# Paper 1, Section II, A

(a) Use suffix notation to prove that

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

(b) Show that the equation of the plane through three non-colinear points with position vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ is

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

where $\mathbf{r}$ is the position vector of a point in this plane.

Find a unit vector normal to the plane in the case $\mathbf{a}=(2,0,1), \mathbf{b}=(0,4,0)$ and $\mathbf{c}=(1,-1,2)$.

(c) Let $\mathbf{r}$ be the position vector of a point in a given plane. The plane is a distance $d$ from the origin and has unit normal vector $\mathbf{n}$, where $\mathbf{n} \cdot \mathbf{r} \geqslant 0$. Write down the equation of this plane.

This plane intersects the sphere with centre at $\mathbf{p}$ and radius $q$ in a circle with centre at $\mathbf{m}$ and radius $\rho$. Show that

$\mathbf{m}-\mathbf{p}=\gamma \mathbf{n}$

Find $\gamma$ in terms of $q$ and $\rho$. Hence find $\rho$ in terms of $\mathbf{n}, d, \mathbf{p}$ and $q$.