Question

The angle between the planes
$\overrightarrow{r}(\hat{i}+2\hat{j}+\hat{k})=4$ and $\overrightarrow{r}(\widehat{-i}+\hat{j}+2\hat{k})=9$

• A. ${30}^o$
• B. ${60}^o$
• C. ${45}^o$
• D. ${90}^0$

Answer 1

The correct option is B ${60}^o$

Angle between two planes is the angle between their normal vectors.

For the first plane, normal vector is $\overrightarrow{n_0}=(1,2,1)$

For second plane, normal vector is $\overrightarrow{n_1}=(-1,1,2)$

Let $\theta$ be the angle between the planes, it is also the angle between their normals.
$⟹\cos \theta =\frac{n_1.n_2}{|n_1||n_2|}$

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

$⟹\cos \theta =\frac{-1+2+2}{6}=\frac{1}{2}$
$⟹\theta$ = ${60}^o$