Vector Cross Product is not Associative

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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 $\R^3$.

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

$\blacksquare$