Question

prove |uxv| = |u||v|sin theta

Answer 1

usually |uxv| is defined to be the area of the parallelogram whose sides are u and v.
If you draw such a parallelogram, the altitude to u is |v| sinθ
since A = bh, that is |u||v| sinθ

There are other proofs online, depending on which definition of uxv you start with.