| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2018 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Shortest distance from point to line |
| Difficulty | Standard +0.3 This is a standard multi-part vectors question covering routine techniques: showing lines don't meet (equating components and checking for consistency), finding an angle using dot product, and finding perpendicular distance from point to line. All methods are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication1.10f Distance between points: using position vectors1.10g Problem solving with vectors: in geometry |
8. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations
$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r }
1 \\
- 3 \\
2
\end{array} \right) + \lambda \left( \begin{array} { l }
1 \\
2 \\
3
\end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }
6 \\
4 \\
1
\end{array} \right) + \mu \left( \begin{array} { r }
1 \\
1 \\
- 1
\end{array} \right)$$
where $\lambda$ and $\mu$ are scalar parameters.
\begin{enumerate}[label=(\alph*)]
\item Show that $l _ { 1 }$ and $l _ { 2 }$ do not meet.
The point $P$ is on $l _ { 1 }$ where $\lambda = 0$, and the point $Q$ is on $l _ { 2 }$ where $\mu = - 1$
\item Find the acute angle between the line segment $P Q$ and $l _ { 1 }$, giving your answer in degrees to 2 decimal places.
\item Find the shortest distance from the point $Q$ to the line $l _ { 1 }$, giving your answer to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C34 2018 Q8 [11]}}