A2 June 2022 Paper 1 Q4
4 Determine the acute angle between the line \(\mathbf{r} = \begin{pmatrix} -\sqrt{3} \\ 1 \\ 3 \end{pmatrix} + \lambda\begin{pmatrix} 1 \\ 2\sqrt{3} \\ -\sqrt{3} \end{pmatrix}\) and the \(y\)-axis. [4]
| Scheme | Marks | AO |
|---|---|---|
| Direction of \(y\)-axis is \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) | B1 | 3.1a |
| \(\begin{pmatrix} 1 \\ 2\sqrt{3} \\ -\sqrt{3} \end{pmatrix} \cdot \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = 2\sqrt{3}\) | M1 | 1.1 |
| \(\cos\theta = \dfrac{2\sqrt{3}}{1 \times \sqrt{1 + \left(2\sqrt{3}\right)^2 + \left(-\sqrt{3}\right)^2}}\) \(= \dfrac{2\sqrt{3}}{4}\) | M1 | 1.1 |
| \(\Rightarrow \theta = \dfrac{\pi}{6}\) or \(30^\circ\) | A1 | 1.1 |
| [4] |
Notes
B1: Correct direction vector representation of the \(y\)-axis.
M1: Correct use of dot product with \(\begin{pmatrix} 1 \\ 2\sqrt{3} \\ -\sqrt{3} \end{pmatrix}\) and their direction vector for \(y\)-axis. soi
M1: Correct use of dot product with their vectors to find cosine of angle soi. Condone eg. \(\left(\sqrt{3}\right)^2\) in place of \(\left(-\sqrt{3}\right)^2\).
A1: Or \(\cos\varphi = -\dfrac{\sqrt{3}}{2} \Rightarrow \theta = \pi - \varphi = \dfrac{\pi}{6}\) or \(30^\circ\). Accept \(0.524^{\mathrm{c}}\)
Mark the final answer
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