C4 January 2011 Q4
4. Relative to a fixed origin \(O\), the point \(A\) has position vector \(\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}\) and the point \(B\) has position vector \(-2\mathbf{i} + 2\mathbf{j} - \mathbf{k}\). The points \(A\) and \(B\) lie on a straight line \(l\).
(a) Find \(\overrightarrow{AB}\). (2)
(b) Find a vector equation of \(l\). (2)
The point \(C\) has position vector \(2\mathbf{i} + p\mathbf{j} - 4\mathbf{k}\) with respect to \(O\), where \(p\) is a constant. Given that \(AC\) is perpendicular to \(l\), find
(c) the value of \(p\), (4)
(d) the distance \(AC\). (2)
| Scheme | Marks |
|---|---|
| \(\overrightarrow{AB} = -2\mathbf{i} + 2\mathbf{j} - \mathbf{k} - (\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}) = -3\mathbf{i} + 5\mathbf{j} - 3\mathbf{k}\) | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathbf{r} = \mathbf{i} - 3\mathbf{j} + 2\mathbf{k} + \lambda(-3\mathbf{i} + 5\mathbf{j} - 3\mathbf{k})\) or \(\mathbf{r} = -2\mathbf{i} + 2\mathbf{j} - \mathbf{k} + \lambda(-3\mathbf{i} + 5\mathbf{j} - 3\mathbf{k})\) | M1 A1ft |
| (2) |
| Scheme | Marks |
|---|---|
| \(\overrightarrow{AC} = 2\mathbf{i} + p\mathbf{j} - 4\mathbf{k} - (\mathbf{i} - 3\mathbf{j} + 2\mathbf{k})\) \(= \mathbf{i} + (p + 3)\mathbf{j} - 6\mathbf{k}\) or \(\overrightarrow{CA}\) | B1 |
| \(\overrightarrow{AC}.\overrightarrow{AB} = \begin{pmatrix}1\\p + 3\\-6\end{pmatrix}.\begin{pmatrix}-3\\5\\-3\end{pmatrix} = 0\) | M1 |
| \(-3 + 5p + 15 + 18 = 0\) Leading to \(p = -6\) | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(AC^2 = (2 - 1)^2 + (-6 + 3)^2 + (-4 - 2)^2\quad (= 46)\) | M1 |
| \(AC = \sqrt{46}\) accept awrt 6.8 | A1 |
| (2) | |
| (10 marks) |