Cross Product - Revision history

Cyan3 at 00:40, 6 April 2010

2010-04-06T00:40:57Z

← Older revision		Revision as of 20:40, 5 April 2010
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	[[Category:~~Graphics~~]]		[[Category:Scripting Reference]]
	The Cross Product is a property of [[Vectors\|vectors]]. The cross product takes two vectors and returns a third, which is perpendicular, or orthogonal, to the passed vectors. This vector is referred to as the normal vector, which normally is corrected to have a total magnitude of one in mathematics.		The Cross Product is a property of [[Vectors\|vectors]]. The cross product takes two vectors and returns a third, which is perpendicular, or orthogonal, to the passed vectors. This vector is referred to as the normal vector, which normally is corrected to have a total magnitude of one in mathematics.

Cyan3 at 00:38, 6 April 2010

2010-04-06T00:38:31Z

← Older revision		Revision as of 20:38, 5 April 2010
Line 1:		Line 1:
			[[Category:Graphics]]
	The Cross Product is a property of [[Vectors\|vectors]]. The cross product takes two vectors and returns a third, which is perpendicular, or orthogonal, to the passed vectors. This vector is referred to as the normal vector, which normally is corrected to have a total magnitude of one in mathematics.		The Cross Product is a property of [[Vectors\|vectors]]. The cross product takes two vectors and returns a third, which is perpendicular, or orthogonal, to the passed vectors. This vector is referred to as the normal vector, which normally is corrected to have a total magnitude of one in mathematics.

Tolnaftate2004: /* General Application of the Cross Product */ Spelling

2007-12-13T22:01:54Z

General Application of the Cross Product: Spelling

← Older revision		Revision as of 18:01, 13 December 2007
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	== General Application of the Cross Product ==		== General Application of the Cross Product ==
	It is common practice in mathematics to write ~~vecotrs~~ as sums of ~~th unti~~ vectors, '''i''', '''j''', and '''k'''. These each have a magnitude of one and, from the origin, move precisely 1 unit on the x, y, and z axes respectively.		It is common practice in mathematics to write vectors as sums of the unit vectors, '''i''', '''j''', and '''k'''. These each have a magnitude of one and, from the origin, move precisely 1 unit on the x, y, and z axes respectively.

	Say, then, that we have two vectors, u and v.		Say, then, that we have two vectors, u and v.
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	= (u<sub>2</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>2</sub>)'''i''' + (u<sub>3</sub>v<sub>1</sub>-u<sub>1</sub>v<sub>3</sub>)'''j''' + (u<sub>1</sub>v<sub>2</sub>-u<sub>2</sub>v<sub>1</sub>)'''k'''		= (u<sub>2</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>2</sub>)'''i''' + (u<sub>3</sub>v<sub>1</sub>-u<sub>1</sub>v<sub>3</sub>)'''j''' + (u<sub>1</sub>v<sub>2</sub>-u<sub>2</sub>v<sub>1</sub>)'''k'''
	= (u<sub>2</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>2</sub>)'''i''' - (u<sub>1</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>1</sub>)'''j''' + (u<sub>1</sub>v<sub>2</sub>-u<sub>2</sub>v<sub>1</sub>)'''k''' (u-components are in order)		= (u<sub>2</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>2</sub>)'''i''' - (u<sub>1</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>1</sub>)'''j''' + (u<sub>1</sub>v<sub>2</sub>-u<sub>2</sub>v<sub>1</sub>)'''k''' (u-components are in order)
	This describes the direction of the normal vector to the vectors u and v. This sum can be expressed in [[Matrix\|~~matricies~~]]		This describes the direction of the normal vector to the vectors u and v. This sum can be expressed in [[Matrix\|matrices]]
	<table>		<table>
	<tr>		<tr>

Stefan: Reverted edits by Anti-Up (Talk); changed back to last version by Stefan

2007-07-01T15:38:50Z

Reverted edits by Anti-Up (Talk); changed back to last version by Stefan

← Older revision		Revision as of 11:38, 1 July 2007
Line 1:		Line 1:
	{{~~antiunix~~}}		The Cross Product is a property of [[Vectors\|vectors]]. The cross product takes two vectors and returns a third, which is perpendicular, or orthogonal, to the passed vectors. This vector is referred to as the normal vector, which normally is corrected to have a total magnitude of one in mathematics.

