Class 12 Exam > Class 12 Questions > What is the relation between the plane ax + b...
Start Learning for Free
What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?
- a)a1a2 . b1b2 . c1c2 = 0
- b)a1a + b1b + c1c = 0
- c)a1a2 + b1b2 – c1c2 = 0
- d)a1a + b1b2 – c1c2 = 0
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
What is the relation between the plane ax + by + cz + d = 0 and a1, b1...
Understanding the Relationship Between a Plane and a Line
When we have a plane defined by the equation \( ax + by + cz + d = 0 \) and a line characterized by direction ratios \( a_1, b_1, c_1 \), it is essential to understand how their orientations relate when the plane and the line are parallel.
Condition for Parallelism
For a line to be parallel to a plane, the direction ratios of the line must be perpendicular to the normal vector of the plane. The normal vector of the plane given by \( ax + by + cz + d = 0 \) is represented by the coefficients \( (a, b, c) \).
Mathematical Representation
The mathematical condition for the perpendicularity of the line's direction ratios \( (a_1, b_1, c_1) \) to the normal vector \( (a, b, c) \) can be expressed as:
\[ a_1a + b_1b + c_1c = 0 \]
This equation states that the dot product of the line's direction ratios and the plane's normal vector equals zero, confirming that they are perpendicular. Hence, if this condition holds, the line is parallel to the plane.
Why Option 'B' is Correct
- **Option (b)**: \( a_1a + b_1b + c_1c = 0 \) indicates that the line is parallel to the plane since the dot product condition for perpendicular vectors is satisfied.
Other options do not satisfy this critical condition for parallelism.
Conclusion
Understanding this relationship is vital in geometry and vector analysis, ensuring accurate interpretations of the spatial relationships between various geometric entities. Thus, option 'B' is the correct choice, confirming the necessary condition for the parallelism of the line to the plane.
When we have a plane defined by the equation \( ax + by + cz + d = 0 \) and a line characterized by direction ratios \( a_1, b_1, c_1 \), it is essential to understand how their orientations relate when the plane and the line are parallel.
Condition for Parallelism
For a line to be parallel to a plane, the direction ratios of the line must be perpendicular to the normal vector of the plane. The normal vector of the plane given by \( ax + by + cz + d = 0 \) is represented by the coefficients \( (a, b, c) \).
Mathematical Representation
The mathematical condition for the perpendicularity of the line's direction ratios \( (a_1, b_1, c_1) \) to the normal vector \( (a, b, c) \) can be expressed as:
\[ a_1a + b_1b + c_1c = 0 \]
This equation states that the dot product of the line's direction ratios and the plane's normal vector equals zero, confirming that they are perpendicular. Hence, if this condition holds, the line is parallel to the plane.
Why Option 'B' is Correct
- **Option (b)**: \( a_1a + b_1b + c_1c = 0 \) indicates that the line is parallel to the plane since the dot product condition for perpendicular vectors is satisfied.
Other options do not satisfy this critical condition for parallelism.
Conclusion
Understanding this relationship is vital in geometry and vector analysis, ensuring accurate interpretations of the spatial relationships between various geometric entities. Thus, option 'B' is the correct choice, confirming the necessary condition for the parallelism of the line to the plane.
Free Test
FREE
| Start Free Test |
Community Answer
What is the relation between the plane ax + by + cz + d = 0 and a1, b1...
The relation between the plane ax + by +cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other is a1a + b1b + c1c = 0.
|
Explore Courses for Class 12 exam
|
|
Top Courses for Class 12View all
Question Description
What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?a)a1a2 . b1b2 . c1c2 = 0b)a1a + b1b + c1c = 0c)a1a2 + b1b2 – c1c2 = 0d)a1a+ b1b2 – c1c2 = 0Correct answer is option 'B'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?a)a1a2 . b1b2 . c1c2 = 0b)a1a + b1b + c1c = 0c)a1a2 + b1b2 – c1c2 = 0d)a1a+ b1b2 – c1c2 = 0Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?a)a1a2 . b1b2 . c1c2 = 0b)a1a + b1b + c1c = 0c)a1a2 + b1b2 – c1c2 = 0d)a1a+ b1b2 – c1c2 = 0Correct answer is option 'B'. Can you explain this answer?.
What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?a)a1a2 . b1b2 . c1c2 = 0b)a1a + b1b + c1c = 0c)a1a2 + b1b2 – c1c2 = 0d)a1a+ b1b2 – c1c2 = 0Correct answer is option 'B'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?a)a1a2 . b1b2 . c1c2 = 0b)a1a + b1b + c1c = 0c)a1a2 + b1b2 – c1c2 = 0d)a1a+ b1b2 – c1c2 = 0Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?a)a1a2 . b1b2 . c1c2 = 0b)a1a + b1b + c1c = 0c)a1a2 + b1b2 – c1c2 = 0d)a1a+ b1b2 – c1c2 = 0Correct answer is option 'B'. Can you explain this answer?.
Solutions for What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?a)a1a2 . b1b2 . c1c2 = 0b)a1a + b1b + c1c = 0c)a1a2 + b1b2 – c1c2 = 0d)a1a+ b1b2 – c1c2 = 0Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Class 12.
Download more important topics, notes, lectures and mock test series for Class 12 Exam by signing up for free.
Here you can find the meaning of What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?a)a1a2 . b1b2 . c1c2 = 0b)a1a + b1b + c1c = 0c)a1a2 + b1b2 – c1c2 = 0d)a1a+ b1b2 – c1c2 = 0Correct answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?a)a1a2 . b1b2 . c1c2 = 0b)a1a + b1b + c1c = 0c)a1a2 + b1b2 – c1c2 = 0d)a1a+ b1b2 – c1c2 = 0Correct answer is option 'B'. Can you explain this answer?, a detailed solution for What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?a)a1a2 . b1b2 . c1c2 = 0b)a1a + b1b + c1c = 0c)a1a2 + b1b2 – c1c2 = 0d)a1a+ b1b2 – c1c2 = 0Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?a)a1a2 . b1b2 . c1c2 = 0b)a1a + b1b + c1c = 0c)a1a2 + b1b2 – c1c2 = 0d)a1a+ b1b2 – c1c2 = 0Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice What is the relation between the plane ax + by + cz + d = 0 and a1, b1, c1 the direction ratios of a line, if the plane and line are parallel to each other?a)a1a2 . b1b2 . c1c2 = 0b)a1a + b1b + c1c = 0c)a1a2 + b1b2 – c1c2 = 0d)a1a+ b1b2 – c1c2 = 0Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice Class 12 tests.
|
|
Explore Courses for Class 12 exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup on EduRev and stay on top of your study goals
10M+ students crushing their study goals daily