Lines of Intersection Between Two Planes

Sometimes we want to calculate the line at which two planes intersect each other. We can accomplish this with a system of equations to determine where these two planes intersect. Note that this will result in a system with parameters from which we can determine parametric equations from.

# Lines of Intersection Between Planes

Let's hypothetically say that we want to find the equation of the line of intersection between the following lines $L_1$ and $L_2$:

(1)
\begin{align} L_1: 2x - y - 4z + 2 = 0 \\ L_2: -3x + 2y - z + 2 = 0 \end{align}

We will begin by first setting up a system of linear equations. Note that we have more variables (3) than the number of equations (2), so there will be a column of zeroes after we convert the matrix of lines $L_1$ and $L_2$ into reduced row echelon form. The following matrix represents our two lines: $\begin{bmatrix}2 & -1 & -4 & -2 \\ -3& 2 & -1 & -2 \end{bmatrix}$.

We will thus convert this matrix intro reduced row echelon form by Gauss-Jordan Elimination:

(2)
\begin{align} \frac{1}{2} R_1 \to R_1 \\ \begin{bmatrix} 1 & -\frac{1}{2} & -2 & -1 \\ -3& 2 & -1 & -2 \end{bmatrix} \end{align}
(3)
\begin{align} -\frac{1}{3} R_2 \to R_2 \\ \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{6}{3} & -\frac{3}{3} \\ 1& -\frac{2}{3} & \frac{1}{3} & \frac{2}{3} \end{bmatrix} \end{align}
(4)
\begin{align} R_2 - R_1 \to R_2 \\ \begin{bmatrix} 1 & \frac{-1}{2} & -\frac{6}{3} & -\frac{3}{3} \\ 0 & -\frac{1}{6} & \frac{7}{3} & \frac{5}{3} \end{bmatrix} \end{align}
(5)
\begin{align} -6R_2 \to R_2 \\ \begin{bmatrix} 1 & \frac{-1}{2} & -\frac{6}{3} & -\frac{3}{3} \\ 0 & 1 & -14 & -10 \end{bmatrix} \end{align}
(6)
\begin{align} R_1 + \frac{1}{2} R_2 \to R_1 \\ \begin{bmatrix} 1 & 0 & -9 & -6 \\ 0 & 1 & -14 & -10 \end{bmatrix} \end{align}

We now have the system in reduced row echelon form. We can see that we have a free parameter for $z$, so let's parameterize this variable. Let $z = t$ for $(-\infty < t < \infty)$. Therefore, we can determine the equation of the line as a set of parameterized equations:

(7)