[In this question, $$\(\mathbf{i}\)$$ and $$\(\mathbf{j}\)$$ are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin $$\(O\)$$.] Two boats, $$\(P\)$$ and $$\(Q\)$$, are moving with constant velocities. The velocity of $$\(P\)$$ is $$\(15 \mathbf{i} \mathrm{ms}^{-1}\)$$ and the velocity of $$\(Q\)$$ is $$\((20 \mathbf{i}-20 \mathbf{j}) \mathrm{m} \mathrm{s}^{-1}\)$$ (a) Find the direction in which $$\(Q\)$$ is travelling, giving your answer as a bearing. (2) The boats are modelled as particles. At time $$\(t=0, P\)$$ is at the origin $$\(O\)$$ and $$\(Q\)$$ is at the point with position vector $$\(200 \mathbf{j} \mathrm{m}\)$$. At time $$\(t\)$$ seconds, the position vector of $$\(P\)$$ is $$\(\mathbf{p} \mathrm{m}\)$$ and the position vector of $$\(Q\)$$ is $$\(\mathbf{q} \mathrm{m}\)$$. (b) Show that $$\[ \overrightarrow{P Q}=[5 t \mathbf{i}+(200-20 t) \mathbf{j}] \mathrm{m} \]$$ (5) (c) Find the bearing of $$\(P\)$$ from $$\(Q\)$$ when $$\(t=10\)$$ (2) (d) Find the distance between $$\(P\)$$ and $$\(Q\)$$ when $$\(Q\)$$ is north east of $$\(P\)$$ (5) (e) Find the times when $$\(P\)$$ and $$\(Q\)$$ are $$\(200 \mathrm{~m}\)$$ apart. (3)

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