# Paper 1, Section II, $5 \mathrm{C}$

Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be unit vectors. By using suffix notation, prove that

$(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{a} \times \mathbf{c})=\mathbf{b} \cdot \mathbf{c}-(\mathbf{a} \cdot \mathbf{b})(\mathbf{a} \cdot \mathbf{c})$

and

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

The three distinct points $A, B, C$ with position vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ lie on the surface of the unit sphere centred on the origin $O$. The spherical distance between the points $A$ and $B$, denoted $\delta(A, B)$, is the length of the (shorter) arc of the circle with centre $O$ passing through $A$ and $B$. Show that

$\cos \delta(A, B)=\mathbf{a} \cdot \mathbf{b}$

A spherical triangle with vertices $A, B, C$ is a region on the sphere bounded by the three circular arcs $A B, B C, C A$. The interior angles of a spherical triangle at the vertices $A, B, C$ are denoted $\alpha, \beta, \gamma$, respectively.

By considering the normals to the planes $O A B$ and $O A C$, or otherwise, show that

$\cos \alpha=\frac{(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{a} \times \mathbf{c})}{|\mathbf{a} \times \mathbf{b} \| \mathbf{a} \times \mathbf{c}|}$

Using identities (1) and (2), prove that

$\cos \delta(B, C)=\cos \delta(A, B) \cos \delta(A, C)+\sin \delta(A, B) \sin \delta(A, C) \cos \alpha$

and

$\frac{\sin \alpha}{\sin \delta(B, C)}=\frac{\sin \beta}{\sin \delta(A, C)}=\frac{\sin \gamma}{\sin \delta(A, B)}$

For an equilateral spherical triangle show that $\alpha>\pi / 3$.