Equation of a New Plane After 90-Degree Rotation

Understanding the geometry of planes and how they transform under various operations is a crucial concept in advanced mathematics. In this article, we will explore the rotation of a plane through a right angle about its line of intersection with another plane. This topic is particularly relevant in the context of the JEE Mains examination, one of the most competitive competitive exams in India for students aiming to pursue undergraduate engineering programs.

Understanding the Problem

The problem at hand involves determining the equation of a new plane after rotating the plane (4x 7y 4z - 81 0) through a right angle (90 degrees) about its line of intersection with the plane (5x - 3y 10z - 25 0).

Theoretical Background

To solve this problem, we need to understand several key concepts:

Equation of a Plane: A plane in 3D space is defined by the equation (Ax By Cz D 0), where (A, B, C) are the normal vector components, and (D) is a constant. Line of Intersection: The line of intersection of two planes is the set of points that lie on both planes. Vector Algebra: Vectors are used to describe directions and points in space. We will use vectors to represent the normal vectors of the planes and the direction vectors of the line of intersection. 90-Degree Rotation: A 90-degree rotation of a vector can be achieved using a rotation matrix or by understanding the geometric properties of the rotation.

Step-by-Step Solution

Step 1: Determine the Normal Vectors

The normal vector to the plane (4x 7y 4z - 81 0) is (mathbf{n}_1 [4, 7, 4]).

The normal vector to the plane (5x - 3y 10z - 25 0) is (mathbf{n}_2 [5, -3, 10]).

Step 2: Find the Direction Vector of the Line of Intersection

The direction vector (mathbf{d}) of the line of intersection can be found by taking the cross product of the normal vectors (mathbf{n}_1) and (mathbf{n}_2).

(mathbf{d} mathbf{n}_1 times mathbf{n}_2)

(mathbf{d} begin{vmatrix} mathbf{i} mathbf{j} mathbf{k} 4 7 4 5 -3 10 end{vmatrix} (70 12)mathbf{i} - (40 - 20)mathbf{j} (-20 - 35)mathbf{k} [82, 20, -55])

Step 3: Rotate the Plane Through 90 Degrees

To rotate the plane through a 90-degree angle, we need to rotate the normal vector (mathbf{n}_1) by 90 degrees about the direction vector (mathbf{d}).

A rotation matrix for a 90-degree counterclockwise rotation about an axis defined by a unit vector (mathbf{u} [u_x, u_y, u_z]) is given by:

[mathbf{R} begin{bmatrix}cos theta u_x^2(1 - cos theta) u_x u_y (1 - cos theta) - u_z sin theta u_x u_z (1 - cos theta) u_y sin theta u_y u_x (1 - cos theta) u_z sin theta cos theta u_y^2(1 - cos theta) u_y u_z (1 - cos theta) - u_x sin theta u_z u_x (1 - cos theta) - u_y sin theta u_z u_y (1 - cos theta) u_x sin theta cos theta u_z^2(1 - cos theta) end{bmatrix}]

Given that the rotation is about the direction vector (mathbf{d} [82, 20, -55]), we first normalize (mathbf{d}) to get the unit vector (mathbf{u}).

(mathbf{u} frac{mathbf{d}}{|mathbf{d}|} frac{[82, 20, -55]}{sqrt{82^2 20^2 (-55)^2}} frac{[82, 20, -55]}{sqrt{8465}})

Now, using the rotation matrix (mathbf{R}) and applying it to (mathbf{n}_1), we get the new normal vector (mathbf{n}_1').

Step 4: Form the Equation of the New Plane

The equation of the new plane will have the form (A'x B'y C'z D' 0), where (mathbf{n}_1' [A', B', C']).

To find (D'), we need a point on the original plane. We can use the point of intersection of the original plane with the line of intersection. Let's denote the point of intersection as (mathbf{P} [x_0, y_0, z_0]). This point satisfies both plane equations.

The equation of the new plane can then be written as:

[A'(x - x_0) B'(y - y_0) C'(z - z_0) 0]

Conclusion

By carefully applying the above steps, we can determine the equation of the new plane after the rotation. This problem requires a strong understanding of vector algebra and the rotation of geometric objects in 3D space. It is a challenging but crucial topic for students preparing for exams like the JEE Mains.

For a more detailed and practical approach, students can use software tools such as MATLAB or Mathematica to perform the necessary calculations and visualize the rotations and intersections. These tools can help in understanding the geometric transformations more clearly.

Feel free to ask if you have any more questions or need further clarification!