A2 June 2024 Paper 1 Q5
5
(a) Given that \(\mathbf{u} = \begin{pmatrix} -2 \\ 1 \\ 2 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} a \\ 0 \\ 1 \end{pmatrix}\) and \(\mathbf{u} \times \mathbf{v} = \begin{pmatrix} 1 \\ b \\ 3 \end{pmatrix}\), find \(a\) and \(b\). [3]
(b) Using \(\mathbf{u} \times \mathbf{v}\), determine the angle between the vectors \(\mathbf{u}\) and \(\mathbf{v}\), given that this angle is acute. [3]
| Scheme | Marks | AO |
|---|---|---|
| \(\mathbf{u} \times \mathbf{v} = \begin{pmatrix} 1 \\ 2 + 2a \\ -a \end{pmatrix}\) | B1 | 1.1 |
| \(2 + 2a = b, \quad -a = 3\) | M1 | 1.1 |
| \(a = -3, b = -4\) | A1 | 1.1 |
| [3] |
Notes
B1: oe
soi by correct equations
M1: Both equations ft their \(\mathbf{u} \times \mathbf{v}\)
A1: www
| Scheme | Marks | AO |
|---|---|---|
| \(\sin\theta = \dfrac{|\mathbf{u} \times \mathbf{v}|}{|\mathbf{u}||\mathbf{v}|}\) | M1 | 3.1a |
| \(= \dfrac{\sqrt{26}}{\sqrt{9}\sqrt{10}}\) | A1FT | 1.1 |
| \(\theta = 32.51\ldots^\circ\) | A1 | 1.1 |
| [3] |
Notes
M1: \(|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin\theta\) must be used with their \(\mathbf{u}\) and \(\mathbf{v}\)
soi by correct expression for their \(\sin\theta\). Formula alone is M0.
A1FT: ft their \(a\) and \(b\), need not be simplified
A1: \(33^\circ\) or 0.57 rad or better