TITLE: Proof that the cross product is not associative without using components
QUESTION [20 upvotes]: I need to show that the cross product is not associative without using components.
I understand how to do it with components, which leads to an immediate counterexample, but other than that I am not sure how to do it.

REPLY [34 votes]: Consider two non-zero perpendicular vectors $\def\v#1{{\bf#1}}\v a$ and $\v b$.  We have
$$(\v a \times\v a)\times\v b=\v0\times\v b=\v0\ .$$
However $\v a\times\v b$ is perpendicular to $\v a$, and is not the zero vector, so
$$\v a\times(\v a\times \v b)\ne\v 0\ .$$
Therefore
$$(\v a \times\v a)\times\v b\ne\v a\times(\v a\times \v b)\ .$$