			== Definition ==
			If vectors u and v are crossed ("u cross v"), the magnitude of the normal vector is equal to \|u\|\|v\|sin(θ). Thus,
			\|u × v\| = \|u\|\|v\|\|sin(θ)\|\|'''n'''\|
			\|u × v\| = \|u\|\|v\|\|sin(θ)\|
			In other applications, this magnitude has been found to be equal to the area of the parallelogram whose vertecies are at '''0''', '''u''', '''u'''+'''v''', and '''v'''.

			In addition,
			u × '''0''' = '''0'''
			u × v = -(v × u)
			(ru) × (sv) = (rs) u × v (where r and s are scalars)

			== General Application of the Cross Product ==
			It is common practice in mathematics to write vecotrs as sums of th unti vectors, '''i''', '''j''', and '''k'''. These each have a magnitude of one and, from the origin, move precisely 1 unit on the x, y, and z axes respectively.

			Say, then, that we have two vectors, u and v.
			u = <u<sub>1</sub>,u<sub>2</sub>,u<sub>3</sub>>
			v = <v<sub>1</sub>,v<sub>2</sub>,v<sub>3</sub>>
			We can rewrite these vectors in terms of '''i''', '''j''', and '''k'''.
			u = u<sub>1</sub>'''i''' + u<sub>2</sub>'''j''' + u<sub>3</sub>'''k'''
			v = v<sub>1</sub>'''i''' + v<sub>2</sub>'''j''' + v<sub>3</sub>'''k'''
			We can then "multiply" them in an algebraic manner.
			u × v = (u<sub>1</sub>v<sub>1</sub>)'''i''' × '''i''' + (u<sub>1</sub>v<sub>2</sub>)'''i''' × '''j''' + (u<sub>1</sub>v<sub>3</sub>)'''i''' × '''k''' + (u<sub>2</sub>v<sub>1</sub>)'''j''' × '''i''' + (u<sub>2</sub>v<sub>2</sub>)'''j''' × '''j''' +
			(u<sub>2</sub>v<sub>3</sub>)'''j''' × '''k''' + (u<sub>3</sub>v<sub>1</sub>)'''k''' × '''i''' + (u<sub>3</sub>v<sub>2</sub>)'''k''' × '''j''' + (u<sub>3</sub>v<sub>3</sub>)'''k''' × '''k'''
			We can remove immediately any terms where the vector is crossing itself. The angle between two parallel vectors, a and b, is 0, thus a × b = \|a\|\|b\|sin(0)\|n\| = 0 . So, the above statement simplifies to

