A2 October 2021 Paper 1 Q11
11
(a) Given that \(\mathbf{u} = \lambda\mathbf{i} + \mathbf{j} - 3\mathbf{k}\) and \(\mathbf{v} = \mathbf{i} + 2\mathbf{j} - 2\mathbf{k}\), find the following, giving your answers in terms of \(\lambda\).
(i) \(\mathbf{u}.\mathbf{v}\) [1]
(ii) \(\mathbf{u} \times \mathbf{v}\) [2]
(b) Hence determine
(i) the acute angle between the planes \(2x + y - 3z = 10\) and \(x + 2y - 2z = 10\), [3]
(ii) the shortest distance between the lines \(\dfrac{x - 3}{3} = \dfrac{y}{1} = \dfrac{z - 2}{-3}\) and \(\dfrac{x}{1} = \dfrac{y - 4}{2} = \dfrac{z + 2}{-2}\), giving your answer as a multiple of \(\sqrt{2}\). [3]
| Scheme | Marks | AO |
|---|---|---|
| (i) \(\mathbf{u}.\mathbf{v} = \lambda \times 1 + 1 \times 2 + (-3) \times (-2)\) \(= \lambda + 8\) | B1 | 1.1 |
| [1] | ||
| (ii) \(\mathbf{u} \times \mathbf{v} = \begin{pmatrix} \lambda \\ 1 \\ -3 \end{pmatrix} \times \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix} = \begin{pmatrix} 4 \\ 2\lambda - 3 \\ 2\lambda - 1 \end{pmatrix}\) oe \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) form | B1 B1 | 1.1a 1.1 |
| [2] |
Notes
(a)(ii)
B1: For one of \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) correct
B1: For all \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) correct
| Scheme | Marks | AO |
|---|---|---|
| (i) angle between \(2\mathbf{i} + \mathbf{j} - 3\mathbf{k}\) and \(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}\) taking \(\lambda = 2\), | M1 | 3.1a |
| \(\cos\theta = \dfrac{10}{\sqrt{14}\sqrt{9}}\) | M1 | 1.1 |
| \(\theta = 27.0^\circ\) or 0.472 rad | A1 | 1.1 |
| [3] | ||
| (ii) direction vectors are \(3\mathbf{i} + \mathbf{j} - 3\mathbf{k}\) and \(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}\), so take \(\lambda = 3\) | M1 | 3.1a |
| with \(\lambda = 3\), \(\mathbf{u} \times \mathbf{v} = 4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}\) \(\text{distance} = \dfrac{1}{\sqrt{50}}\left|\begin{pmatrix} 4 \\ 3 \\ 5 \end{pmatrix}.\begin{pmatrix} 3 - 0 \\ 0 - 4 \\ 2 - (-2) \end{pmatrix}\right| = \dfrac{20}{5\sqrt{2}} = 2\sqrt{2}\) | M1 A1 | 1.1 1.1 |
| [3] |