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

NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options

Interview Questions and Answers, Database Interview Questions and Answers for Freshers and Experience

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

cosθ=\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

cosθ=\left |\frac{2(1)+2(-1)+1(1)}{|\sqrt{2^2+2^2+1^2} \sqrt{1^2+(-1)^2+1^2}|}\right |

cosθ=\left |\frac{1}{\sqrt{9}.\sqrt{3}}\right |

θ=cos^{-1}\frac{1}{3\sqrt{3}}.