RD Sharma Class 12 Solutions Chapter 29 - The Plane (Ex 29.9) Exercise 29.9 - Free PDF
FAQs on RD Sharma Class 12 Solutions Chapter 29 - The Plane (Ex 29.9) Exercise 29.9
1. What is the primary concept a student must master for solving problems in RD Sharma Class 12 Maths Exercise 29.9?
For Exercise 29.9 of Chapter 29, The Plane, the primary concept is to calculate the angle between two planes. Students must be proficient in identifying the normal vectors to the planes from their equations (both in vector and Cartesian form) and applying the dot product formula to find the cosine of the angle between these normals.
2. What is the step-by-step method to find the angle between two planes given in Cartesian form as per RD Sharma solutions?
To find the angle θ between two planes, A₁x + B₁y + C₁z + D₁ = 0 and A₂x + B₂y + C₂z + D₂ = 0, follow these steps:
First, identify the direction ratios of the normal vectors for both planes. For the first plane, the normal vector n₁ has direction ratios (A₁, B₁, C₁). For the second plane, the normal vector n₂ has direction ratios (A₂, B₂, C₂).
Apply the dot product formula: cos θ = |(A₁A₂ + B₁B₂ + C₁C₂) / (√(A₁² + B₁² + C₁²) * √(A₂² + B₂² + C₂²))|.
Calculate the value of cos θ and then find the angle θ using the inverse cosine function, θ = arccos(cos θ).
3. How do you determine if two planes are perpendicular using the method from RD Sharma Chapter 29?
Two planes are perpendicular to each other if the angle between them is 90°. This occurs when the dot product of their normal vectors is zero. If the planes are A₁x + B₁y + C₁z + D₁ = 0 and A₂x + B₂y + C₂z + D₂ = 0, the condition for them to be perpendicular is A₁A₂ + B₁B₂ + C₁C₂ = 0. This is a crucial check used in many problems in this exercise.
4. What is the condition for two planes to be parallel as covered in this chapter?
Two planes are parallel if their normal vectors are parallel. This means the direction ratios of their normals must be proportional. For two planes with normals having direction ratios (A₁, B₁, C₁) and (A₂, B₂, C₂), the condition for them to be parallel is A₁/A₂ = B₁/B₂ = C₁/C₂.
5. Why is the angle between two planes defined as the angle between their normal vectors and not the planes themselves?
This is a fundamental concept in 3D geometry. A plane is an infinite two-dimensional surface, making it ambiguous to directly measure an angle. However, the normal vector to a plane has a fixed direction, perpendicular to every line on that plane. Therefore, the angle between the fixed normal vectors provides a consistent and well-defined measure for the angle of inclination between the two planes.
6. What is a common mistake when calculating the angle between a line and a plane, and how does it differ from the method in Exercise 29.9?
A common mistake is using the wrong trigonometric function. The method in Exercise 29.9 (angle between two planes) uses the dot product with cos θ. However, when finding the angle (say, φ) between a line and a plane, you are finding the angle between the line's direction vector and the plane's normal vector. This gives you the complement of the required angle. Therefore, the correct formula for the angle between a line and a plane is sin φ = |(b · n) / (|b| |n|)|, where 'b' is the direction vector of the line and 'n' is the normal vector of the plane.
7. How does the vector form of a plane's equation simplify finding the angle between them?
The vector form, r · n₁ = d₁ and r · n₂ = d₂, makes the process more direct. The normal vectors, n₁ and n₂, are explicitly stated in the equations. You can immediately take these vectors and apply the vector dot product formula: cos θ = |(n₁ · n₂) / (|n₁| |n₂|)|. This avoids the intermediate step of extracting coefficients as required in the Cartesian form and directly reinforces the geometric concept.
