# Paper 1, Section II, C

(a) Let $A, B$, and $C$ be three distinct points in the plane $\mathbb{R}^{2}$ which are not collinear, and let $\mathbf{a}, \mathbf{b}$, and $\mathbf{c}$ be their position vectors.

Show that the set $L_{A B}$ of points in $\mathbb{R}^{2}$ equidistant from $A$ and $B$ is given by an equation of the form

$\mathbf{n}_{A B} \cdot \mathbf{x}=p_{A B},$

where $\mathbf{n}_{A B}$ is a unit vector and $p_{A B}$ is a scalar, to be determined. Show that $L_{A B}$ is perpendicular to $\overrightarrow{A B}$.

Show that if $\mathbf{x}$ satisfies

$\mathbf{n}_{A B} \cdot \mathbf{x}=p_{A B} \quad \text { and } \quad \mathbf{n}_{B C} \cdot \mathbf{x}=p_{B C}$

then

$\mathbf{n}_{C A} \cdot \mathbf{x}=p_{C A} .$

How do you interpret this result geometrically?

(b) Let $\mathbf{a}$ and $\mathbf{u}$ be constant vectors in $\mathbb{R}^{3}$. Explain why the vectors $\mathbf{x}$ satisfying

$\mathbf{x} \times \mathbf{u}=\mathbf{a} \times \mathbf{u}$

describe a line in $\mathbb{R}^{3}$. Find an expression for the shortest distance between two lines $\mathbf{x} \times \mathbf{u}_{k}=\mathbf{a}_{k} \times \mathbf{u}_{k}$, where $k=1,2$.