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Introduction to Vector Cross Product Properties

The vector cross product of two vectors `veca` and `vecb` is defined as`vecaxxvecb = lvecal lvecblsinthetahatn` .

Let us see the properties of cross product vector:

Let `veca ` and `vec b` be two vectors with `theta` as the angle between them.

  • Vector cross product is non commutative

 `veca xx vecb = lvecal lvecbl sin theta hatn`

`vecbxxveca = lvecbllvecalsin(-theta)hatn`

`theta ` angle between `veca` and `vecb` . So the angle between `vecb` and `veca` is `-theta` as shown in figure.

  Angle Between Vectors

`sin(-theta) = -sintheta`

`vecbxxveca = -lvecbllvecalsin theta hatn = - veca xx vecb` 

  • Vector product of collinear(Parallel vectors:

 If the vectors `veca` and `vecb` are collinear or parallel then `veca xx vecb =veco`

The vectors `veca ` and `vecb` are parallel or cillinear means the angle between them is `theta=0 "or" pi`

Thus `veca xx vecb =lvecallvecblsin(theta)hatn = lvecal lvecbl(0)hatn = veco`

  • Cross product of equal vectors:

`veca xx veca =lvecal lvecalsintheta hatn = lvecallvecal(0) hatn="veco`

         Equal vectors means both vectors are the same , so `theta=0`

We will discuss some more properties of vector cross product.

Check this awesome Scalar Product and Vector Product Of Two Vecrots video it may help you.

Vector Cross Product Properties :- Continued

  • Cross product of unit vectors `veci,vecj,veck` 

 By the last property

`vecixxveci = vecj xx vecj="veckxxveck" = veco`

`veci,vecj,veck` are unit vectors in the X,Y and Z plane

`vecixxvecj = lvecil lvectjlsin90^0veck=(1)(1)(1)veck=veck`

Similarly `vecjxxveck=veci,veckxxveci=vecj`

And by the first property of cross product of vectors

`vecjxxveci=-veck , veck xx vecj="-veci" , vecixxveck="-vecj`

  • Disributivity of cross product over vector addition:

 Let `veca,vecb,vecc` be any three vectors, then

`veca xx (vecb+vecc) = (veca xx vecb) + (veca xxvecc)` (left distributivity)
`(vecb+vecc)xxveca= (vecb xx veca) +(vecc xx veca)` 

More Properties on Vector Cross Product:

  • Vector product in determinant form:

`veca = a_1veci +a_2vecj+a_3veck` and

`vecb=b_1veci +b_2vecj+b_3veck`

The `veca xx vecb = [ [veci,vecj,veck],[a_1,a_2,a_3],[b_1,b_2,b_3]]` the determinant for of the matrix

  • Angle between two vectors:

 We know `veca xx vecb = lveca l l vecb l sin theta hatn`

`rArr lvecaxxvecbl = l lvecal lvecbl sintheta hatnl`

         `rArr sintheta = (l veca xx vecb l )/ ( lvecal lvecbl)`

`rArr theta = sin^(-1) ( (lveca xx vecbl)/(lvecallvecbl))`

  • Unit vectors perpendicular to two given vectors

 `hatn = (veca xx vecb)/ (lveca xx vecbl)`

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