Class 12 Math Chapter 29 PDF Download
FAQs on RD Sharma Solutions for Class 12 Math Chapter 29 - The Plane - Free PDF Download
1. What are the key concepts covered in RD Sharma Class 12 Maths Chapter 29, The Plane?
The RD Sharma solutions for Chapter 29, The Plane, meticulously cover the core concepts aligned with the CBSE syllabus for the 2025-26 session. Key topics include:
The definition and general equation of a plane.
Finding the equation of a plane in normal form.
Deriving the equation of a plane passing through three non-collinear points.
Understanding and applying the intercept form of the equation of a plane.
Calculating the angle between two planes and the distance of a point from a plane.
2. How do the RD Sharma solutions for Chapter 29 (The Plane) help in preparing for the CBSE board exams?
Practising the RD Sharma solutions for this chapter provides a significant advantage for board exams. It offers a wide variety of problems that build a strong foundation in three-dimensional geometry. By working through these detailed, step-by-step solutions, students can master different problem types, understand the correct application of formulas, and learn how to present their answers as per the CBSE marking scheme, which is crucial for scoring well.
3. What is the step-by-step method to find the equation of a plane in normal form?
To find the equation of a plane in normal form, you need two things: the perpendicular distance (p) of the plane from the origin and a unit vector (n̂) normal to the plane. The correct method is:
Step 1: Identify or calculate the normal vector to the plane.
Step 2: Convert this into a unit vector (n̂) by dividing the vector by its magnitude.
Step 3: The vector equation is given by r ⋅ n̂ = p. The Cartesian form is lx + my + nz = p, where l, m, and n are the direction cosines of the normal vector.
4. How do you solve for the equation of a plane passing through three non-collinear points?
To find the equation of a plane passing through three non-collinear points A, B, and C with position vectors a, b, and c, follow these steps:
Step 1: Form two vectors lying in the plane, for example, vector AB (b - a) and vector AC (c - a).
Step 2: Calculate the normal vector (N) to the plane by taking the cross product of these two vectors: N = (b - a) × (c - a).
Step 3: The equation of the plane is then given by (r - a) ⋅ N = 0, where 'r' is the position vector of any general point on the plane. This confirms that the vector from point A to any point on the plane is perpendicular to the normal.
5. What is the difference between the vector and Cartesian equations of a plane, and when should each be used?
The primary difference lies in their representation. The vector equation (e.g., r ⋅ n = d) describes the plane using position vectors and is geometrically intuitive. The Cartesian equation (e.g., ax + by + cz = d) uses x, y, z coordinates. You should use the vector form when problems involve points and directions given as vectors. The Cartesian form is often more convenient for algebraic manipulations, finding intercepts, and calculating the distance of a specific point from the plane.
6. How can you determine if a given line lies in a plane, is parallel to it, or intersects it?
To determine the relationship between a line (with direction vector 'b') and a plane (with normal vector 'n'), check the following conditions:
Intersects the plane: If the dot product b ⋅ n ≠ 0, the line is not perpendicular to the normal, so it must intersect the plane.
Parallel to the plane: If b ⋅ n = 0, the line is parallel to the plane. To confirm it doesn't lie within the plane, check if any point on the line fails to satisfy the plane's equation.
Lies in the plane: If b ⋅ n = 0 and a known point on the line also satisfies the equation of the plane, the entire line lies within it.
7. Why is the cross product used to find the normal vector to a plane?
The cross product of two vectors results in a new vector that is, by definition, perpendicular to both original vectors. When we define two non-parallel vectors that lie within a plane, their cross product generates a third vector that is perpendicular to both. Since this resulting vector is perpendicular to two different directions in the plane, it must be perpendicular (or normal) to the plane itself. This property is essential for defining a plane's orientation in 3D space.
8. What are some real-world applications of the concepts learned in the chapter 'The Plane'?
The concept of a plane in 3D geometry has numerous practical applications. For instance, in computer graphics and animation, surfaces of objects are modelled as a collection of planes (polygons). In architecture and engineering, the floors, walls, and ceilings of a building are all examples of planes. Additionally, concepts from this chapter are fundamental in fields like aerospace engineering for defining flight paths and orientations of aircraft.
