| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795 (Pre-U Further Mathematics) |
| Session | Specimen |
| Marks | 12 |
| Topic | Vectors: Cross Product & Distances |
| Type | Acute angle between two planes |
| Difficulty | Standard +0.3 This is a standard multi-part 3D vector geometry question testing routine techniques: cross products for perpendicular vectors, angle between planes using normal vectors, skew line distance formula, and point-to-line distance. While it has 12 marks total and requires careful calculation, all parts use well-practiced A-level Further Maths methods with no novel problem-solving or geometric insight required. Slightly easier than average due to its straightforward application of formulas. |
| 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 |
With respect to an origin $O$, the points $A, B, C, D$ have position vectors
$$\mathbf{2i - j + k}, \quad \mathbf{i - 2k}, \quad \mathbf{-i + 3j + 2k}, \quad \mathbf{-i + j + 4k},$$
respectively. Find
\begin{enumerate}[label=(\roman*)]
\item a vector perpendicular to the plane $OAB$, [2]
\item the acute angle between the planes $OAB$ and $OCD$, correct to the nearest $0.1°$, [3]
\item the shortest distance between the line which passes through $A$ and $B$ and the line which passes through $C$ and $D$, [4]
\item the perpendicular distance from the point $A$ to the line which passes through $C$ and $D$. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q12 [12]}}