7. [In this question, the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are due east and due north respectively. Position vectors are relative to a fixed origin $O$.]

A boat $P$ is moving with constant velocity $( - 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Calculate the speed of $P$.

When $t = 0$, the boat $P$ has position vector $( 2 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }$. At time $t$ hours, the position vector of $P$ is $\mathbf { p ~ k m }$.
\item Write down $\mathbf { p }$ in terms of $t$.

A second boat $Q$ is also moving with constant velocity. At time $t$ hours, the position vector of $Q$ is $\mathbf { q } \mathrm { km }$, where

$$\mathbf { q } = 18 \mathbf { i } + 12 \mathbf { j } - t ( 6 \mathbf { i } + 8 \mathbf { j } )$$

Find
\item the value of $t$ when $P$ is due west of $Q$,
\item the distance between $P$ and $Q$ when $P$ is due west of $Q$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M1 2012 Q7 [9]}}