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

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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

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 |

Given that, θ=90°

∴cos90°=\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.