Question

Find the angle between the planes $2x+y-z=9$ and $x+2y+z=7$.

Answer 1

Given equations of planes are $2x+y-z=9$ and $x+2y+z=7$

Here, ${\overrightarrow{n}}_1=2\overrightarrow{i}+\overrightarrow{j}-\overrightarrow{k}$
${\overrightarrow{n}}_2=\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k}$
${\overrightarrow{n}}_1{\overrightarrow{n}}_2=2+2-1=3$
$\left|{\overrightarrow{n}}_1\right|=\sqrt{4+1+1}=\sqrt{6}$
$\left|{\overrightarrow{n}}_2\right|=\sqrt{1+4+1}=\sqrt{6}$
Let $\theta$ be angle between the plane
$\cos \theta =\frac{{\overrightarrow{n}}_1\cdot {\overrightarrow{n}}_2}{\left|{\overrightarrow{n}}_1\right|\left|{\overrightarrow{n}}_2\right|}=\frac{3}{\sqrt{6}\sqrt{6}}=\frac{3}{6}=\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3}$