A2 October 2020 Paper 1 Q6
6 The equations of two non-intersecting lines, \(l_1\) and \(l_2\), are
\[l_1 : \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}, \qquad l_2 : \mathbf{r} = \begin{pmatrix} 2 \\ 2 \\ -3 \end{pmatrix} + \mu\begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}.\]Find the shortest distance between lines \(l_1\) and \(l_2\). [5]
| Scheme | Marks | AO |
|---|---|---|
| \(\mathbf{n} = \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix} \times \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 2 \\ -10 \\ -3 \end{pmatrix},\) | M1 A1 | 3.1a 1.1 |
| \(|\mathbf{n}| = \sqrt{113}\) | A1 | 1.1 |
| \(\mathbf{b} - \mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} - \begin{pmatrix} 2 \\ 2 \\ -3 \end{pmatrix} = \begin{pmatrix} -1 \\ 0 \\ 2 \end{pmatrix}\) \(\Rightarrow d = \dfrac{\left|\begin{pmatrix} -1 \\ 0 \\ 2 \end{pmatrix}.\begin{pmatrix} 2 \\ -10 \\ -3 \end{pmatrix}\right|}{|\mathbf{n}|}\) | M1 | 1.1 |
| \(= \dfrac{8}{\sqrt{113}} = 0.753\) to 3sf | A1 | 1.1 |
| [5] |
Notes
M1: Cross product
A1: Modulus
M1: Using correct formula for \(d\)
A1: Accept exact or correct to 3sf (0.75257669…)
FT an exact answer