			u × v = (u<sub>1</sub>v<sub>2</sub>)'''i''' × '''j''' + (u<sub>1</sub>v<sub>3</sub>)'''i''' × '''k''' + (u<sub>2</sub>v<sub>1</sub>)'''j''' × '''i''' + (u<sub>2</sub>v<sub>3</sub>)'''j''' × '''k''' + (u<sub>3</sub>v<sub>1</sub>)'''k''' × '''i''' + (u<sub>3</sub>v<sub>2</sub>)'''k''' × '''j'''
			Because the cross product reflects the normal vector, the following hold:
			i × j = -(j × i) = k
			j × k = -(k × j) = i
			k × i = -(i × k) = j
			Then, we can simplify our equation further:
			u × v = (u<sub>1</sub>v<sub>2</sub>)'''k''' + (u<sub>1</sub>v<sub>3</sub>)(-'''j''') + (u<sub>2</sub>v<sub>1</sub>)(-'''k''') + (u<sub>2</sub>v<sub>3</sub>)'''i''' + (u<sub>3</sub>v<sub>1</sub>)'''j''' + (u<sub>3</sub>v<sub>2</sub>)(-'''i''')
			= (u<sub>2</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>2</sub>)'''i''' + (u<sub>3</sub>v<sub>1</sub>-u<sub>1</sub>v<sub>3</sub>)'''j''' + (u<sub>1</sub>v<sub>2</sub>-u<sub>2</sub>v<sub>1</sub>)'''k'''
			= (u<sub>2</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>2</sub>)'''i''' - (u<sub>1</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>1</sub>)'''j''' + (u<sub>1</sub>v<sub>2</sub>-u<sub>2</sub>v<sub>1</sub>)'''k''' (u-components are in order)
			This describes the direction of the normal vector to the vectors u and v. This sum can be expressed in [[Matrix\|matricies]]
			<table>
			<tr>
			<td>        =</td>
			<td>
			<table frame="vsides" style="border: 1px solid black;">
			<tr>
			<td>u<sub>2</sub></td>
			<td>u<sub>3</sub></td>
			</tr>
			<tr>
			<td>v<sub>2</sub></td>
			<td>v<sub>3</sub></td>
			</tr>
			</table>
			</td>
			<td>'''i''' - </td>
			<td>
			<table frame="vsides" style="border: 1px solid black;">
			<tr>
			<td>u<sub>1</sub></td>
			<td>u<sub>3</sub></td>
			</tr>
			<tr>
			<td>v<sub>1</sub></td>
			<td>v<sub>3</sub></td>
			</tr>
			</table>
			</td>
			<td>'''j''' + </td>
			<td>
			<table frame="vsides" style="border: 1px solid black;">
			<tr>
			<td>u<sub>1</sub></td>
			<td>u<sub>2</sub></td>
			</tr>
			<tr>
			<td>v<sub>1</sub></td>
			<td>v<sub>2</sub></td>
			</tr>
			</table>
			</td>
			<td>'''k'''</td>
			</tr>
			</table>
			<table><tr><td>        =</td><td><table frame="vsides" style="border: 1px solid black;"><tr> <td>'''i'''</td><td>'''j'''</td><td>'''k'''</td> </tr><tr> <td>u<sub>1</sub></td><td>u<sub>2</sub></td><td>u<sub>3</sub></td> </tr><tr> <td>v<sub>1</sub></td><td>v<sub>2</sub></td><td>v<sub>3</sub></td> </tr></table></td></tr></table>

			== The Cross Product in GraalScript ==
			The cross product of two vector arrays in Graal can be obtained through the vectorcross({{graycourier\|u}},{{graycourier\|v}}) function. It's return value is a vector, w<sub>1</sub>'''i''' + w<sub>2</sub>'''j''' + w<sub>3</sub>'''k''', where the vector w is calculated from the determinant of the above [[Matrix\|matrix]].

Anti-Up: Replacing page with '{{antiunix}}'

2007-07-01T14:44:55Z

Replacing page with '{{antiunix}}'

← Older revision		Revision as of 10:44, 1 July 2007
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	The Cross Product is a property of [[Vectors\|vectors]]. The cross product takes two vectors and returns a third, which is perpendicular, or orthogonal, to the passed vectors. This vector is referred to as the normal vector, which normally is corrected to have a total magnitude of one in mathematics.		{{antiunix}}

	~~== Definition ==~~
	~~If vectors u and v are crossed ("u cross v"), the magnitude of the normal vector is equal to \|u\|\|v\|sin(θ). Thus,~~
	~~\|u × v\| = \|u\|\|v\|\|sin(θ)\|\|'''n'''\|~~
	~~\|u × v\| = \|u\|\|v\|\|sin(θ)\|~~
	~~In other applications, this magnitude has been found to be equal to the area of the parallelogram whose vertecies are at '''0''', '''u''', '''u'''+'''v''', and '''v'''.~~

	~~In addition,~~
	~~u × '''0''' = '''0'''~~
	~~u × v = -(v × u)~~
	~~(ru) × (sv) = (rs) u × v (where r and s are scalars)~~

	~~== General Application of the Cross Product ==~~
	It is common practice in mathematics to write vecotrs as sums of th unti vectors, '''i''', '''j''', and '''k'''. These each have a magnitude of one and, from the origin, move precisely 1 unit on the x, y, and z axes respectively.

