Intersection of Two Lines Calculator

Find the intersection point of two lines in 2D or 3D space with step-by-step solutions

Calculate Line Intersection

First Line

Equation: y = 0x + 0

Second Line

Equation: y = 0x + 0

Intersection Results

Infinite Intersections
Lines are coincident (same line)

Step-by-Step Solution

  1. 1.Line 1: y = 0x + 0
  2. 2.Line 2: y = 0x + 0
  3. 3.Slopes are equal: 0 = 0
  4. 4.Intercepts are equal: 0 = 0
  5. 5.Lines are the same (coincident)

Example Calculation

2D Example

Line 1: y = x + 3

Line 2: y = 2x + 1

Solution: x + 3 = 2x + 1

Solving: x = 2, y = 5

Intersection: (2, 5)

3D Example

Line 1: x = 6 + 6t, y = 8 + 7t, z = 2 + 4t

Line 2: x = 6 + 6s, y = 8 + 7s, z = 4

System: 6t - 6s = 0, 7t - 7s = 0, 4t = 2

Solution: t = s = 0.5

Intersection: (9, 11.5, 4)

Line Intersection Types

Intersecting

Lines cross at exactly one point

Parallel

Lines never meet (same slope, different intercepts)

Coincident

Same line (infinite intersection points)

Skew (3D only)

Lines don't intersect and aren't parallel

Key Formulas

2D Slope-Intercept

x₀ = (b₂ - b₁)/(a₁ - a₂)

y₀ = a₁x₀ + b₁

2D General Form

x₀ = (B₁C₂ - B₂C₁)/(A₁B₂ - A₂B₁)

y₀ = (C₁A₂ - C₂A₁)/(A₁B₂ - A₂B₁)

3D Parametric

Solve system of 3 equations in 2 unknowns (t, s)

Understanding Line Intersections

What is Line Intersection?

The intersection of two lines is the point where they cross each other. In 2D space, two lines can either intersect at one point, be parallel (no intersection), or be coincident (same line). In 3D space, lines can also be skew, meaning they don't intersect and aren't parallel.

Applications

  • Computer graphics and 3D modeling
  • