Line of Intersection of Two Planes Calculator

Two nonparallel planes in three-dimensional space meet along a straight line. This calculator accepts each plane in the standard form ax + by + cz + d = 0 and returns a point on their shared line, a direction vector, and a parametric vector equation. The four coefficients describe the orientation and offset of each plane. Multiplying every coefficient in one equation by the same nonzero number does not change that plane. The direction of the intersection line comes from the cross product of the two plane normals. A plane written as ax + by + cz + d = 0 has normal vector (a, b, c). A vector perpendicular to both normals must lie parallel to both planes, so n1 cross n2 gives a valid line direction. The calculator then fixes one coordinate and solves the remaining two simultaneous equations to find a convenient point on the line. Any nonzero scalar multiple of the displayed direction vector represents the same geometric line, and a different point may also describe the same answer. If the cross product is the zero vector, the normals are parallel. In that case the planes are either distinct and never meet, or coincident and share infinitely many lines. Neither case has one unique line of intersection, so the calculator reports that distinction instead of inventing a result. Very small values are treated carefully to avoid displaying negative zero caused by floating-point arithmetic. Intersection lines appear throughout analytic geometry, computer graphics, CAD, structural modeling, and multivariable calculus. You can verify a result by substituting the displayed point into both original equations. You can also add any multiple of the direction vector to that point; every resulting coordinate triple should satisfy both plane equations. The parametric form is especially useful for finding points, testing intersections with other objects, or converting the line into symmetric form when all direction components are nonzero.