the list of what &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Unixmad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]'&lt;/ins&gt;, &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Stefan&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt;, and they're servers are doing 24/7 for nearly 4 years now , anyone that likes &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;or thinks he is a good person even after all this is clearly a fool, everything in this document is the truth and why i hate &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;so much, if you like &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;after all these sickening things he does, then you are no better than a KKK or Neo Nazi member and should go kill yourselfs, i do not tolerate facists, racists, or nazis, they are all bad people, and &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;is one of the worst of the worst, and no person with any morales could tolerate or support such a person like Stephane Portha or &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Stefan&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;Knorr. The people that spin, lie and bend the truth in support of &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;will always exist, but i will always reveal the truth about &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]]&lt;/ins&gt;, and EVERYTHING said in this document is true, despite what any assclowns try and say about this, and this supriseingly enough is only 1% of the bad things &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;Unixmad&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]] &lt;/ins&gt;has done, how some people even tolerate, none the less support such a bad person is beyond me... I fight against bad people like &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]]&lt;/ins&gt;, why? Because unlike many others in this world i actually care about other people, and i will do everything in my power to stop &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;from hurting &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]] &lt;/ins&gt;or its innocent players, many say i want to destroy and kill &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]&lt;/ins&gt;, this is NOT true, i used to love &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]&lt;/ins&gt;, i only wish to stop &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]]&lt;/ins&gt;'s evil sadistic illegal actions and restore &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]] &lt;/ins&gt;to the great game it used to be when it was &amp;quot;for the players, by the players&amp;quot; , but if i have to destroy &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]]Online &lt;/ins&gt;in exchange to stop &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]]&lt;/ins&gt;'s tyranny, it will be regreteable, but sometimes some sacrifices have to be made for the greater good, lets just hope that &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;drops dead and that sacrifice never has to be made... I love &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Graal]] &lt;/ins&gt;and lots of its players, and i wish to someday restore it to the grandness it once had, but bad people like &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Unixmad]] &lt;/ins&gt;must be gotten rid of if that is to be achieved...&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;

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&lt;/table&gt;</summary>
		<author><name>Anti-Up</name></author>
	</entry>
	<entry>
		<id>https://graalonline.net/index.php?title=Vectors&amp;diff=8458&amp;oldid=prev</id>
		<title>Anti-Up at 19:35, 30 June 2007</title>
		<link rel="alternate" type="text/html" href="https://graalonline.net/index.php?title=Vectors&amp;diff=8458&amp;oldid=prev"/>
		<updated>2007-06-30T19:35:08Z</updated>

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		<author><name>Anti-Up</name></author>
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