Question

Describe a method for finding the line of intersection of two planes.

Short answer

The line of intersection is found by first, understanding the plane equations. Second, solve these equations simultaneously to express two variables in terms of a parameter. Third, find the direction vector which is the cross product of the normals of the two planes. Fourth, find a point on the line. Lastly, write the parametric equations using the direction vector and the point on the line.

Step-by-step solution

Understand the Plane Equations

The standard form of a plane equation is \(Ax + By + Cz = D\), where A, B, C are the coefficients of x, y, z and they are the direction numbers that are perpendicular to the plane, and D is a constant. Suppose we have two plane equations, Plane 1: \(A1x + B1y + C1z = D1\) and Plane 2: \(A2x + B2y + C2z = D2\).

Solve Simultaneously

To find the line of intersection, we first need to find an equivalent system of equations that have two of the variables (suppose y and z) in terms of a parameter (usually denoted as t). This can be done by subtracting the equations (if necessary) and factoring out the common variable.

Find the Direction Vector

The direction vector of the line of intersection is the cross product of the normal vectors of the two planes. For Plane 1, the normal vector is \(N1 = (A1, B1, C1)\) and for Plane 2, the normal vector is \(N2 = (A2, B2, C2)\). The direction vector \(D = N1 × N2\).

Find a Point on the Line

We need a point on our line of intersection. Generally, this can be done by setting a variable (like z) equal to zero and then solving the two plane equations for x and y. That will give us a point \(P = (x0, y0, z0)\) that lies on our line of intersection.

Write the Parametric Equations

With the direction vector from Step 3, and a point on the line from Step 4, we can now write the parametric form of the line of intersection. The parametric form of this line will be \(x = x0 + Dt\), \(y = y0 + Dt\), \(z = z0 + Dt\) where \(D = (Dx, Dy, Dz)\) is the direction vector.