# 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

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

 $\ds \mathbf a \times \paren {\mathbf b \times \mathbf c}$ $=$ $\ds \mathbf a \times \paren {\begin {bmatrix} 1 \\ 1 \\ 0 \end {bmatrix} \times \begin {bmatrix} 1 \\ 1 \\ 1 \end {bmatrix} }$ $\ds$ $=$ $\ds \mathbf a \times \begin {bmatrix} 1 \\ -1 \\ 0 \end {bmatrix}$ $\ds$ $=$ $\ds \begin {bmatrix} 1 \\ 0 \\ 0 \end {bmatrix} \times \begin {bmatrix} 1 \\ -1 \\ 0 \end {bmatrix}$ $\ds$ $=$ $\ds \begin {bmatrix} 0 \\ 0 \\ -1 \end {bmatrix}$ $\ds \paren {\mathbf a \times \mathbf b} \times \mathbf c$ $=$ $\ds \paren {\begin {bmatrix} 1 \\ 0 \\ 0 \end {bmatrix} \times \begin {bmatrix} 1 \\ 1 \\ 0 \end {bmatrix} } \times \mathbf c$ $\ds$ $=$ $\ds \begin {bmatrix} 0 \\ 0 \\ 1 \end {bmatrix} \times \mathbf c$ $\ds$ $=$ $\ds \begin {bmatrix} 0 \\ 0 \\ 1 \end {bmatrix} \times \begin {bmatrix} 1 \\ 1 \\ 1 \end {bmatrix}$ $\ds$ $=$ $\ds \begin {bmatrix} -1 \\ 1 \\ 0 \end {bmatrix}$

$\blacksquare$