	~~Say, then, that we have two vectors, u and v.~~
	~~u = <u<sub>1</sub>,u<sub>2</sub>,u<sub>3</sub>>~~
	~~v = <v<sub>1</sub>,v<sub>2</sub>,v<sub>3</sub>>~~
	~~We can rewrite these vectors in terms of '''i''', '''j''', and '''k'''.~~
	~~u = u<sub>1</sub>'''i''' + u<sub>2</sub>'''j''' + u<sub>3</sub>'''k'''~~
	~~v = v<sub>1</sub>'''i''' + v<sub>2</sub>'''j''' + v<sub>3</sub>'''k'''~~
	~~We can then "multiply" them in an algebraic manner.~~
	u × v = (u<sub>1</sub>v<sub>1</sub>)'''i''' × '''i''' + (u<sub>1</sub>v<sub>2</sub>)'''i''' × '''j''' + (u<sub>1</sub>v<sub>3</sub>)'''i''' × '''k''' + (u<sub>2</sub>v<sub>1</sub>)'''j''' × '''i''' + (u<sub>2</sub>v<sub>2</sub>)'''j''' × '''j''' +
	(u<sub>2</sub>v<sub>3</sub>)'''j''' × '''k''' + (u<sub>3</sub>v<sub>1</sub>)'''k''' × '''i''' + (u<sub>3</sub>v<sub>2</sub>)'''k''' × '''j''' + (u<sub>3</sub>v<sub>3</sub>)'''k''' × '''k'''
	We can remove immediately any terms where the vector is crossing itself. The angle between two parallel vectors, a and b, is 0, thus a × b = \|a\|\|b\|sin(0)\|n\| = 0 . So, the above statement simplifies to

	u × v = (u<sub>1</sub>v<sub>2</sub>)'''i''' × '''j''' + (u<sub>1</sub>v<sub>3</sub>)'''i''' × '''k''' + (u<sub>2</sub>v<sub>1</sub>)'''j''' × '''i''' + (u<sub>2</sub>v<sub>3</sub>)'''j''' × '''k''' + (u<sub>3</sub>v<sub>1</sub>)'''k''' × '''i''' + (u<sub>3</sub>v<sub>2</sub>)'''k''' × '''j'''
	~~Because the cross product reflects the normal vector, the following hold:~~
	~~i × j = -(j × i) = k~~
	~~j × k = -(k × j) = i~~
	~~k × i = -(i × k) = j~~
	~~Then, we can simplify our equation further:~~
	u × v = (u<sub>1</sub>v<sub>2</sub>)'''k''' + (u<sub>1</sub>v<sub>3</sub>)(-'''j''') + (u<sub>2</sub>v<sub>1</sub>)(-'''k''') + (u<sub>2</sub>v<sub>3</sub>)'''i''' + (u<sub>3</sub>v<sub>1</sub>)'''j''' + (u<sub>3</sub>v<sub>2</sub>)(-'''i''')
	~~= (u<sub>2</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>2</sub>)'''i''' + (u<sub>3</sub>v<sub>1</sub>-u<sub>1</sub>v<sub>3</sub>)'''j''' + (u<sub>1</sub>v<sub>2</sub>-u<sub>2</sub>v<sub>1</sub>)'''k'''~~
	= (u<sub>2</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>2</sub>)'''i''' - (u<sub>1</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>1</sub>)'''j''' + (u<sub>1</sub>v<sub>2</sub>-u<sub>2</sub>v<sub>1</sub>)'''k''' (u-components are in order)
	~~This describes the direction of the normal vector to the vectors u and v. This sum can be expressed in [[Matrix\|matricies]]~~
	~~<table>~~
	~~<tr>~~
	~~<td>        =</td>~~
	~~<td>~~
	~~<table frame="vsides" style="border: 1px solid black;">~~
	~~<tr>~~
	~~<td>u<sub>2</sub></td>~~
	~~<td>u<sub>3</sub></td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td>v<sub>2</sub></td>~~
	~~<td>v<sub>3</sub></td>~~
	~~</tr>~~
	~~</table>~~
	~~</td>~~
	~~<td>'''i''' - </td>~~
	~~<td>~~
	~~<table frame="vsides" style="border: 1px solid black;">~~
	~~<tr>~~
	~~<td>u<sub>1</sub></td>~~
	~~<td>u<sub>3</sub></td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td>v<sub>1</sub></td>~~
	~~<td>v<sub>3</sub></td>~~
	~~</tr>~~
	~~</table>~~
	~~</td>~~
	~~<td>'''j''' + </td>~~
	~~<td>~~
	~~<table frame="vsides" style="border: 1px solid black;">~~
	~~<tr>~~
	~~<td>u<sub>1</sub></td>~~
	~~<td>u<sub>2</sub></td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td>v<sub>1</sub></td>~~
	~~<td>v<sub>2</sub></td>~~
	~~</tr>~~
	~~</table>~~
	~~</td>~~
	~~<td>'''k'''</td>~~
	~~</tr>~~
	~~</table>~~
	<table><tr><td>        =</td><td><table frame="vsides" style="border: 1px solid black;"><tr> <td>'''i'''</td><td>'''j'''</td><td>'''k'''</td> </tr><tr> <td>u<sub>1</sub></td><td>u<sub>2</sub></td><td>u<sub>3</sub></td> </tr><tr> <td>v<sub>1</sub></td><td>v<sub>2</sub></td><td>v<sub>3</sub></td> </tr></table></td></tr></table>

