| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2017 |
| Session | December |
| Marks | 4 |
| Topic | Vector Product and Surfaces |
| Type | Geometric interpretation of vector product |
| Difficulty | Standard +0.3 Part (i) requires recall of a standard geometric property of the vector product (parallel vectors). Part (ii) is a routine application: find direction vector PQ, then write the standard vector equation of a line using the cross product form. This is straightforward Further Maths content with no novel insight required, making it slightly easier than average A-level difficulty overall. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04g Vector product: a x b perpendicular vector |
| Answer | Marks | Guidance |
|---|---|---|
| (i) x and y are parallel | B1 [1] | 1.2 |
| (ii) E.g. \(d = p - q = i + 4j - 5k\); \(a = p\); so \((r - 2i - 7j + k) \times (i + 4j - 5k)\) | M1 A1 B1 [3] | \(d = \pm(p - q) = \pm(i + 4j - 5k)\); \(a = \) either p or q; i.e. \((r - \text{their } a) \times \text{their } d = 0\); Accept a, d clearly stated (as form is given) |
**(i)** x and y are parallel | B1 [1] | 1.2
**(ii)** E.g. $d = p - q = i + 4j - 5k$; $a = p$; so $(r - 2i - 7j + k) \times (i + 4j - 5k)$ | M1 A1 B1 [3] | $d = \pm(p - q) = \pm(i + 4j - 5k)$; $a = $ either p or q; i.e. $(r - \text{their } a) \times \text{their } d = 0$; Accept a, d clearly stated (as form is given)2 (i) For non-zero vectors $\mathbf { x }$ and $\mathbf { y }$, explain the geometrical significance of the statement $\mathbf { x } \times \mathbf { y } = \mathbf { 0 }$.\\
(ii) The points $P$ and $Q$ have position vectors $\mathbf { p } = 2 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }$ and $\mathbf { q } = \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }$ respectively. Find, in the form $( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }$, the equation of line $P Q$.
\hfill \mbox{\textit{OCR Further Additional Pure AS 2017 Q2 [4]}}