M1 January 2013 Q6
6. [In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.]
A ship sets sail at 9 am from a port \(P\) and moves with constant velocity. The position vector of \(P\) is \((4\mathbf{i} - 8\mathbf{j})\) km. At 9.30 am the ship is at the point with position vector \((\mathbf{i} - 4\mathbf{j})\) km.
(a) Find the speed of the ship in km h\(^{-1}\). (4)
(b) Show that the position vector \(\mathbf{r}\) km of the ship, \(t\) hours after 9 am, is given by \(\mathbf{r} = (4 - 6t)\mathbf{i} + (8t - 8)\mathbf{j}\). (2)
At 10 am, a passenger on the ship observes that a lighthouse \(L\) is due west of the ship. At 10.30 am, the passenger observes that \(L\) is now south-west of the ship.
(c) Find the position vector of \(L\). (5)
| Scheme | Marks |
|---|---|
| \(\dfrac{(\mathbf{i} - 4\mathbf{j}) - (4\mathbf{i} - 8\mathbf{j})}{0.5};\ (\pm 6\mathbf{i} \pm 8\mathbf{j})\) | M1 A1 |
| \(\sqrt{(\pm 6)^2 + (\pm 8)^2} = 10\) | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\mathbf{r} = (4\mathbf{i} - 8\mathbf{j}) + t(-6\mathbf{i} + 8\mathbf{j})\) | M1 |
| \(= (4\mathbf{i} - 8\mathbf{j}) - 6t\mathbf{i} + 8t\mathbf{j}\) \(= (4 - 6t)\mathbf{i} + (8t - 8)\mathbf{j}\) \(\ast\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| At 10 am, \(\mathbf{r} = -2\mathbf{i}\) | M1 A1 |
| At 10.30 am, \(\mathbf{r} = -5\mathbf{i} + 4\mathbf{j}\) | A1 |
| \(\mathbf{l} = k\mathbf{i},\ k \lt -2\) \(k = -5 - 4 = -9\) | DM1 |
| \(\mathbf{l} = -9\mathbf{i}\) | A1 |
| (5) | |
| (11 marks) |