| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2025 |
| Session | February |
| Marks | 8 |
| Topic | Vectors 3D & Lines |
| Type | Show lines intersect and find intersection point |
| Difficulty | Standard +0.3 This is a standard Further Maths vector geometry question requiring routine techniques: equating components to find intersection (solving simultaneous equations), then using the scalar product formula for angle between direction vectors. While it involves multiple steps and careful algebra, it follows a well-practiced procedure with no novel insight required, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
The equations of two lines are
$\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \lambda(2\mathbf{i} + \mathbf{j} + 3\mathbf{k})$ and $\mathbf{r} = 6\mathbf{i} + 8\mathbf{j} + \mathbf{k} + \mu(\mathbf{i} + 4\mathbf{j} - 5\mathbf{k})$.
\begin{enumerate}[label=(\roman*)]
\item Show that these lines meet, and find the coordinates of the point of intersection.
[5]
\item Find the acute angle between these lines.
[3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2025 Q6 [8]}}