| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Plane containing line and point/vector |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question covering routine techniques: finding intersection conditions by equating components, determining a plane equation from two direction vectors and a point, and applying the skew lines distance formula. While it requires multiple steps and careful algebra, all methods are textbook procedures with no novel insight required, making it slightly easier than average for FP3. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting4.04f Line-plane intersection: find point4.04h Shortest distances: between parallel lines and between skew lines |
7. The lines $l _ { 1 }$ and $l _ { 2 }$ have equations
$$\mathbf { r } = \left( \begin{array} { r }
1 \\
- 1 \\
2
\end{array} \right) + \lambda \left( \begin{array} { r }
- 1 \\
3 \\
4
\end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r }
\alpha \\
- 4 \\
0
\end{array} \right) + \mu \left( \begin{array} { l }
0 \\
3 \\
2
\end{array} \right) .$$
If the lines $l _ { 1 }$ and $l _ { 2 }$ intersect, find
\begin{enumerate}[label=(\alph*)]
\item the value of $\alpha$,
\item an equation for the plane containing the lines $l _ { 1 }$ and $l _ { 2 }$, giving your answer in the form $a x + b y + c z + d = 0$, where $a , b , c$ and $d$ are constants.
For other values of $\alpha$, the lines $l _ { 1 }$ and $l _ { 2 }$ do not intersect and are skew lines.\\
Given that $\alpha = 2$,
\item find the shortest distance between the lines $l _ { 1 }$ and $l _ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q7 [9]}}