CBSE Class 11-science Answered
plz give me the derivation of the formula of cross product mentioned in my previous question???
Asked by Avi Wadhwa | 17 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 ) :
The cross product is not commutative , as shown in the next image :
Use the right hand rule to determine the direction of the resulting cross product :
Move your fingers from the first vector to the second vector, and your thumb points in the direction of the resulting vector.
if v1 = [x1, y1, z1] and v2 = [x2, y2, z2]
then v3 = cross ( v1, v2) = [ y1*z2 - y2*z1 , z1*x2 - z2*x1 , x1*y2 - x2*y1 ]
The following mnemonic is useful if you know how to calculate matrix determinants :
In this matrix the x, y & z vectors are the unit vectors (of length 1) of the coordinate system :
if v3 = cross ( v1, v2 )
Then the length of the vector v3 :
| v3 |= | v1 | * | v2 | * sin ( ? )
The usefulness for this formula is limited , because the length of the vector v3 can also be calculated with formula 3.
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 |
4.1 Cross product formula
The formula for the cross product is the following :if v1 = [x1, y1, z1] and v2 = [x2, y2, z2]
then v3 = cross ( v1, v2) = [ y1*z2 - y2*z1 , z1*x2 - z2*x1 , x1*y2 - x2*y1 ]
The following mnemonic is useful if you know how to calculate matrix determinants :
Formula 6 : determinant mnemonic |
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 :
| v3 |= | v1 | * | v2 | * sin ( ? )
The usefulness for this formula is limited , because the length of the vector v3 can also be calculated with formula 3.
Answered by | 18 Apr, 2012, 10:24: AM
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