Question

If two planes \(a_{1}x+b_{1}y+c_{1}z+d_{1}=0\) and \(a_{2}x+b_{2}y+c_{2}z+d_{2}=0\) are mutually perpendicular, then

• A. \(\dfrac{a_{1}}{a_{2}}=\dfrac{b_{1}}{b_{2}}=\dfrac{c_{1}}{c_{2}}\)
• B. \(\dfrac{a_{1}}{a_{2}}+\dfrac{b_{1}}{b_{2}}+\dfrac{c_{1}}{c_{2}}=0\)
• C. \(a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}=0\)
• D. none of these

Hint:

Angles between planes equal angles between their normal vectors.

Answer 1

The Correct Option is C

Normals of the planes are \(\vec{n}_{1}=(a_{1},b_{1},c_{1})\) and \(\vec{n}_{2}=(a_{2},b_{2},c_{2})\). Planes are perpendicular \(\iff\) their normals are perpendicular: \(\vec{n}_{1}\cdot\vec{n}_{2}=0\Rightarrow a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}=0\).