Angles between planes and between lines and planesAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

How do you find the angle between two planes?
What is the normal vector of ax+by+cz=d?
State the formula for the angle between a line and a plane.
Why does the line-plane formula use sine?
What does the modulus in the formulae do?
When are two planes perpendicular?
When is a line parallel to a plane?
What angle does \cos^{-1}\frac{|\mathbf b\cdot\mathbf n|}{|\mathbf b||\mathbf n|} give for a line and a plane?
Normals (2,1,-2) and (2,3,6): acute angle between the planes?
Line direction (2,-1,2), plane normal (1,2,2): angle between line and plane?
How do you find the equation of a plane through (4,0,0), (0,3,0), (0,0,2)?
What is the angle between the plane z=0 and a line with direction (a,b,c)?

Exam questions on Angles between planes and between lines and planes

Plane $\Pi_1$ has equation $2x+y-2z=5$ and plane $\Pi_2$ has equation $2x+3y+6z=11$ .
A third plane $\Pi_3$ has equation $x+ky+2z=1$ . Given that $\Pi_1$ and $\Pi_3$ are perpendicular, find the value of $k$ .2 marks
The line $l$ has equation $\mathbf r=\begin{pmatrix}1 \\ -1 \\ 3\end{pmatrix}+\lambda\begin{pmatrix}2 \\ -1 \\ 2\end{pmatrix}$ and the plane $\Pi$ has equation $x+2y+2z=9$ .
Find the acute angle between $l$ and $\Pi$ .2 marks
Plane $\Pi_1$ has equation $x+2y+2z=3$ and plane $\Pi_2$ has equation $4x-3y=7$ .
Find the acute angle between $\Pi_1$ and $\Pi_2$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).