Find the angle between the two planes 2x+2y+z=2 and x-y+z=1?

Category: QuestionsFind the angle between the two planes 2x+2y+z=2 and x-y+z=1?
Editor">Editor Staff asked 11 months ago

Find the angle between the two planes 2x+2y+z=2 and x-y+z=1?
 
(a) cos^{-1}\frac{⁡1}{3}
 
(b) cos^{-1}⁡\sqrt{3}
 
(c) cos^{-1}⁡\frac{1}{3}
 
(d) cos^{-1}⁡\frac{1}{3\sqrt{3}}
 
I had been asked this question in an online interview.
 
Enquiry is from Three Dimensional Geometry in portion Three Dimensional Geometry of Mathematics – Class 12
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1 Answers
Editor">Editor Staff answered 11 months ago

The correct answer is (d) cos^{-1}⁡\frac{1}{3\sqrt{3}}
 
Best explanation: The angle between two planes of the form A_1 x+B_1 y+C_1 z+D_1=0 and A_2 x+B_2 y+C_2 z+D_2=0 is given by
 
cos⁡θ=\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2} \sqrt{A_2^2+B_2^2+C_2^2}}\right |
 
According to the given question, A_1=2,B_1=2,C_1=1 \,and \,A_2=1,B_2=-1,C_2=1
 
cos⁡θ=\left |\frac{2(1)+2(-1)+1(1)}{|\sqrt{2^2+2^2+1^2} \sqrt{1^2+(-1)^2+1^2}|}\right |
 
cos⁡θ=\left |\frac{1}{\sqrt{9}.\sqrt{3}}\right |
 
θ=cos^{-1}⁡\frac{1}{3\sqrt{3}}.