Question

If the given planes \[ax + by + cz + d = 0\] and \[a'x + b'y + c'z + d' = 0\] be mutually perpendicular. Then, which of the following equations is true?
A. \[\dfrac{a}{{a'}} = \dfrac{b}{{b'}} = \dfrac{c}{{c'}}\]
B. \[\dfrac{a}{{a'}} + \dfrac{b}{{b'}} + \dfrac{c}{{c'}} = 0\]
C. \[aa' + bb' + cc' + dd' = 0\]
D. \[aa' + bb' + cc' = 0\]

Answer 1

Hint: Here, the equations of the two mutually perpendicular planes are given. Apply the condition required for the two planes to be perpendicular to each other on the given equations of the plane. Solve it and get the required answer.

Formula used: 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 perpendicular to each other then, \[{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0\].

Complete step by step solution: Given: The planes \[ax + by + cz + d = 0\] and \[a'x + b'y + c'z + d' = 0\] be mutually perpendicular to each other.

In the given planes, \[\left( {a,b,c} \right)\] are the direction ratios of the plane \[ax + by + cz + d = 0\] and \[\left( {a',b',c'} \right)\] are the direction ratios of the plane \[a'x + b'y + c'z + d' = 0\].

Apply the condition required for the perpendicular planes, 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 perpendicular to each other then \[{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0\].
So, for the given planes we get
\[aa' + bb' + cc' = 0\]

Thus, Option (D) is correct.

Note: The condition required for the perpendicular planes \[{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0\] is obtained from the formula of the angle between the two planes.
The angle between the plane \[{a_1}x + {b_1}y + {c_1}z + {d_1} = 0\] and \[{a_2}x + {b_2}y + {c_2}z + {d_2} = 0\] is \[\cos \theta = \left| {\dfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {{a_1}^2 + {b_1}^2 + {c_1}^2} \sqrt {{a_2}^2 + {b_2}^2 + {c_2}^2} }}} \right|\].
When the planes are perpendicular, the angle between them is \[{90^ \circ }\].
We know that \[\cos \left( {{{90}^ \circ }} \right) = 0\]. So, the numerator must be 0.
Thus, \[{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0\].