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How to Find the Intersection Between Two Planes

Two intersecting planes

If you have two different planes, they will either be parallel or intersect along a line. The parametric equation for the line can be found by using a point on the line and a directional vector. You can find the directional vector ${\vec{r}}_{l}$ by taking the cross product of the normal vectors to the planes:

\begin{array}{l} α : a x + b y + c z = 0 \\ β : f x + g y + h z = 0 \end{array}

This recipe will help you find the line:

Rule

The Line of Intersection Between Two Planes

1.: Find the directional vector by taking the cross product of ${\vec{n}}_{α}$ and ${\vec{n}}_{β}$ , such that ${\vec{r}}_{l} = {\vec{n}}_{α} \times {\vec{n}}_{β}$ . If the directional vector is $(0, 0, 0)$ , that means the two planes are parallel. Then they won’t have a line of intersection, and you do not have to do any more calculations.
2.: To find a point on the line, you set one of the coordinates in the equations of both planes equal to zero and solve the system of equations you end up with. Note! If you set $z = 0$ and the line of intersection is perpendicular to the $z$ -axis, no points on the line have $z = 0$ . This applies to the $x$ - and $y$ -axes as well. If this happens, you just set one of the other variables equal to $0$ , and solve the new system of equations. If that does not work either, you try to set the last coordinate equal to $0$ .
3.: Now, use the point and the directional vector you’ve found to make a parametric equation for the line.