How to Find the Line of Intersection of Two Planes
Two planes in 3D space, each written in the general form ax + by + cz = d, intersect in a straight line as long as they are not parallel. That line is fully described by two pieces of information: a direction vector (which way the line runs) and a point that lies on the line (where it is). This calculator finds both directly from the two plane equations you enter.
Finding the direction vector with the cross product
Every plane ax + by + cz = d has a normal vector n = (a, b, c) that points perpendicular to the plane. The line where two planes meet lies in both planes, so it must be perpendicular to both normal vectors — which means its direction is the cross product of the two normals: d = n₁ × n₂ = (b₁c₂ − c₁b₂, c₁a₂ − a₁c₂, a₁b₂ − b₁a₂). If this cross product is the zero vector, the two normals are parallel, so the planes themselves are parallel (either the same plane or two planes that never meet), and there is no single intersection line.
Finding a point on the line by elimination
Two plane equations in three unknowns (x, y, z) have one degree of freedom, so infinitely many points satisfy both — all of them lying on the line. To pick one, this calculator sets whichever coordinate corresponds to a nonzero component of the direction vector to 0, then solves the remaining 2×2 system for the other two coordinates with Cramer's rule. For example, if the z-component of the direction vector is nonzero, setting z = 0 turns the two plane equations into a solvable 2×2 linear system in x and y alone. Combined with the direction vector, that point gives the full parametric line: (x, y, z) = (x₀, y₀, z₀) + t(dx, dy, dz).
When the planes don't meet in a line
If n₁ × n₂ = (0, 0, 0), the two normal vectors are parallel, so the planes are parallel too. There are only two possibilities in that case: the equations describe the exact same plane (every point that satisfies one satisfies the other — infinitely many shared points, not a single line), or they describe two distinct parallel planes that never touch at all. The calculator checks which case applies by comparing the ratio of the constants (d₁, d₂) to the ratio of the coefficients, and explains it instead of returning a line.
Checking your result
A quick way to verify the output: plug the point (x₀, y₀, z₀) back into both original plane equations — it should satisfy both. Then pick any value of t, compute the resulting (x, y, z), and confirm that point also satisfies both plane equations. The direction vector should also be perpendicular to both normal vectors, so each dot product n·d should equal 0.
Applications
Finding the intersection line of two planes comes up throughout 3D geometry and engineering: computing the edge where two flat surfaces (walls, roof panels, machined faces) meet, finding the axis of a dihedral angle, setting up constraints in CAD and robotics, and as a building block for related problems like finding the single point where three planes intersect.