Given and.

To draw inferences from this, we shall analyze these two equations one at a time.

First, let us consider.

We know the cross product of two vectors and forming an angle θ is

where is a unit vector perpendicular to and .

So, if, we have at least one of the following true –

(a)

(b)

(c) and

(d) is parallel to

Now, let us consider.

We have the dot product of two vectors and forming an angle θ is

So, if, we have at least one of the following true –

(a)

(b)

(c) and

(d) is perpendicular to

Given both these conditions are true.

Hence, the possibility (d) cannot be true as can’t be both parallel and perpendicular to at the same time.

Thus, either one or both of and are zero vectors if we have as well as.