
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
161.1k+ views
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, \[{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0\].
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, \[{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0\].
Recently Updated Pages
If there are 25 railway stations on a railway line class 11 maths JEE_Main

Minimum area of the circle which touches the parabolas class 11 maths JEE_Main

Which of the following is the empty set A x x is a class 11 maths JEE_Main

The number of ways of selecting two squares on chessboard class 11 maths JEE_Main

Find the points common to the hyperbola 25x2 9y2 2-class-11-maths-JEE_Main

A box contains 6 balls which may be all of different class 11 maths JEE_Main

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Free Radical Substitution Mechanism of Alkanes for JEE Main 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations
