| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Reflection in plane |
| Difficulty | Standard +0.8 This is a comprehensive Further Maths vectors question requiring multiple techniques: finding normal vectors via cross product, vector equations of lines, scalar product form of planes, point-plane distance, and reflection in a plane. While each individual step uses standard FP3 methods, the multi-part structure (18 marks total) and requirement to synthesize several concepts makes it moderately challenging, though not requiring novel insight beyond textbook techniques. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04f Line-plane intersection: find point4.04g Vector product: a x b perpendicular vector4.04j Shortest distance: between a point and a plane |
The plane $\Pi$ passes through the points
$$A(-1, -1, 1), B(4, 2, 1) \text{ and } C(2, 1, 0).$$
\begin{enumerate}[label=(\alph*)]
\item Find a vector equation of the line perpendicular to $\Pi$ which passes through the point $D(1, 2, 3)$. [3]
\item Find the volume of the tetrahedron $ABCD$. [3]
\item Obtain the equation of $\Pi$ in the form $\mathbf{r} \cdot \mathbf{n} = p$. [3]
\end{enumerate}
The perpendicular from $D$ to the plane $\Pi$ meets $\Pi$ at the point $E$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the coordinates of $E$. [4]
\item Show that $DE = \frac{11\sqrt{35}}{35}$. [2]
\end{enumerate}
The point $D'$ is the reflection of $D$ in $\Pi$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{5}
\item Find the coordinates of $D'$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q9 [18]}}