| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2011 |
| Session | June |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Reflection and symmetry |
| Difficulty | Hard +2.3 This AEA question requires multiple sophisticated 3D vector techniques: finding reflections in lines (perpendicular distance, foot of perpendicular), angle calculations with dot products, using geometric constraints (kite properties, area formula) to find unknown positions, and finding circle centers in 3D. The multi-part structure with 19 marks and interconnected parts requiring extended reasoning places it well above typical A-level questions, though the individual techniques are accessible to strong Further Maths students. |
| Spec | 1.10c Magnitude and direction: of vectors4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04d Angles: between planes and between line and plane |
The line $L$ has equation
$$\mathbf{r} = \begin{pmatrix} 13 \\ -3 \\ -8 \end{pmatrix} + t \begin{pmatrix} -5 \\ 3 \\ 4 \end{pmatrix}$$
The point $P$ has position vector $\begin{pmatrix} -7 \\ 2 \\ 7 \end{pmatrix}$.
The point $P'$ is the reflection of $P$ in $L$.
\begin{enumerate}[label=(\alph*)]
\item Find the position vector of $P'$.
[6]
\item Show that the point $A$ with position vector $\begin{pmatrix} -7 \\ 9 \\ 8 \end{pmatrix}$ lies on $L$.
[1]
\item Show that angle $PAP' = 120°$.
[3]
\end{enumerate}
% Figure 3 shows kite APBP' with angle at A = 120°
\includegraphics{figure_3}
Figure 3
The point $B$ lies on $L$ and $APBP'$ forms a kite as shown in Figure 3.
The area of the kite is $50\sqrt{3}$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the position vector of the point $B$.
[5]
\item Show that angle $BPA = 90°$.
[2]
\end{enumerate}
The circle $C$ passes through the points $A$, $P$, $P'$ and $B$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{5}
\item Find the position vector of the centre of $C$.
[2]
\end{enumerate}
[Total 19 marks]
\hfill \mbox{\textit{Edexcel AEA 2011 Q6 [19]}}