Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 1 / 1A (Algebra and Geometry)

The three spherical basis vectors form an orthonormal basis: their displayed Cartesian components have unit lengths and pairwise zero dot products. For an orthonormal basis, a vector is the sum of its projections onto the basis vectors. Taking the dot product of each Cartesian unit vector with each spherical unit vector therefore gives the inverse change of basis:Equivalently, the change-of-basis matrix is orthogonal, so its matrix inverse is its transpose. These formulas use the stated angular convention for spherical polar coordinates; at a pole the tangential basis vectors still depend on the chosen azimuth.

Use suffix notation, summing repeated indices, and the Levi-Civita symbol convention $(A\times B{)}_i={ϵ}_{ijk}{A}_j{B}_k$. Contracting the outer symbol with the symbol from the second cross product gives the epsilon-delta identitywhere $\delta$ is the Kronecker delta. ThusThe second term is zero because $A_p{A}_{\ell }$ is symmetric in $p,\ell$ and the Levi-Civita symbol is antisymmetric. The cyclic rearrangement of the remaining symbol has positive sign. Its contraction is the scalar triple product $A\cdot (B\times C)$. HenceThis cross products with a common vector identity requires no independence assumption, so it also covers zero or parallel vectors.