Vector Cross Product is not Associative

Theorem
The vector cross product is not associative.

That is, in general:


 * $\mathbf a \times \paren {\mathbf b \times \mathbf c} \ne \paren {\mathbf a \times \mathbf b} \times \mathbf c$

for $\mathbf a, \mathbf b, \mathbf c \in \R^3$.

Proof
Proof by Counterexample:

Let $\mathbf a = \begin {bmatrix} 1 \\ 0 \\ 0 \end {bmatrix}$, $\mathbf b = \begin {bmatrix} 1 \\ 1 \\ 0 \end {bmatrix}$, $\mathbf c = \begin {bmatrix} 1 \\ 1 \\ 1 \end {bmatrix}$

be vectors in $\R^3$.