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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.
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| == Definition ==
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| If vectors u and v are crossed ("u cross v"), the magnitude of the normal vector is equal to |u||v|sin(θ). Thus,
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| |u × v| = |u||v||sin(θ)||'''n'''|
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| |u × v| = |u||v||sin(θ)|
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| 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'''.
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| In addition,
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| u × '''0''' = '''0'''
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| u × v = -(v × u)
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| (ru) × (sv) = (rs) u × v (where r and s are scalars)
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| == General Application of the Cross Product ==
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| 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.
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| Say, then, that we have two vectors, u and v.
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| u = <u<sub>1</sub>,u<sub>2</sub>,u<sub>3</sub>>
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| v = <v<sub>1</sub>,v<sub>2</sub>,v<sub>3</sub>>
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| We can rewrite these vectors in terms of '''i''', '''j''', and '''k'''.
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| u = u<sub>1</sub>'''i''' + u<sub>2</sub>'''j''' + u<sub>3</sub>'''k'''
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| v = v<sub>1</sub>'''i''' + v<sub>2</sub>'''j''' + v<sub>3</sub>'''k'''
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| We can then "multiply" them in an algebraic manner.
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| 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''' +
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| (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'''
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| 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
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| 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'''
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| Because the cross product reflects the normal vector, the following hold:
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| i × j = -(j × i) = k
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| j × k = -(k × j) = i
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| k × i = -(i × k) = j
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| Then, we can simplify our equation further:
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| 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''')
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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'''
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| = (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)
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| This describes the direction of the normal vector to the vectors u and v. This sum can be expressed in [[Matrix|matricies]]
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| <table>
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| <tr>
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| <td> =</td>
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| <td>
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| <table frame="vsides" style="border: 1px solid black;">
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| <tr>
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| <td>u<sub>2</sub></td>
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| <td>u<sub>3</sub></td>
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| </tr>
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| <tr>
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| <td>v<sub>2</sub></td>
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| <td>v<sub>3</sub></td>
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| </tr>
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| </table>
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| </td>
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| <td>'''i''' - </td>
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| <td>
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| <table frame="vsides" style="border: 1px solid black;">
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| <tr>
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| <td>u<sub>1</sub></td>
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| <td>u<sub>3</sub></td>
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| </tr>
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| <tr>
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| <td>v<sub>1</sub></td>
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| <td>v<sub>3</sub></td>
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| </tr>
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| </table>
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| </td>
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| <td>'''j''' + </td>
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| <td>
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| <table frame="vsides" style="border: 1px solid black;">
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| <tr>
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| <td>u<sub>1</sub></td>
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| <td>u<sub>2</sub></td>
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| </tr>
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| <tr>
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| <td>v<sub>1</sub></td>
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| <td>v<sub>2</sub></td>
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| </tr>
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| </table>
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| </td>
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| <td>'''k'''</td>
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| </tr>
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| </table>
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| <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>
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| == The Cross Product in GraalScript ==
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| 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]].
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