3.38: Relative position of two boats with constant velocity

In this question all lengths are in kilometres and time is in hours.

Boat $A$ sails, with constant velocity, from a point $O$ with position vector $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$. After $3$ hours $A$ is at the point with position vector $\begin{pmatrix} -12 \\ 9 \end{pmatrix}$.

Find the position vector, $\overrightarrow{OP}$, of $A$ at time $t$.

At the same time as $A$ sails from $O$, boat $B$ sails from a point with position vector $\begin{pmatrix} 12 \\ 6 \end{pmatrix}$, with constant velocity $\begin{pmatrix} -5 \\ 8 \end{pmatrix}$.

Find the position vector, $\overrightarrow{OQ}$, of $B$ at time $t$.

Show that at time $t$, $|\overrightarrow{PQ}|^{2}=26t^{2}+36t+180$.

Hence show that $A$ and $B$ do not collide.