(a) Use index notation to prove $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$.

Hence simplify

(i) $(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})$,

(ii) $(\mathbf{a} \times \mathbf{b}) \cdot[(\mathbf{b} \times \mathbf{c}) \times(\mathbf{c} \times \mathbf{a})]$.

(b) Give the general solution for $\mathbf{x}$ and $\mathbf{y}$ of the simultaneous equations

$$\mathbf{x}+\mathbf{y}=2 \mathbf{a}, \quad \mathbf{x} \cdot \mathbf{y}=c \quad(c<\mathbf{a} \cdot \mathbf{a})$$

Show in particular that $\mathbf{x}$ and $\mathbf{y}$ must lie at opposite ends of a diameter of a sphere whose centre and radius should be found.

(c) If two pairs of opposite edges of a tetrahedron are perpendicular, show that the third pair are also perpendicular to each other. Show also that the sum of the lengths squared of two opposite edges is the same for each pair.