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

Exam questions on Angles between planes and between lines and planes

1. 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
2. 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
3. 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