Question

What is the angle between two planes having normal vectors as
$\overrightarrow{n_1}=\hat{i}+2\hat{j}+2\hat{k}$ and $\overrightarrow{n_2}=4\hat{i}-4\hat{j}+2\hat{k}$ ?

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

Answer 1

The correct option is B

${90}^{\circ }$
We know that angle between two planes is equal to the angle between their normal vector.
So,
$\mathrm{cos\theta }=\frac{\overrightarrow{n_{1.}}\overrightarrow{n_2}}{\left(\overrightarrow{n_1}\right|.\left(\overrightarrow{n_2}\right|}=\frac{(\hat{i}+2\hat{j}+2\hat{k}).(4\hat{i}-4\hat{j}+2\hat{k})}{\left(\sqrt{1^2+2^2+2^2}\right)({\sqrt{4^2+{(-4)}^2+2^2}}^{}}=0$

$\therefore \theta ={90}^{\circ }$
So, the angle between the planes is ${90}^{\circ }.$