M4 June 2013 (R) Q1
1. [In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively.]
Boat \(A\) is moving with velocity \((3\mathbf{i} + 4\mathbf{j})\) km h\(^{-1}\) and boat \(B\) is moving with velocity \((6\mathbf{i} - 5\mathbf{j})\) km h\(^{-1}\). Find
(a) the magnitude of the velocity of \(A\) relative to \(B\), (3)
(b) the direction of the velocity of \(A\) relative to \(B\), giving your answer as a bearing. (2)
| Scheme | Marks |
|---|---|
| \({}_A\mathbf{v}_B = \mathbf{v}_A - \mathbf{v}_B\) | |
| \(= -3\mathbf{i} + 9\mathbf{j}\) km h\(^{-1}\) | M1 |
| Mag \(= \sqrt{9 + 81} = 3\sqrt{10}\) | M1A1 |
| (3) |
Notes
A1 9.5 or better
| Scheme | Marks |
|---|---|
| \(\tan\theta = \dfrac{3}{9}\) | M1 |
| \(\theta = 18.4^\circ\) | |
| Direction \(= 360 - 18.4\) | |
| \(= 342^\circ\) | A1 |
| (2) | |
| (5 marks) |
Notes
M1 Allow \(\pm\) or reciprocal
Or 71.6\(^\circ\)
A1 Allow 341.6\(^\circ\)