Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 1 / 5B / a / ii / Solution

Put $\tau =a\cdot (b\times c)$. The vector triple product identity givesCyclic invariance of the scalar triple product gives $(a\times b)\cdot c=\tau$. Hence the squared scalar triple product from cyclic cross products isIf an expansion entirely in pairwise dot products is desired, expand the Gram determinant of $a,b,c$:Both expressions remain valid for degenerate, including linearly dependent, vectors.