Question

If the 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 are at right angles to each other, then which of the following is true?

(a) \(\frac{A_1+B_1+C_1}{A_2+B_2+C_2}\)=0

(b) \(A_1+A_2+B_1 +B_2+C_1+C_2\)=0

(c) \(A_1+B_1+C_1=A_2 B_2 C_2\)

(d) \(A_1 A_2+B_1 B_2+C_1 C_2\)=0

This question was addressed to me by my college professor while I was bunking the class.

Question is from Three Dimensional Geometry in chapter Three Dimensional Geometry of Mathematics – Class 12

Answer 1

The correct choice is (d) \(A_1 A_2+B_1 B_2+C_1 C_2\)=0

The best I can explain: We know that the angle between two planes is given by

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

Given that, θ=90°

∴cos⁡90°=\(\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right |\)

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

⇒\(A_1 A_2+B_1 B_2+C_1 C_2\)=0.