Question

The acute angle between the planes $2x-y+z=6$ and $x+y+2z=3$ is _____.

Answer 1

Two planes given:

$P_1:2x-y+z=6$

$P_2:x+y+2z=3$

Normals are:

$\overrightarrow{n_1}:2\hat{i}-\hat{j}+\hat{k}$

$\overrightarrow{n_2}:\hat{i}+\hat{j}+2\hat{k}$

Acute angle will be $\theta$ and

$\theta ={\cos }^{-1}\frac{\overrightarrow{n_1}\cdot \overrightarrow{n_2}}{|\overrightarrow{n_1}||\overrightarrow{n_2}|}$

$\Rightarrow \theta ={\cos }^{-1}\left(\frac{1}{2}\right)=60˚$