= Solution
For the specified <standard basis> vectors, $\mathbf a\times\mathbf b=\hat{\mathbf k}$, so
$$
(\mathbf a\times\mathbf b)\times\mathbf c=\hat{\mathbf k}\times\hat{\mathbf k}=0.
$$
On the other hand, $\mathbf b\times\mathbf c=\hat{\mathbf i}-\hat{\mathbf j}$ and hence
$$
\mathbf a\times(\mathbf b\times\mathbf c)
=\hat{\mathbf i}\times(\hat{\mathbf i}-\hat{\mathbf j})=-\hat{\mathbf k}.
$$
These results are unequal, proving that the <cross product> does not satisfy the <associative property>.
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