Line of Intersection of Two Planes Calculator

A plane in geometry is a flat, two-dimensional surface extending infinitely in 3D space. It has zero curvature and is uniquely defined by a point on the plane and a normal vector perpendicular to it. Algebraically, a plane is expressed as ax + by + cz = d, where (a, b, c) is the normal vector and d is a constant.

No — two planes in 3D space cannot intersect at a single point. Their intersection is either a line (when the planes are non-parallel and not coincident), no intersection (when the planes are parallel but distinct), or the entire plane itself (when both planes are identical/coincident).

The direction vector of the intersection line is found by taking the cross product of the two planes' normal vectors: d = n₁ × n₂. Then, find a specific point on the line by solving the two plane equations simultaneously (setting one variable to zero to simplify). Together, the point and direction vector fully define the intersection line.

For x + y = 0 (normal n₁ = ⟨1,1,0⟩) and z = 3 (normal n₂ = ⟨0,0,1⟩), the direction vector is n₁ × n₂ = ⟨1,−1,0⟩. A point on the line can be found by setting x = 0: y = 0, z = 3, giving P = (0, 0, 3). The parametric equations are x = t, y = −t, z = 3.

Once you have a point P = (px, py, pz) on the line and the direction vector d = ⟨dx, dy, dz⟩, the parametric equations are: x = px + dx·t, y = py + dy·t, z = pz + dz·t, where t is a free real-number parameter. Different values of t trace out all points along the intersection line.

A zero direction vector (0, 0, 0) means the cross product of the two normal vectors is zero, which means the planes are parallel (their normals point in the same direction). Parallel planes either never intersect (if d₁ ≠ d₂ proportionally) or are the same plane (if all coefficients are proportional including d).

The symmetric form expresses the line as (x − px)/dx = (y − py)/dy = (z − pz)/dz, where (px, py, pz) is a point on the line and (dx, dy, dz) is the direction vector. This form is useful when none of the direction vector components are zero; if a component is zero, that variable is held constant in the symmetric representation.

The intersection line lies within both planes simultaneously, meaning it must be perpendicular to both normal vectors. The cross product n₁ × n₂ naturally produces a vector perpendicular to both normals, which is exactly why it gives the direction of the intersection line. This is the geometric foundation of the calculation.