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

In suffix notation, repeated indices are summed from one to three. The Levi-Civita symbol and Kronecker delta contraction isFor distinct free index pairs this follows from the two possible matching permutations and their signs; if either pair repeats an index both sides vanish. ThereforeRecognizing the dot products proves the vector triple product identity:

Use the Levi-Civita symbol contraction ${ϵ}_{ijk}{ϵ}_{i\ell m}={\delta }_{j\ell }{\delta }_{km}-{\delta }_{jm}{\delta }_{k\ell }$ in suffix notation:Thus the cross product contraction expands as

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.