Question

Find the angle between the normals to the planes 2x-y+z=6 and x+y+2z=7.

Answer 1

Let ${\overrightarrow{n}}_1$ and ${\overrightarrow{n}}_2$ be vectors normal to the plane $2x-y+z=6$ and $x+y+2z=7$.

${\overrightarrow{n}}_1=2\hat{i}-\hat{j}+\hat{k}$

${\overrightarrow{n}}_2=\hat{i}+\hat{j}+2\hat{k}$

Let $\theta$ be the angle between the normals ${\overrightarrow{n}}_1$ and ${\overrightarrow{n}}_2$. Then,

$\cos \theta =\frac{{\overrightarrow{n}}_1.{\overrightarrow{n}}_2}{|{\overrightarrow{n}}_1||{\overrightarrow{n}}_2|}$

$\cos \theta =\frac{2\times 1+(-1)\times 1+1\times 2}{\sqrt{2^2+(-1{)}^2+1^2}\sqrt{1^2+1^2+2^2}}$

$\cos \theta =\frac{1}{2}$

$\theta =\frac{\pi }{3}$