	~~== The Cross Product in GraalScript ==~~
	~~The cross product of two vector arrays in Graal can be obtained through the vectorcross(~~{{~~graycourier\|u~~}},{{graycourier\|v}}) function. It's return value is a vector, w<sub>1</sub>'''i''' + w<sub>2</sub>'''j''' + w<sub>3</sub>'''k''', where the vector w is calculated from the determinant of the above [[Matrix\|matrix]].

Stefan at 13:18, 1 July 2007

2007-07-01T13:18:55Z

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Anti-up at 08:16, 1 July 2007

2007-07-01T08:16:07Z

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Anti-Up at 19:07, 30 June 2007

2007-06-30T19:07:32Z

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Tolnaftate2004: /* Definition */

2006-12-28T18:28:50Z

Definition

← Older revision		Revision as of 14:28, 28 December 2006
Line 3:		Line 3:
	== Definition ==		== Definition ==
	If vectors u and v are crossed ("u cross v"), the magnitude of the normal vector is equal to \|u\|\|v\|sin(θ). Thus,		If vectors u and v are crossed ("u cross v"), the magnitude of the normal vector is equal to \|u\|\|v\|sin(θ). Thus,
	u × v = \|u\|\|v\|sin(θ)\|'''n'''\|		\|u × v\| = \|u\|\|v\|\|sin(θ)\|\|'''n'''\|
	\|u × v\| = \|u\|\|v\|\|sin(θ)\|		\|u × v\| = \|u\|\|v\|\|sin(θ)\|
	In other applications, this magnitude has been found to be equal to the area of the parallelogram whose vertecies are at '''0''', '''u''', '''u'''+'''v''', and '''v'''.		In other applications, this magnitude has been found to be equal to the area of the parallelogram whose vertecies are at '''0''', '''u''', '''u'''+'''v''', and '''v'''.

Tolnaftate2004: /* General Application of the Cross Product */

2006-11-20T03:04:31Z

General Application of the Cross Product

← Older revision		Revision as of 23:04, 19 November 2006
Line 35:		Line 35:
	= (u<sub>2</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>2</sub>)'''i''' + (u<sub>3</sub>v<sub>1</sub>-u<sub>1</sub>v<sub>3</sub>)'''j''' + (u<sub>1</sub>v<sub>2</sub>-u<sub>2</sub>v<sub>1</sub>)'''k'''		= (u<sub>2</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>2</sub>)'''i''' + (u<sub>3</sub>v<sub>1</sub>-u<sub>1</sub>v<sub>3</sub>)'''j''' + (u<sub>1</sub>v<sub>2</sub>-u<sub>2</sub>v<sub>1</sub>)'''k'''
	= (u<sub>2</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>2</sub>)'''i''' - (u<sub>1</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>1</sub>)'''j''' + (u<sub>1</sub>v<sub>2</sub>-u<sub>2</sub>v<sub>1</sub>)'''k''' (u-components are in order)		= (u<sub>2</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>2</sub>)'''i''' - (u<sub>1</sub>v<sub>3</sub>-u<sub>3</sub>v<sub>1</sub>)'''j''' + (u<sub>1</sub>v<sub>2</sub>-u<sub>2</sub>v<sub>1</sub>)'''k''' (u-components are in order)
	This describes the direction of the normal vector to the vectors u and v. This sum can be expressed in matricies		This describes the direction of the normal vector to the vectors u and v. This sum can be expressed in [[Matrix\|matricies]]
	<table>		<table>
	<tr>		<tr>