| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2023 |
| Session | February |
| Marks | 8 |
| Topic | Vectors 3D & Lines |
| Type | Show lines intersect and find intersection point |
| Difficulty | Standard +0.8 This is a multi-part Further Maths vectors question requiring: (a) solving simultaneous equations to find intersection point, (b) using dot product formula for angle between lines, and (c) geometric reasoning about perpendicular distances involving projections. While each individual technique is standard, the combination across three parts with the final part requiring spatial visualization and understanding of perpendicular projections makes this moderately challenging, though still within typical FM scope. |
| 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 |
Two lines, $l_1$ and $l_2$, have the following equations.
$$l_1: \mathbf{r} = \begin{pmatrix} -1 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$
$$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$
P is the point of intersection of $l_1$ and $l_2$.
\begin{enumerate}[label=(\alph*)]
\item Find the position vector of P. [3]
\item Find, correct to 1 decimal place, the acute angle between $l_1$ and $l_2$. [3]
\end{enumerate}
Q is a point on $l_1$ which is 12 metres away from P. R is the point on $l_2$ such that QR is perpendicular to $l_1$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Determine the length QR. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2023 Q7 [8]}}