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Line of Intersection of Two Planes Calculator

Find the parametric and symmetric forms of the line where two planes intersect

Enter Plane Equations

Enter the coefficients for each plane in the form: ax + by + cz = d

First Plane

Plane 1: x + y = 0

Second Plane

Plane 2: z = 3

Results

Normal Vectors

n₁ = ⟨1, 1, 0⟩
n₂ = ⟨0, 0, 1⟩

Direction Vector (n₁ × n₂)

r = ⟨1.0000, -1.0000, 0.0000⟩

Common Point

P₀ = (0.0000, 0.0000, 3.0000)

Parametric Form

l: (0.0000, 0.0000, 3.0000) + λ(1.0000, -1.0000, 0.0000)
Individual equations:
x(λ) = 0.0000 + 1.0000λ
y(λ) = 0.0000 -1.0000λ
z(λ) = 3.0000 + 0.0000λ

Symmetric Form

(x - 0.0000)/1.0000 = (y - 0.0000)/-1.0000

Example: Finding Line of Intersection

Given Planes

Plane A: -2x + 3y + 4z = -1
Plane B: 2x - y - 3z = 2

Step-by-Step Solution

Step 1: Identify normal vectors: n₁ = ⟨-2, 3, 4⟩, n₂ = ⟨2, -1, -3⟩

Step 2: Calculate direction vector: r = n₁ × n₂ = ⟨-5, 2, -4⟩

Step 3: Find common point by setting x = 0: P₀ = (0, 1, -1)

Step 4: Form parametric equation: l: ⟨0, 1, -1⟩ + λ⟨-5, 2, -4⟩

Plane Relationships

I

Intersecting

Two planes meet along a line

Most common case

P

Parallel

Planes never intersect

Same orientation, different positions

C

Coincident

Planes are identical

Same equation, infinite intersection

Key Formulas

Plane Equation:
ax + by + cz = d
Normal Vector:
n = ⟨a, b, c⟩
Direction Vector:
r = n₁ × n₂
Parametric Form:
l: P₀ + λr

Tips

✓

Direction vector is perpendicular to both normal vectors

✓

If cross product is zero, planes are parallel

✓

Any point on the line satisfies both plane equations

✓

Parameter λ can be any real number

Understanding Plane Intersections

What is a Plane?

A plane in 3D geometry is a flat, two-dimensional surface that extends infinitely in all directions. It can be uniquely defined by a point and a normal vector (perpendicular to the plane).

Plane Equation

Every plane can be expressed as: ax + by + cz = d

  • •(a, b, c) forms the normal vector
  • •d is the constant term
  • •(x, y, z) represents any point on the plane

Cross Product Method

The direction vector of the intersection line is found using the cross product of the normal vectors:

  • •r = n₁ × n₂
  • •This vector is perpendicular to both planes
  • •If r = 0, planes are parallel

Finding the Intersection Line

The intersection of two planes (when they're not parallel) is always a straight line. We express this line in parametric form.

Parametric Form

The line equation: l: P₀ + λr

  • •P₀ is any point on both planes
  • •r is the direction vector
  • •λ is a parameter (any real number)

Applications

Engineering

Structural analysis, intersecting surfaces

Computer Graphics

3D modeling, collision detection

Geology

Rock layer intersections, fault lines

Related Math Calculators

Cross Product Calculator

Calculate cross product of two vectors

Line Intersection Calculator

Find intersection of two lines in 2D/3D

Point to Plane Distance

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