Problem

With respect to a fixed origin $O$,

• the line $l$ has equation

$$\mathbf{r}=\begin{pmatrix}4\\2\\-3\end{pmatrix}+\lambda \begin{pmatrix}-4\\-3\\5\end{pmatrix}$$

where $\lambda$ is a scalar constant

• the point $A$ has position vector $9\mathbf{i}-3\mathbf{j}+2\mathbf{k}$

Given that $X$ is the point on $l$ nearest to $A$,

(a) find

(i) the coordinates of $X$

(ii) the shortest distance from $A$ to $l$.

Give your answer in the form $\sqrt{d}$, where $d$ is an integer.

The point $B$ is the image of $A$ after reflection in $l$.

(b) Find the position vector of $B$.