0606 P12 - Mar 2021 - Q5 - 7 marks
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 \(\begin{pmatrix}-12\\9\end{pmatrix}\).
(a) 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 \(\begin{pmatrix}-5\\8\end{pmatrix}\).
(b) Find the position vector, \(\overrightarrow{OQ}\), of \(B\) at time \(t\).
(c) Show that at time \(t\), \(|\overrightarrow{PQ}|^2=26t^2+36t+180\).
(d) Hence show that \(A\) and \(B\) do not collide.
