Question

Which of the following is the correct formula for the angle between two planes \(A_1 x+B_1 y+C_1 z+D_1\)=0 and \(A_2 x+B_2 y+C_2 z+D_2\)=0?

(a) cos⁡θ=\(\frac{A_1 B_1 C_1}{A_2 B_2 C_2}\)

(b) cos⁡θ=\(\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right |\)

(c) sin⁡θ=\(\left |\frac{A_1 A_2-B_1 B_2-C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right |\)

(d) cos⁡θ=\(A_1 A_2+B_1 B_2+C_1 C_2\)

I had been asked this question in an online interview.

I would like to ask this question from Three Dimensional Geometry topic in division Three Dimensional Geometry of Mathematics – Class 12

Answer 1

Right option is (b) cos⁡θ=\(\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right |\)

For explanation I would say: If the planes are in the Cartesian form i.e. \(A_1 x+B_1 y+C_1 z+D_1\)=0 and \(A_2 x+B_2 y+C_2 z+D_2\)=0, where \(A_1,B_1,C_1 \,and \,A_2,B_2,C_2\) are the direction ratios of the planes, then the angle between them is given by

cos⁡θ=\(\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2} \sqrt{A_2^2+B_2^2+C_2^2}}\right |\)