plz give me the derivation of the formula of cross product mentioned in my previous question???

Asked by Avi Wadhwa | 17th Apr, 2012, 06:14: PM

Expert Answer:

The cross product calculates a new vector from 2 existing vectors. The resulting vector is perpendicular to the 2 original vectors ( The 2 original vectors can not be parallel ) :

Image 7 : cross ( v1, v2 )  = v3

The cross product is not commutative , as shown in the next image :
Image 8: cross ( v2, v1 ) = v4

Use the right hand rule to determine the direction of the resulting cross product :
Image 9 : Right hand rule
Move your fingers from the first vector to the second vector, and your thumb points in the direction of the resulting vector.

4.1 Cross product formula 

The formula for the cross product is the following :

if v= [x1, y1, z1] and v2 = [x2, y2, z2]

then v3 = cross ( v1,  v2) = [ y1*z2 - y2*z1 , z1*x2 - z2*x, x1*y2 - x2*y1 ]

The following mnemonic is useful if you know how to calculate matrix determinants :
Formula 6 : determinant mnemonic
In this matrix the x, y & z vectors are the unit vectors (of length 1) of the coordinate system :

Image 10 : Unit vectors

4.2 Length of the cross product vector

There is a formula to calculate the length of the resulting cross product from the length of the original vectors and the angle between the two vectors v1 and v2 :

Image 11 :  length of vector v3

if v3 = cross ( v1, v2 )

Then the length of the vector v3 : 

| v|= | v1 | * | v2 | * sin ( ? )

The usefulness for this formula is limited , because the length of the vector vcan also be calculated with formula 3.

Answered by  | 18th Apr, 2012, 10:24: AM

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