Is the Vector Equation AxB=AxC Only True When A=0 or B=C?

[cp255]

True or False, if AxB = AxC then either A=0 or B=C.

A, B, and C are vectors and I thought this statement would be true. However the answer key says it is not. Why?

 

[bhanesh]

This can be explained by following example

 

[cp255]

Ok that makes sense. Now if the the magnitude of the cross product was not zero then B would have to equal C Right?

 

[Chestermiller]

The magnitude of AxB is equal to the magnitude of A times the magnitude of B times the sine of the angle between the vectors A and B. If the magnitude of C is larger than the magnitude of B, but the sine of the angle between A and C is less, AxC can be equal to AxB.

 

[D H]

Ok that makes sense. Now if the the magnitude of the cross product was not zero then B would have to equal C Right?

Not at all. All that AxB=AxC says is that either A=0 (in which case any two vectors B and C satisfy AxB=AxC), or that the components of B and C normal to A are equal.

 

[Dick]

Or that since AxB=AxC is the same as AxB-AxC=Ax(B-C)=0 then A and B-C are parallel.