| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2022 |
| Session | February |
| Marks | 8 |
| Topic | Vectors 3D & Lines |
| Type | Area of triangle from given side vectors or coordinates |
| Difficulty | Standard +0.3 This is a straightforward Further Maths vectors question requiring standard techniques: finding direction vectors AB and AC, computing their dot product and magnitudes for the angle, then using the cross product (or ½|AB||AC|sin θ) for area. All steps are routine applications of formulas with no conceptual challenges or novel insights required. |
| Spec | 4.04c Scalar product: calculate and use for angles4.04g Vector product: a x b perpendicular vector |
The position vectors of three points $A$, $B$ and $C$ relative to an origin $O$ are given respectively by
$$\overrightarrow{OA} = 7\mathbf{i} + 3\mathbf{j} - 3\mathbf{k},$$
$$\overrightarrow{OB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$$
and
$$\overrightarrow{OC} = 5\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$
\begin{enumerate}[label=(\roman*)]
\item Find the angle between $AB$ and $AC$. [6]
\item Find the area of triangle $ABC$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2022 Q9 [8]}}