3.36: Two particles and no-collision argument

Find the velocity vector of $P$.

When $t = 0$, $P$ passes through $A$ with position vector $\begin{pmatrix} 3 \\ 6 \end{pmatrix}$. Write down the position vector of $P$ at time $t$.

At the same time, particle $Q$ has position vector $\begin{pmatrix} 8t - 5 \\ 2 - 25t \end{pmatrix}$. The distance between $P$ and $Q$ is $d$. Show that $d^{2} = mt^{2} + nt + r$, where $m$, $n$ and $r$ are integers to be found.

Hence show that $P$ and $Q$ do not collide.