| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2022 |
| Session | February |
| Marks | 9 |
| Topic | Vectors 3D & Lines |
| Type | Line-plane intersection and related angle/perpendicularity |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question testing routine techniques: finding line-plane intersection by substitution, calculating angle between line and plane using dot product formula, and finding a line perpendicular to a given direction and parallel to a plane using cross product. All parts follow textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane |
The line $l_1$ has equation $\mathbf{r} = \begin{pmatrix} 1 \\ -3 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 2 \\ -2 \end{pmatrix}$.
The plane $\Pi$ has equation $\mathbf{r} \cdot \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix} = 4$.
\begin{enumerate}[label=(\alph*)]
\item Find the position vector of the point of intersection of $l_1$ and $\Pi$. [3]
\item Find the acute angle between $l_1$ and $\Pi$. [3]
\end{enumerate}
$A$ is the point on $l_1$ where $\lambda = 1$.
$l_2$ is the line with the following properties.
• $l_2$ passes through $A$
• $l_2$ is perpendicular to $l_1$
• $l_2$ is parallel to $\Pi$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find, in vector form, the equation of $l_2$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q3 [9]}}