Question

- Show that (A x B) is perpendicular to both A and B.​

Answer 1

Answer:

Assume A⃗ ≠kB⃗ .

Let A be the first of your basis vectors.

That makes A⃗ =⎡⎣⎢100⎤⎦⎥

The second basis vector will be B.

That makes B⃗ =⎡⎣⎢010⎤⎦⎥

A⃗ ×B⃗ =⎡⎣⎢001⎤⎦⎥

A⃗ −B⃗ =⎡⎣⎢1−10⎤⎦⎥

So, in a vector space where A and B are the basis vectors, A×B is perpendicular to A-B.

But this is a transformed space, we want to look at a space that is not transformed and is agnostic to choice of basis vectors. So let's define the plane where A and B are as the xy-plane, with x being the same direction as A.

The transform of that would change the xy coordinates of A and B, but the z coordinate is unchanged, and still is 0, which means that A×B would have x and y coordinates of 0 but a nonzero z coordinate.

A-B would also have a z coordinate of 0, which means that A×B would still be perpendicular to A-B.

Rotating the basis vectors would not change this relationship so thus it is proven.