Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 1 / 5C (Vectors and Matrices)

Using the Levi-Civita symbol and the epsilon-delta identity,This proves the vector triple product identityApplying it inside the scalar triple product gives Lagrange's identity

The vector triple product identity givesTaking the dot product with $\mathbf{a}\times \mathbf{b}$ therefore yields

WriteThe first equation is then automatic, while the second becomesThus the general solution isAs $\mathbf{u}$ varies, $\mathbf{x}$ and $\mathbf{y}$ are antipodal points of the sphere with center $\mathbf{a}$ and radius $\sqrt{\mathbf{a}\cdot \mathbf{a}-c}$, so they are opposite ends of a diameter.

Let the vertex position vectors be $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}$. SupposeSubtracting the second scalar product from the first givesso the remaining pair of opposite edges of a tetrahedron is also perpendicular.

PutDirect expansion gives, cyclically,with the other two differences equal to minus twice the scalar products of the other opposite-edge pairs. All three scalar products vanish, hence