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

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

 $\displaystyle \mathbf a \times \left({\mathbf b \times \mathbf c}\right)$ $=$ $\displaystyle \mathbf a \times \left({\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} \times \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} }\right)$ $\displaystyle$ $=$ $\displaystyle \mathbf a \times \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}$ $\displaystyle$ $=$ $\displaystyle \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \times \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}$ $\displaystyle$ $=$ $\displaystyle \begin{bmatrix} 0 \\ 0 \\ -1 \end{bmatrix}$ $\displaystyle \left({\mathbf a \times \mathbf b}\right) \times \mathbf c$ $=$ $\displaystyle \left({\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \times \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} }\right) \times \mathbf c$ $\displaystyle$ $=$ $\displaystyle \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \times \mathbf c$ $\displaystyle$ $=$ $\displaystyle \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \times \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ $\displaystyle$ $=$ $\displaystyle \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}$

$\blacksquare$