AS June 2024 Paper 1 Q5
5 The vectors \(\mathbf{a}\) and \(\mathbf{b}\) are given by
\[\mathbf{a} = 3\mathbf{i} + 4\mathbf{j} - 2\mathbf{k} \qquad \text{and} \qquad \mathbf{b} = 2\mathbf{i} - \mathbf{j} - 5\mathbf{k}\](a) Calculate \(\mathbf{a}.\mathbf{b}\) [1 mark]
(b) Calculate \(|\mathbf{a}|\) and \(|\mathbf{b}|\) [2 marks]
(c) Calculate the acute angle between \(\mathbf{a}\) and \(\mathbf{b}\)
Give your answer to the nearest degree. [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| Obtains 12 | B1 | 1.1b |
| (1) |
Typical solution
\[\begin{aligned}\mathbf{a}.\mathbf{b} &= 3 \times 2 + 4 \times -1 + -2 \times -5 \\ &= 12\end{aligned}\]| Scheme | Marks | AO |
|---|---|---|
| Obtains correct expression for at least one of \(|\mathbf{a}|\) or \(|\mathbf{b}|\) Condone missing brackets on the negative values. | M1 | 1.1a |
| Obtains \(\sqrt{29}\) and \(\sqrt{30}\) Condone AWRT 5.39 and 5.48 | A1 | 1.1b |
| (2) |
Typical solution
\[|\mathbf{a}| = \sqrt{3^2 + 4^2 + (-2)^2} = \sqrt{29}\]\[|\mathbf{b}| = \sqrt{2^2 + (-1)^2 + (-5)^2} = \sqrt{30}\]| Scheme | Marks | AO |
|---|---|---|
| Writes a correct equation in \(\theta\) Use of \(\mathbf{a} \times \mathbf{b}\) must at least proceed to a correct calculation of \(|\mathbf{a} \times \mathbf{b}|\) leading to an equation in \(\theta\) PI by 65.99… FT their \(\mathbf{a}.\mathbf{b}\) and \(|\mathbf{a}|\) and \(|\mathbf{b}|\) | M1 | 1.1a |
| Obtains 66 | A1 | 1.1b |
| (2) | ||
| (5 marks) |