| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | 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 from FP3. Part (a) requires computing a cross product using the determinant method, (b) is direct substitution into the scalar triple product, (c) uses the formula area = ½|b×c|, and (d) uses volume = ⅙|a·(b×c)|. All parts follow directly from learned formulas with no problem-solving insight required, making it slightly easier than average despite being Further Maths content. |
| Spec | 4.04c Scalar product: calculate and use for angles4.04g Vector product: a x b perpendicular vector |
2.
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\caption{Figure 1}
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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
$$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } .$$
Calculate
\begin{enumerate}[label=(\alph*)]
\item $\mathbf { b } \times \mathbf { c }$,
\item a.(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 2009 Q2 [8]}}