5. [In this question $\mathbf { i }$ and $\mathbf { j }$ are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin $O$.]

A particle $P$ is moving in a straight line with constant velocity. At 9 am, the position vector of $P$ is $( 7 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }$ and at 9.20 am , the position vector of $P$ is $6 \mathbf { i } \mathrm {~km}$. At time $t$ hours after 9 am , the position vector of $P$ is $\mathbf { r } _ { P } \mathrm {~km}$.
\begin{enumerate}[label=(\alph*)]
\item Find, in $\mathrm { kmh } ^ { - 1 }$, the speed of $P$.
\item Show that $\mathbf { r } _ { P } = ( 7 - 3 t ) \mathbf { i } + ( 5 - 15 t ) \mathbf { j }$.
\item Find the value of $t$ when $\mathbf { r } _ { P }$ is parallel to $16 \mathbf { i } + 5 \mathbf { j }$.

The position vector of another particle $Q$, at time $t$ hours after 9 am , is $\mathbf { r } _ { Q } \mathrm {~km}$, where $\mathbf { r } _ { Q } = ( 5 + 2 t ) \mathbf { i } + ( - 3 + 5 t ) \mathbf { j }$
\item Show that $P$ and $Q$ will collide and find the position vector of the point of collision.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M1 2018 Q5 [15]}}