| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Acute angle between two planes |
| Difficulty | Standard +0.3 This is a standard FP3 3D vectors question involving finding a plane equation, angle between planes, and intersection of line with plane. All parts use routine methods (cross product for normal, dot product for angle, parametric line intersection) with straightforward cuboid geometry. While it requires multiple techniques, each step follows textbook procedures without requiring novel insight. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04d Angles: between planes and between line and plane4.04f Line-plane intersection: find point |
\includegraphics{figure_6}
The cuboid $OABCDEFG$ shown in the diagram has $\overrightarrow{OA} = 4\mathbf{i}, \overrightarrow{OC} = 2\mathbf{j}, \overrightarrow{OD} = 3\mathbf{k}$, and $M$ is the mid-point of $GF$.
\begin{enumerate}[label=(\roman*)]
\item Find the equation of the plane $ACGE$, giving your answer in the form $\mathbf{r} \cdot \mathbf{n} = p$. [4]
\item The plane $OEFC$ has equation $\mathbf{r} \cdot (3\mathbf{i} - 4\mathbf{k}) = 0$. Find the acute angle between the planes $OEFC$ and $ACGE$. [4]
\item The line $AM$ meets the plane $OEFC$ at the point $W$. Find the ratio $AW : WM$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q6 [13]}}