| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795 (Pre-U Further Mathematics) |
| Session | Specimen |
| Topic | Vectors: Cross Product & Distances |
| Type | Shortest distance between two skew lines |
| Difficulty | Challenging +1.2 This is a comprehensive multi-part vectors question requiring cross products, angles between planes, and distance calculations. While it involves several techniques (finding normals via cross product, angle between planes, skew line distance formula, point-to-line distance), each part follows standard Further Maths procedures without requiring novel insight. The calculations are moderately lengthy but routine for Pre-U/Further Maths students who have learned these formulas. Slightly above average difficulty due to the multi-step nature and Further Maths content, but not exceptionally challenging. |
| Spec | 4.04d Angles: between planes and between line and plane4.04g Vector product: a x b perpendicular vector4.04i Shortest distance: between a point and a line4.04j Shortest distance: between a point and a plane |
11 With respect to an origin $O$, the points $A , B , C$ and $D$ have position vectors
$$\mathbf { a } = 2 \mathbf { i } - \mathbf { j } + \mathbf { k } , \quad \mathbf { b } = \mathbf { i } - 2 \mathbf { k } , \quad \mathbf { c } = - \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { d } = - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ,$$
respectively. Find\\
(i) a vector perpendicular to the plane $O A B$,\\
(ii) the acute angle between the planes $O A B$ and $O C D$, correct to the nearest $0.1 ^ { \circ }$,\\
(iii) the shortest distance between the line $A B$ and the line $C D$,\\
(iv) the perpendicular distance from the point $A$ to the line $C D$.
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q11}}