| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2010 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with line |
| Difficulty | Challenging +1.2 This is a structured 3D vectors question with standard techniques: (a) uses dot product for angle between vectors, (b) requires perpendicular projection onto a line, (c) involves finding line intersection by equating parametric forms. While it requires multiple steps and careful coordinate work, all techniques are standard A-level Further Maths content with no novel insights needed. The cuboid setup makes visualization straightforward. Slightly above average due to the 3D context and multi-part nature, but well within typical AEA difficulty. |
| Spec | 4.04c Scalar product: calculate and use for angles4.04d Angles: between planes and between line and plane4.04e Line intersections: parallel, skew, or intersecting4.04f Line-plane intersection: find point |
4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-3_643_332_246_870}
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\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a cuboid $O A B C D E F G$, where $O$ is the origin, $A$ has position vector $5 \mathbf { i } , C$ has position vector $10 \mathbf { j }$ and $D$ has position vector $20 \mathbf { k }$.
\begin{enumerate}[label=(\alph*)]
\item Find the cosine of angle $C A F$.
Given that the point $X$ lies on $A C$ and that $F X$ is perpendicular to $A C$,
\item find the position vector of point $X$ and the distance $F X$.
The line $l _ { 1 }$ passes through $O$ and through the midpoint of the face $A B F E$. The line $l _ { 2 }$ passes through $A$ and through the midpoint of the edge $F G$.
\item Show that $l _ { 1 }$ and $l _ { 2 }$ intersect and find the coordinates of the point of intersection.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2010 Q4 [16]}}