M4 June 2008 Q1
1. [In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively.]
A ship \(P\) is moving with velocity \((5\mathbf{i} - 4\mathbf{j})\) km h\(^{-1}\) and a ship \(Q\) is moving with velocity \((3\mathbf{i} + 7\mathbf{j})\) km h\(^{-1}\).
Find the direction that ship \(Q\) appears to be moving in, to an observer on ship \(P\), giving your answer as a bearing. (5)
| Scheme | Marks |
|---|---|
| \({}_{Q}\mathbf{V}_{P} = \mathbf{V}_{Q} - \mathbf{V}_{P} = (3\mathbf{i} + 7\mathbf{j}) - (5\mathbf{i} - 4\mathbf{j})\) | M1 |
| \(= (-2\mathbf{i} + 11\mathbf{j})\) | A1 |
| \(\tan\theta = \dfrac{11}{2} \Rightarrow \theta = 79.690\ldots^\circ\) | M1 A1 |
| Bearing is \(350^\circ\) | A1 |
| (5 marks) |