| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2014 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Vector geometry in 3D shapes |
| Difficulty | Challenging +1.8 This AEA question requires systematic 3D vector manipulation across multiple parts: computing six vectors, verifying a square base, finding angles using dot products, determining pyramid height through perpendicularity conditions, and locating apex positions. While conceptually straightforward, it demands careful algebraic work, spatial reasoning about pyramid geometry, and understanding that the apex lies on a perpendicular through the base center. The octahedron extension requires recognizing symmetry. The length and multi-step nature elevate this above standard A-level but it follows established techniques without requiring novel geometric insight. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication4.04c Scalar product: calculate and use for angles |
The square-based pyramid $P$ has vertices $A, B, C, D$ and $E$. The position vectors of $A, B, C$ and $D$ are $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ and $\mathbf{d}$ respectively where
$$\mathbf{a} = \begin{pmatrix} -2 \\ 3 \\ -1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 8 \\ -6 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 2 \\ 5 \\ 3 \end{pmatrix}, \quad \mathbf{d} = \begin{pmatrix} 6 \\ 1 \\ 1 \end{pmatrix}$$
(a) Find the vectors $\overrightarrow{AB}$, $\overrightarrow{AC}$, $\overrightarrow{AD}$, $\overrightarrow{BC}$, $\overrightarrow{BD}$ and $\overrightarrow{CD}$.
[3]
(b) Find
\begin{enumerate}[label=(\roman*)]
\item the length of a side of the square base of $P$,
\item the cosine of the angle between one of the slanting edges of $P$ and its base,
\item the height of $P$,
\item the position vector of $E$.
\end{enumerate}
[9]
A second pyramid, identical to $P$, is attached by its square base to the base of $P$ to form an octahedron.
(c) Find the position vector of the other vertex of this octahedron.
[3]
\hfill \mbox{\textit{Edexcel AEA 2014 Q5 [15]}}