AS October 2020 Paper 1 Q10
10 A vector \(\mathbf{v}\) has magnitude 1 unit. The angle between \(\mathbf{v}\) and the positive \(z\)-axis is \(60^\circ\), and \(\mathbf{v}\) is parallel to the plane \(x - 2y = 0\).
Given that \(\mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}\), where \(a\), \(b\) and \(c\) are all positive, find \(\mathbf{v}\). [7]
| Scheme | Marks | AO |
|---|---|---|
| \(a^2 + b^2 + c^2 = 1\) | B1 | 3.1a |
| \(\cos 60^\circ = \dfrac{(a\mathbf{i} + b\mathbf{j} + c\mathbf{k}).\mathbf{k}}{1 \times \sqrt{a^2 + b^2 + c^2}}\) | M1 | 3.1a |
| \(\Rightarrow c = \dfrac{1}{2}\sqrt{a^2 + b^2 + c^2}\) | A1 | 1.1 |
| \((a\mathbf{i} + b\mathbf{j} + c\mathbf{k}).(\mathbf{i} - 2\mathbf{j}) = 0\) | M1 | 3.1a |
| \(\Rightarrow a - 2b = 0\) | A1 | 1.1 |
| \(c = \dfrac{1}{2},\quad b = \sqrt{\dfrac{3}{20}},\quad a = \sqrt{\dfrac{3}{5}}\) | B2,1,0 | 1.1, 3.1a |
| [7] |
Notes
B1: soi
M1: (1st) oe, e.g. \(\sin 30^\circ = c/\sqrt{a^2 + b^2 + c^2}\) as line makes \(30^\circ\) angle with O\(xy\) plane
A1: (1st) [so \(c = \tfrac{1}{2}\)]
A1: (2nd) or equiv arguments
B2,1,0: oe or 0.387 or 0.775 or better