Prove the area of parallelogram

Question

How to prove the length of the cross product axb is equal to the area of parallelogram determined by a and b?

Answer 1

Presumably you've defined $$a\times b=\det\begin{pmatrix}i&j&k\\ a_1&a_2&a_3\\ b_1&b_2&b_3\end{pmatrix}.$$ Now write that out, calculate $$\|a\times b\|^2=(a_2b_3-a_3b_2)^2+(a_1b_3-a_1b_3)^2+(a_1b_2-a_2b_1)^2$$ and show that this squared length equals $\|a\|^2\|b\|^2-\langle a,b\rangle^2$, which is known (?) to be the squared area of the parallelogram spanned by $a$ and $b$.