| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Volume of tetrahedron using scalar triple product |
| Difficulty | Standard +0.3 This is a straightforward application of standard vector product formulas with minimal problem-solving required. Parts (a)-(b) are direct calculations of cross product and scalar triple product, while (c)-(d) apply standard formulas (area = ½|b×c|, volume = ⅙|a·(b×c)|). The arithmetic is routine and all vectors have simple integer components. This is slightly easier than average as it's purely procedural with no conceptual challenges. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication4.04c Scalar product: calculate and use for angles4.04g Vector product: a x b perpendicular vector |
2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{045545c7-06d9-40b6-9d01-fc792ab0aa07-01_222_241_525_2042}
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\caption{Figure 1}
\end{center}
\end{figure}
The points $A , B$ and $C$ have position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$ respectively, relative to a fixed origin $O$, as shown in Figure 1.
It is given that
$$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = \mathbf { 3 i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = \mathbf { 2 i } + \mathbf { j } - \mathbf { k } .$$
Calculate
\begin{enumerate}[label=(\alph*)]
\item $\mathbf { b } \times \mathbf { c }$,
\item $\mathbf { a . } ( \mathbf { b } \times \mathbf { c } )$,
\item the area of triangle $O B C$,
\item the volume of the tetrahedron $O A B C$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q2 [7]}}