**2022 TMJC J1 MYE Q8**

Relative to the origin $O$, the position vectors of the points $A$, $B$ and $C$ are $4\mathbf{i}+\mathbf{j}+3\mathbf{k}$, $-6\mathbf{i}+3\mathbf{j}-7\mathbf{k}$ and $-2\mathbf{i}+7\mathbf{j}-6\mathbf{k}$ respectively. The plane $p$ contains the points $A$ and $C$. $p$ is also parallel to the vector $\left( \begin{matrix}

-3 \\

6 \\

2 \\

\end{matrix} \right)$.

(i)

Show that the cartesian equation of $p$ can be expressed as $2x-y+6z=25$.

[4]

(ii)

Find the coordinates of the foot of the perpendicular from $B$ to $p$.

[4]

(iii)

Hence, or otherwise, find the coordinates of the point of reflection of $B$ about $p$.

[2]

(iv)

The point $M$ lies on $AB$ extended such that $AB:AM=3:5$. Find $\overrightarrow{AB}$ and determine the exact shortest distance between $M$ and $p$.

[4]

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