OCR MEI
Further Pure Core
2020
November
— Question 15
Exam Board
OCR MEI
Module
Further Pure Core (Further Pure Core)
Year
2020
Session
November
Topic
Vectors: Lines & Planes
15
Show that the three planes with equations
$$\begin{aligned}
x + \lambda y + 3 z & = - 12
2 x + y + 5 z & = - 11
x - 2 y + 2 z & = - 9
\end{aligned}$$
where \(\lambda\) is a constant, meet at a unique point except for one value of \(\lambda\) which is to be determined.
In the case \(\lambda = - 2\), use matrices to find the point of intersection P of the planes, showing your method clearly.
The line \(l\) has equation \(\frac { x - 1 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z + 2 } { - 2 }\).
Find a vector equation of \(l\).
Find the shortest distance between the point P and \(l\).
Show that \(l\) is parallel to the plane \(x - 2 y + 2 z = - 9\).
Find the distance between \(l\) and the plane \(x - 2 y + 2 z = - 9\).