Coordinate geometry — practice question

The diagram depicts a solid cylinder on a horizontal circular base with centre $O$ and radius $4$ units. Points $A$, $B$ and $C$ are on the circumference of the base, with $AB$ as a diameter and angle $BOC = 90^\circ$. Points $P$, $Q$ and $R$ are on the top face of the cylinder directly above $A$, $B$ and $C$ respectively. The cylinder has height $12$ units. $M$ is the midpoint of $CR$ and $N$ is on $BQ$ with $BN = 4$ units. Unit vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $OB$ and $OC$ respectively, and unit vector $\mathbf{k}$ points vertically upwards.