Last updated October 10, 2026 · Report an error
Two non-parallel planes in 3D space always intersect in exactly one straight line (never a single point, never a curve — planes are flat, so their overlap is always linear). That line's direction is found from the cross product of the two planes' normal vectors, and any one specific point on the line comes from solving the two plane equations together as a system. This calculation shows up in 3D CAD and modeling (finding the edge where two flat faces of a solid meet), computer graphics (clipping geometry against multiple planes), and structural engineering (finding the seam line where two flat panels or roof sections join).
How it works
Enter both planes' A, B, C, D coefficients (from Ax+By+Cz+D=0). The calculator takes the cross product of the two planes' normal vectors, (A₁,B₁,C₁) × (A₂,B₂,C₂), to get the intersection line's direction vector — since that direction must be perpendicular to both normals simultaneously (lying in both planes at once), and the cross product is exactly the vector construction guaranteed to do that. To find one concrete point on the line, the calculator identifies which of the three direction-vector components has the largest magnitude, sets that corresponding coordinate to zero, and solves the two remaining plane equations as a simple two-variable system for the other two coordinates — choosing the largest component avoids a numerically unstable or unsolvable case where the eliminated variable's coefficient happens to vanish.
- Enter plane 1 — A.
- Enter plane 1 — B.
- Enter plane 1 — C.
- Enter plane 1 — D.
- Enter plane 2 — A.
- Enter plane 2 — B.
- Enter plane 2 — C.
- Enter plane 2 — D.
- Click Calculate to see your results.
Examples
x+y+z=6 and x−y=0
Normals (1,1,1) and (1,-1,0) cross to give direction (1,1,-2). Setting z=0 (largest-magnitude direction component) and solving x+y=6, x-y=0 gives x=3, y=3. The line passes through (3,3,0) with direction (1,1,-2).
2x+y−z=3 and x−y+2z=1
Normals (2,1,-1) and (1,-1,2) cross to give direction (1,-5,-3). Setting y=0 (largest-magnitude component) and solving 2x-z=3, x+2z=1 gives x=1.4, z=-0.2. The line passes through (1.4, 0, -0.2) with direction (1,-5,-3).
x=2 and y=3
Normals (1,0,0) and (0,1,0) cross to give direction (0,0,1) — a purely vertical line. Setting z=0 and solving directly gives the point (2,3,0). This makes intuitive sense: the intersection of the vertical plane x=2 and the vertical plane y=3 is the vertical line running through the point (2,3), extending in the z-direction.
Who should use it
- Finding the edge where two flat faces of a 3D solid meet.
- Multivariable calculus and linear algebra coursework on planes and lines in 3D.
- CAD and structural engineering seam/edge calculations.
Industry applications
- 3D CAD and computer graphics
- Structural and architectural engineering
- Linear algebra education
Advantages
- Finds both the direction vector and a concrete point on the line in one step.
- Automatically selects a numerically stable coordinate to eliminate.
Limitations
- Doesn't handle the degenerate case of two parallel or identical planes — those return a zero direction vector instead of a line.
Common mistakes to avoid
- Mixing up the sign convention for D — this tool expects the plane written in Ax+By+Cz+D=0 form, not Ax+By+Cz=D, which flips D's sign.
- Assuming there's only one "correct" point on the line — any point that satisfies both plane equations is valid, since a line contains infinitely many points.
- Forgetting to check for a zero direction vector, which signals the planes are parallel (or identical) and don't share a unique intersection line.
Best practices
- Double check both plane equations are in consistent Ax+By+Cz+D=0 form before entering coefficients.
- Verify your found point by substituting it back into both original plane equations.
- If the direction vector is (0,0,0), first check whether the planes are truly parallel before troubleshooting further.
Tips
- Need a single plane's equation first? Use the Equation of a Plane Calculator.