Understanding the Intersection of Two Planes

When two planes intersect, they do so along a line. This intersection line can be mathematically described using a set of equations. This article guides you through the process of finding the equation of the line where two planes intersect, providing a step-by-step methodology that is both accurate and easy to follow. Whether you're a student, a mathematician, or a professional in a related field, this guide will help you understand and apply the necessary mathematical concepts.

Identifying the Equations of the Planes

The first step in determining the line of intersection is to identify the equations of the two planes. The general form of the equation of a plane is:

a_1x b_1y c_1z d_1

and

a_2x b_2y c_2z d_2

Here, a_1, b_1, c_1 and a_2, b_2, c_2 are the coefficients of the variables x, y, z, and d_1, d_2 are constants. These equations represent the planes in the three-dimensional space.

Deriving the Normal Vectors of the Planes

The normal vector to a plane is a vector that is perpendicular to the plane. For the two planes identified earlier, the normal vectors can be derived using the coefficients of the variables in the plane equations:

mathbf{n_1} a_1, b_1, c_1

and

mathbf{n_2} a_2, b_2, c_2

The direction of the line of intersection is the same as the cross product of these two normal vectors. This vector represents the direction of the line of intersection and is crucial for further calculations.

Calculating the Direction Vector of the Line of Intersection

To find the direction vector of the line of intersection, you can calculate the cross product of the normal vectors of the two planes:

mathbf{d} mathbf{n_1} times mathbf{n_2}

The cross product of two vectors in three-dimensional space can be represented using a determinant:

mathbf{d} begin{vmatrix} mathbf{i} mathbf{j} mathbf{k} a_1 b_1 c_1 a_2 b_2 c_2 end{vmatrix}

This determinant gives us the components of the direction vector mathbf{d} d_x, d_y, d_z.

Identifying a Point on the Line of Intersection

Once you have the direction vector of the line of intersection, the next step is to identify a point on the line. This can be done by solving the system of equations formed by the two plane equations simultaneously. This can be achieved by setting one variable to a specific value (often zero) or substituting one variable in terms of the others.

Writing the Parametric Equations of the Line

With both a point on the line of intersection and the direction vector, you can now write the parametric equations of the line. If (x_0, y_0, z_0) is a point on the line and the direction vector is mathbf{d} d_x, d_y, d_z, the parametric equations of the line can be expressed as:

x x_0 t cdot d_x

y y_0 t cdot d_y

z z_0 t cdot d_z

where t is a parameter that varies over the real numbers, allowing you to find any point on the line.

Example: Finding the Line of Intersection for Two Planes

Let's consider the following example:

Plane 1: x 2y 3z 6

Plane 2: 2x - y z 4

Step 1: Identify the normal vectors:

Normal vector of Plane 1: mathbf{n_1} 1, 2, 3

Normal vector of Plane 2: mathbf{n_2} 2, -1, 1

Step 2: Calculate the direction vector mathbf{d} mathbf{n_1} times mathbf{n_2}:

mathbf{d} begin{vmatrix} mathbf{i} mathbf{j} mathbf{k} 1 2 3 2 -1 1 end{vmatrix} 5, 5, -5

The direction vector is mathbf{d} 5, 5, -5.

Step 3: Find a point on the line. Set z 0 and solve the system of equations:

x 2y 6

2x - y 4

Solving these equations gives you (x, y) (2, 2). So, the point is (2, 2, 0).

Step 4: Write the parametric equations of the line:

x 2 5t

y 2 5t

z -5t

The parametric equations of the line of intersection are now fully defined. This process will give you the equation of the line where the two planes intersect.

Conclusion

This comprehensive guide has outlined the steps to find the equation of the line where two planes intersect. By identifying the equations of the planes, calculating the normal vectors, determining the direction vector, and finding a point on the line, you can write the parametric equations of the line of intersection. This method is valuable for both educational and professional purposes and can be applied to a wide range of mathematical and physical problems involving the intersection of planes. Whether you need to solve a problem or simply understand the concept, this guide provides a clear and detailed explanation.