In this question all distances are in km.

A ship \(P\) sails from a point \(A\), which has position vector

\(\begin{pmatrix}0\\0\end{pmatrix},\)

with a speed of \(52\text{ km h}^{-1}\) in the direction of

\(\begin{pmatrix}-5\\12\end{pmatrix}.\)

(a) Find the velocity vector of the ship.

(b) Write down the position vector of \(P\) at a time \(t\) hours after leaving \(A\).

At the same time that ship \(P\) sails from \(A\), a ship \(Q\) sails from a point \(B\), which has position vector

\(\begin{pmatrix}12\\8\end{pmatrix},\)

with velocity vector

\(\begin{pmatrix}-25\\45\end{pmatrix}\text{ km h}^{-1}.\)

(c) Write down the position vector of \(Q\) at a time \(t\) hours after leaving \(B\).

(d) Using your answers to parts (b) and (c), find the displacement vector \(\overrightarrow{PQ}\) at time \(t\) hours.

(e) Hence show that

\(PQ=\sqrt{34t^2-168t+208}.\)

(f) Find the value of \(t\) when \(P\) and \(Q\) are first \(2\) km apart.