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)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)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)