| Exam Board | OCR MEI |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 24 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Shortest distance between two skew lines |
| Difficulty | Challenging +1.2 This is a comprehensive multi-part vectors question requiring finding line intersections from planes, verifying parallelism, and computing distances. While it involves several techniques (cross products, scalar products, parametric equations), each part follows standard FP3 procedures without requiring novel insight. The length and multiple parts elevate it slightly above average difficulty, but the methods are all textbook applications. |
| 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 point4.04i Shortest distance: between a point and a line |
| Answer | Marks |
|---|---|
| (i) Show K and L parallel, find shortest distance between them. | [9] |
| (ii) Show shortest distance between line K and plane R is \(5\sqrt{6}\) | [3] |
| (iii) Show lines K and M intersect, find coordinates of intersection | [7] |
| (iv) Find shortest distance between lines L and M | [5] |
## Question 1 (Option 1: Vectors)
**(i)** Show K and L parallel, find shortest distance between them. | [9] |
**(ii)** Show shortest distance between line K and plane R is $5\sqrt{6}$ | [3] |
**(iii)** Show lines K and M intersect, find coordinates of intersection | [7] |
**(iv)** Find shortest distance between lines L and M | [5] |
---1 Three planes $P , Q$ and $R$ have the following equations.
$$\begin{array} { l l }
\text { Plane } P : & 8 x - y - 14 z = 20 \\
\text { Plane } Q : & 6 x + 2 y - 5 z = 26 \\
\text { Plane } R : & 2 x + y - z = 40
\end{array}$$
The line of intersection of the planes $P$ and $Q$ is $K$.\\
The line of intersection of the planes $P$ and $R$ is $L$.\\
(i) Show that $K$ and $L$ are parallel lines, and find the shortest distance between them.\\
(ii) Show that the shortest distance between the line $K$ and the plane $R$ is $5 \sqrt { 6 }$.
The line $M$ has equation $\mathbf { r } = ( \mathbf { i } - 4 \mathbf { j } ) + \lambda ( 5 \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k } )$.\\
(iii) Show that the lines $K$ and $M$ intersect, and find the coordinates of the point of intersection.\\
(iv) Find the shortest distance between the lines $L$ and $M$.
\hfill \mbox{\textit{OCR MEI FP3 2007 Q1 [24]}}