M5 January 2006 Q6
6. The vertices of a tetrahedron \(PQRS\) have position vectors \(\mathbf{p}\), \(\mathbf{q}\), \(\mathbf{r}\) and \(\mathbf{s}\) respectively, where
\[\mathbf{p} = -3\mathbf{i} + 4\mathbf{j} - \mathbf{k},\qquad \mathbf{q} = 4\mathbf{i} + 4\mathbf{j} - 2\mathbf{k},\qquad \mathbf{r} = \mathbf{i} - 2\mathbf{j} + \mathbf{k},\qquad \mathbf{s} = 4\mathbf{i} + \mathbf{k}.\]Forces of magnitude 20 N and \(2\sqrt{13}\) N act along \(SQ\) and \(SR\) respectively. A third force acts at \(P\).
Given that the system of three forces reduces to a couple \(\mathbf{G}\), find
(a) the third force, (6)
(b) the magnitude of \(\mathbf{G}\). (6)
| Scheme | Marks |
|---|---|
| \(\mathbf{SQ} = \mathbf{q} - \mathbf{s} = 4\mathbf{j} - 3\mathbf{k} \;\Rightarrow\; \mathbf{F}_1 = 16\mathbf{j} - 12\mathbf{k}\) | M1 A1 |
| \(\mathbf{SR} = -3\mathbf{i} - 2\mathbf{j} \;\Rightarrow\; \mathbf{F}_2 = -6\mathbf{i} - 4\mathbf{j}\) | M1 A1 |
| Net couple alone \(\Rightarrow\ \Sigma\mathbf{F}_i = \mathbf{0}\) | |
| \(\Rightarrow\ \mathbf{F}_3 = 6\mathbf{i} - 12\mathbf{j} + 12\mathbf{k}\) | M1 A1 |
| (6) |
| Scheme | Marks |
|---|---|
| M(\(S\)) \(\mathbf{G} = \mathbf{SP}\times\mathbf{F}_3\) | M1 |
| \(= (-7\mathbf{i} + 4\mathbf{j} - 2\mathbf{k})\times(6\mathbf{i} - 12\mathbf{j} + 12\mathbf{k})\) (or complete express\(^n\) if about another pt.) | A1 |
| \(\mathbf{G} = 24\mathbf{i} + 72\mathbf{j} + 60\mathbf{k}\) | M1 A1 |
| \(= 12(2\mathbf{i} + 6\mathbf{j} + 5\mathbf{k})\) | |
| \(|\mathbf{G}| = 12\sqrt{65}\) | M1 A1 |
| (6) | |
| (12 marks) |