Line of Intersection of Two Planes

Calculate the vector and parametric equations of the line where two planes intersect.

Enter the coefficients of your two planes in the form Ax + By + Cz + D = 0.

Plane 1 (A₁x + B₁y + C₁z + D₁ = 0)

Plane 2 (A₂x + B₂y + C₂z + D₂ = 0)

Examples

Explore some common scenarios for the intersection of two planes.

General Case

general

Two non-parallel planes intersecting at a line.

P₁: 2x + 3y + -1z + -3 = 0

P₂: 1x + -1y + 2z + -1 = 0

Simple Intersection (z-axis)

simple

The intersection of the xy-plane (z=0) and the yz-plane (x=0) is the y-axis.

P₁: 1x + 0y + 0z + 0 = 0

P₂: 0x + 0y + 1z + 0 = 0

Parallel Planes

parallel

Two parallel planes that never intersect.

P₁: 2x + -3y + 1z + -1 = 0

P₂: 4x + -6y + 2z + -5 = 0

Coincident Planes

coincident

Two equations representing the exact same plane.

P₁: 1x + 1y + 1z + -1 = 0

P₂: 3x + 3y + 3z + -3 = 0

Other Titles
Understanding the Line of Intersection of Two Planes: A Comprehensive Guide
Dive deep into the geometry and algebra of intersecting planes, from basic concepts to mathematical derivations.

What is the Line of Intersection of Two Planes?

  • Geometric Intuition
  • Defining Planes
  • The Intersection Line
In three-dimensional space, planes can interact in three distinct ways: they can be parallel, they can be the exact same plane (coincident), or they can intersect. When two distinct, non-parallel planes cross, they do so along a single straight line. This line is known as the line of intersection. This concept is a fundamental part of vector geometry and linear algebra, providing a bridge between algebraic equations and spatial visualization.
Defining Planes in 3D Space
A plane is a flat, two-dimensional surface that extends infinitely. It can be uniquely defined by a linear equation of the form Ax + By + Cz + D = 0. Here, (x, y, z) represents any point on the plane, and A, B, and C are the components of the normal vector (a vector perpendicular to the plane's surface). The constant D determines the plane's position relative to the origin.
Characterizing the Intersection Line
The line of intersection has two key properties: a direction and a position. The direction of the line is given by a direction vector, and its position is established by finding any single point that lies on the line (and therefore on both planes). The calculator determines both of these to provide a complete description of the line in both vector and parametric forms.

Step-by-Step Guide to Using the Calculator

  • Inputting Plane Equations
  • Calculating the Result
  • Interpreting the Output
Our calculator simplifies the process of finding the intersection line into a few easy steps. The primary goal is to accurately transfer the coefficients from your plane equations into the input fields.
Inputting Your Plane Equations
The calculator requires the coefficients for two planes, based on the standard form Ax + By + Cz + D = 0. For each plane, you need to provide four values: A, B, C, and D. For example, if your plane equation is 2x - y + 5z = 8, you must first rewrite it as 2x - y + 5z - 8 = 0. The coefficients would then be A=2, B=-1, C=5, and D=-8.
Calculating and Interpreting the Results
Once you've entered all eight coefficients, click the 'Calculate' button. The tool will display the results, including: a status message (intersecting, parallel, or coincident), the parametric equations for x(t), y(t), and z(t), a specific point on the line, and the line's direction vector. If the planes do not intersect, a corresponding message will be shown instead.

Mathematical Derivation and Formulas

  • Finding the Direction Vector
  • Finding a Point on the Line
  • Constructing the Line Equations
The calculation relies on fundamental principles of vector algebra. The line of intersection belongs to both planes, which imposes specific geometric constraints that we can exploit algebraically.
1. Finding the Direction Vector (v)
Let the two planes be defined by their normal vectors, N₁ = <A₁, B₁, C₁> and N₂ = <A₂, B₂, C₂>. Since the line of intersection lies in both planes, it must be perpendicular to both of these normal vectors. The cross product of two vectors yields a third vector that is perpendicular to both, so the direction vector 'v' of the line is found by computing the cross product: v = N₁ × N₂.
2. Finding a Point on the Line (P₀)
To find a specific point P₀ = (x₀, y₀, z₀) on the line, we need to solve the system of two linear equations for the planes. Since there are three variables (x, y, z) and only two equations, there are infinitely many solutions (which form the line). We can find one by setting one of the variables to a constant (e.g., z=0) and solving the remaining 2x2 system for x and y. If this system has no unique solution (which happens if the line is parallel to the xy-plane), the calculator intelligently tries setting x=0 or y=0 instead to find a point.
3. Constructing the Line Equations
With a point P₀ and a direction vector v = <l, m, n>, the line can be expressed using a vector equation r = P₀ + tv, or as a set of parametric equations: x(t) = x₀ + lt, y(t) = y₀ + mt, z(t) = z₀ + nt, where 't' is a real parameter.

Real-World Applications and Use Cases

  • Computer Graphics and Game Development
  • Engineering and Architecture
  • Physics and Robotics
The intersection of planes is not just an abstract mathematical exercise; it has numerous practical applications across various fields of science and technology.
Computer-Aided Design (CAD) and 3D Modeling
In architecture and engineering, determining the intersection of planes is crucial for designing structures. For example, calculating the line where a sloped roof meets a vertical wall, or finding the join line between two structural panels.
Collision Detection in Games and Simulations
In computer graphics and physics simulations, objects are often represented by meshes of polygons (planes). Detecting if and where two objects intersect often involves calculating the intersection lines between their constituent planes.

Common Questions and Edge Cases

  • Parallel vs. Coincident Planes
  • What if the Cross Product is Zero?
  • Interpreting Parametric Equations
Understanding the special conditions and nuances of the calculation is key to correctly interpreting the results.
What's the difference between Parallel and Coincident?
Both cases occur when the normal vectors of the planes are parallel (one is a scalar multiple of the other). If the constant terms (D₁ and D₂) also maintain this same scalar relationship, the planes are coincident (the same). If they don't, the planes are parallel and distinct, meaning they never intersect.
What does a zero direction vector mean?
The direction vector is found using the cross product of the planes' normal vectors. If this cross product results in the zero vector <0, 0, 0>, it signifies that the normal vectors are parallel. As explained above, this means the planes are either parallel or coincident, and do not have a unique line of intersection. The calculator will explicitly state which case it is.
How do I use the parametric equations?
The parametric equations provide a way to generate any point on the line. By choosing any real number for the parameter 't' and plugging it into the equations for x, y, and z, you will get the coordinates of a point on the line. For instance, t=0 gives you the point P₀, while t=1 gives you a new point in the direction of the vector v away from P₀.

Mathematical Examples

  • Parallel planes never intersect (result is null)
  • Identical planes intersect everywhere (the plane itself)
  • Intersecting planes result in a parametric line equation
  • Used in 3D graphics for collision detection between flat surfaces