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

Category: QuestionsWhich of the following is the correct formula for the angle between two planes?
Editor">Editor Staff asked 11 months ago

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
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

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 |