Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 1 / 6 / a / i

The scalar triple product is a determinant:A cyclic permutation of its three columns is an even permutation, so it leaves the determinant unchanged. Thus $\det (\mathbf{a},\mathbf{b},\mathbf{c})=\det (\mathbf{b},\mathbf{c},\mathbf{a})=\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})$, proving the identity.