Question

if vector A.B , does it necessarily follow that A = 0 or B = 0? explain
if vector AxB does it necessarily follow that A = 0 or B = 0? explain

Answer 1

You mean: if A.B=0 ... if AxB=0 ...
don't you?

Answer 2

If the dot product is zero, it does not imply that either of the vectors is 0. It implies the vectors are orthogonal. \( \mathbf{A} \cdot \mathbf{B} = \vert A \vert \vert B \vert \cos \theta \) where theta is the angle between them.

If one of them were the zero vector, the dot product would also be zero because the zero vector is both perpendicular to and parallel with every other vector.

The cross product of two vectors is a vector (unlike the dot product which is a number.
\( \mathbf{A} \times \mathbf{B} = \vert A \vert \vert B \vert \sin \theta \mathbf n \) where n is a vector orthogonal to both A and B. it can be zero when sin theta is zero which happens when A and B are parallel or antiparallel (parallel in opposite directions).

So if the dot product is zero or the cross product is the zero vector it is not necessarily true that either A or B is zero.