How to Find a Plane Parallel to a Line and Passing Through the Intersection of Two Planes

Understanding how to find a plane that is parallel to a specific line and passes through the intersection of two given planes is a fundamental concept in three-dimensional geometry. This guide will walk you through the step-by-step process, ensuring you have a clear understanding and can carry out this task accurately. The process involves several mathematical operations, including solving systems of equations, finding direction vectors, and using the cross product.

Step 1: Finding the Line of Intersection of Two Planes

To begin, we need to find the line of intersection for the two given planes. Let's denote the planes as follows:

Plane 1: (a_1x b_1y c_1z d_1) Plane 2: (a_2x b_2y c_2z d_2)

The goal is to find a parametric representation of the line of intersection. To do this, we typically express one variable in terms of a parameter, often denoted as (t), and substitute it back into the equations. This process involves solving the system of equations simultaneously.

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

The direction vector (mathbf{d}) of this line can be found using the cross product of the normal vectors of the two planes. The normal vectors are given by:

(mathbf{n_1} begin{pmatrix} a_1 b_1 c_1 end{pmatrix}) (mathbf{n_2} begin{pmatrix} a_2 b_2 c_2 end{pmatrix})

To find (mathbf{d}), we compute the cross product:

( mathbf{d} mathbf{n_1} times mathbf{n_2} )

Once we have the direction vector (mathbf{d}), we can use it in further calculations.

Step 3: Finding the Equation of the Desired Plane

Now, let's consider a line with a direction vector (mathbf{v} begin{pmatrix} v_x v_y v_z end{pmatrix}). For the plane to be parallel to this line, its normal vector (mathbf{n}) must be perpendicular to (mathbf{v}). We can achieve this by taking the cross product of (mathbf{d}) and (mathbf{v}):

( mathbf{n} mathbf{d} times mathbf{v} )

The normal vector (mathbf{n}) will give us a vector that is perpendicular to (mathbf{v}) and thus defines the plane's orientation.

Step 4: Using a Point on the Line of Intersection

From Step 1, we have a point ((x_0, y_0, z_0)) that lies on the line of intersection. This point will be crucial for defining the plane equation.

Step 5: Writing the Equation of the Plane

The equation of a plane can be expressed as:

( n_x(x - x_0) n_y(y - y_0) n_z(z - z_0) 0 )

Where (mathbf{n} begin{pmatrix} n_x n_y n_z end{pmatrix}) are the components of the normal vector (mathbf{n}).

Summary

To summarize, the process involves several key steps:

Find the normal vectors of the given planes. Calculate the direction vector of their line of intersection using the cross product of the normals. Define the normal vector of the desired plane using the cross product of the line of intersections direction vector and the direction vector of the line you want to be parallel. Use a point from the line of intersection to write the equation of the plane.

By following these steps, you can successfully determine the equation of a plane that is parallel to a line and passes through the line of intersection of two planes. This method ensures a systematic and rigorous approach to solving this geometric problem.