FP3 June 2009 Q2
2.
The points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively, relative to a fixed origin \(O\), as shown in Figure 1.
It is given that \[\mathbf{a} = \mathbf{i} + \mathbf{j}, \quad \mathbf{b} = 3\mathbf{i} - \mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{c} = 2\mathbf{i} + \mathbf{j} - \mathbf{k}.\]
Calculate
(a) \(\mathbf{b} \times \mathbf{c}\), (3)
(b) \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\), (2)
(c) the area of triangle \(OBC\), (2)
(d) the volume of the tetrahedron \(OABC\). (1)
| Scheme | Marks |
|---|---|
| \(\mathbf{b} \times \mathbf{c} = 0\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}\) | M1 A1 A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0 + 5 = 5\) | M1 A1 ft |
| (2) |
| Scheme | Marks |
|---|---|
| Area of triangle \(OBC = \tfrac{1}{2}\left|5\mathbf{j} + 5\mathbf{k}\right| = \tfrac{5}{2}\sqrt{2}\) | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| Volume of tetrahedron \(= \tfrac{1}{6} \times 5 = \tfrac{5}{6}\) | B1 ft |
| (1) | |
| (8 marks) |