Intersection of Two Planes Calculator

Find the line of intersection of two planes given their equations in the form ax + by + cz = d.

Enter Plane Equations

Plane 1: a1x + b1y + c1z = d1
Plane 2: a2x + b2y + c2z = d2

Result

Parametric Equations
--
Direction Vector --
Point on Line --
Plane 1 Normal --
Plane 2 Normal --

Step-by-Step Solution

Direction = n1 x n2 (cross product of normal vectors)

Understanding the Intersection of Two Planes

In three-dimensional geometry, two planes can either be parallel (never intersecting), identical (coincident), or intersecting along a line. When two non-parallel planes meet, they form a straight line of intersection. Finding this line is a common problem in linear algebra, computer graphics, and engineering.

Methods for Finding the Intersection

Cross Product Method

The direction vector of the intersection line is the cross product of the two plane normals.

d = n1 x n2

Parametric Equations

Express x, y, z in terms of a parameter t using a point and the direction vector.

r(t) = P + t * d

Symmetric Form

Eliminate the parameter to express the line in symmetric (Cartesian) form.

(x-x0)/a = (y-y0)/b = (z-z0)/c

Step-by-Step Process

  1. Identify the normal vectors: For plane ax + by + cz = d, the normal vector is n = (a, b, c).
  2. Compute the cross product: d = n1 x n2. If d = (0, 0, 0), the planes are parallel.
  3. Find a point on the line: Set one variable to zero and solve the resulting 2x2 system for the other two.
  4. Write the parametric equations: Use the point and direction vector to express the line.

Special Cases

  • Parallel planes: The normal vectors are scalar multiples of each other. The cross product is zero, and no intersection line exists.
  • Coincident planes: The planes are identical (same equation, possibly scaled). Every point on one plane is on the other.
  • Perpendicular planes: The angle between the normals is 90 degrees. The dot product of the normals is zero.

Applications

Finding plane intersections is essential in computer-aided design (CAD), computer graphics (ray tracing, clipping), robotics (workspace analysis), geology (fault plane intersections), and structural engineering (load analysis at joints). The concept extends to higher dimensions in linear programming and optimization.