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.
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$\begingroup$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 $$(\v a \times\v a)\times\v b\ne\v a\times(\v a\times \v b)\ .$$
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