Question

Which of the following is the correct formula for the angle between two planes?

(a) cos⁡θ=\(\left |\frac{\vec{n_1}.\vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\right |\)

(b) sin⁡θ=\(\left |\frac{\vec{n_1}.\vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\right |\)

(c) cos⁡θ=\(\left |\frac{\vec{n_1}+\vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\right |\)

(d) sin⁡θ=\(\left |\frac{\vec{n_1}+\vec{n_2}}{(|\vec{n_1}|+|\vec{n_2}|}\right |\)

I have been asked this question during a job interview.

This key question is from Three Dimensional Geometry topic in section Three Dimensional Geometry of Mathematics – Class 12

Answer 1

The correct option is (a) cos⁡θ=\(\left |\frac{\vec{n_1}.\vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\right |\)

For explanation I would say: If two planes of the form \(\vec{r}.\vec{n_1}=d_1\) and \(\vec{r}.\vec{n_2}=d_2\) where \(\vec{n_1} \,and \,\vec{n_2}\) are the normals to the plane, then the angle between them is given by

cos⁡θ=\(\left |\frac{\vec{n_1}.\vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\right |\)