M1 January 2012 Q7
7. [In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are due east and due north respectively. Position vectors are relative to a fixed origin \(O\).]
A boat \(P\) is moving with constant velocity \((-4\mathbf{i} + 8\mathbf{j})\) km h\(^{-1}\).
(a) Calculate the speed of \(P\). (2)
When \(t = 0\), the boat \(P\) has position vector \((2\mathbf{i} - 8\mathbf{j})\) km. At time \(t\) hours, the position vector of \(P\) is \(\mathbf{p}\) km.
(b) Write down \(\mathbf{p}\) in terms of \(t\). (1)
A second boat \(Q\) is also moving with constant velocity. At time \(t\) hours, the position vector of \(Q\) is \(\mathbf{q}\) km, where
\[\mathbf{q} = 18\mathbf{i} + 12\mathbf{j} - t(6\mathbf{i} + 8\mathbf{j})\]Find
(c) the value of \(t\) when \(P\) is due west of \(Q\), (3)
(d) the distance between \(P\) and \(Q\) when \(P\) is due west of \(Q\). (3)
| Scheme | Marks |
|---|---|
| \(\sqrt{\left((-4)^2 + 8^2\right)} = \sqrt{80}\ \ (\text{km h}^{-1})\) accept exact equivalents or 8.9 or better | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathbf{p} = (2\mathbf{i} - 8\mathbf{j}) + t(-4\mathbf{i} + 8\mathbf{j})\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| Equating \(\mathbf{j}\) components | |
| \(-8 + 8t = 12 - 8t\) | M1 A1 |
| \(t = \dfrac{5}{4}\) oe | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| Using their \(t\) from (c) to find the \(\mathbf{i}\)-cpts of \(\mathbf{p}\) and \(\mathbf{q}\) and subtract them | M1 |
| \(10\tfrac{1}{2} - (-3) = 13\tfrac{1}{2}\ \ (\text{km})\) | A1 ft A1 |
| (3) | |
| (9 marks) |