= Solution
The <scalar triple product> is the <determinant> with the three vectors as columns. A cyclic permutation of three columns is even, while exchanging two columns reverses the determinant's sign. Consequently
$$
(\mathbf a\times\mathbf b)\cdot\mathbf c=\mathbf a\cdot(\mathbf b\times\mathbf c)=-\mathbf a\cdot(\mathbf c\times\mathbf b).
$$
Equivalently, expand the first expression as $\epsilon_{ijk}a_i b_j c_k$ and interchange the last two indices, using the antisymmetry of the <Levi-Civita symbol>. This argument includes linearly dependent vectors, for which both sides vanish.
Back to article page
