= Solution
The <vector triple product identity> gives
$$
(\mathbf b\times\mathbf c)\times(\mathbf c\times\mathbf a)
=\mathbf c[(\mathbf b\times\mathbf c)\cdot\mathbf a]
-\mathbf a[(\mathbf b\times\mathbf c)\cdot\mathbf c]
=\mathbf c[\mathbf a\cdot(\mathbf b\times\mathbf c)].
$$
Taking the dot product with $\mathbf a\times\mathbf b$ therefore yields
$$
\boxed{(\mathbf a\times\mathbf b)\cdot[(\mathbf b\times\mathbf c)\times(\mathbf c\times\mathbf a)]
=[\mathbf a\cdot(\mathbf b\times\mathbf c)]^2}.
$$
Back to article page
