| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2023 |
| Session | January |
| Marks | 9 |
| Topic | Vectors 3D & Lines |
| Type | Show lines intersect and find intersection point |
| Difficulty | Standard +0.3 This is a straightforward 3D vectors question testing standard techniques: equating parametric equations to find intersection (routine but algebraically careful), checking perpendicularity via dot product (direct calculation), and verifying a point lies on a line (substitution). All parts are textbook exercises requiring no problem-solving insight, though the algebra in part (a) requires care. Slightly easier than average due to the mechanical nature of all steps. |
| 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 |
With respect to a fixed origin $O$, the lines $l_1$ and $l_2$ are given by the equations
\begin{align}
l_1: \quad \mathbf{r} &= (-9\mathbf{i} + 10\mathbf{k}) + \lambda(2\mathbf{i} + \mathbf{j} - \mathbf{k}) \\
l_2: \quad \mathbf{r} &= (3\mathbf{i} + \mathbf{j} + 17\mathbf{k}) + \mu(3\mathbf{i} - \mathbf{j} + 5\mathbf{k})
\end{align}
where $\lambda$ and $\mu$ are scalar parameters.
\begin{enumerate}[label=(\alph*)]
\item Show that $l_1$ and $l_2$ meet and find the position vector of their point of intersection. [6]
\item Show that $l_1$ and $l_2$ are perpendicular to each other. [2]
\end{enumerate}
The point $A$ has position vector $5\mathbf{i} + 7\mathbf{j} + 3\mathbf{k}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Show that $A$ lies on $l_1$. [1]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2023 Q7 [9]}}