Vector Cross Product is not Associative

Theorem
The vector cross product is not associative.

That is, in general:


 * $\mathbf a \times \left({\mathbf b \times \mathbf c}\right) \ne \left({\mathbf a \times \mathbf b}\right) \times \mathbf c$

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 $\R^3$.