| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Volume of tetrahedron using scalar triple product |
| Difficulty | Challenging +1.8 This AEA question requires understanding that three edges of a cube from one vertex are mutually perpendicular with equal length, then applying scalar triple product for volume and using dot product properties. The cube geometry insight and multi-step reasoning (verifying perpendicularity, computing volume, solving for α, finding space diagonal) elevate this above standard A-level but it follows systematic methods once the geometric relationship is recognized. |
| Spec | 4.04c Scalar product: calculate and use for angles4.04g Vector product: a x b perpendicular vector |
4.
$$\mathbf { a } = \left( \begin{array} { r }
- 3 \\
1 \\
4
\end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { r }
5 \\
- 2 \\
9
\end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { r }
8 \\
- 4 \\
3
\end{array} \right)$$
The points $A , B$ and $C$ with position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$ ,respectively,are 3 vertices of a cube.
\begin{enumerate}[label=(\alph*)]
\item Find the volume of the cube.
The points $P , Q$ and $R$ are vertices of a second cube with $\overrightarrow { P Q } = \left( \begin{array} { l } 3 \\ 4 \\ \alpha \end{array} \right) , \overrightarrow { P R } = \left( \begin{array} { l } 7 \\ 1 \\ 0 \end{array} \right)$ and $\alpha$ a positive constant.
\item Given that angle $Q P R = 60 ^ { \circ }$ ,find the value of $\alpha$ .
\item Find the length of a diagonal of the second cube.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2012 Q4 [11]}}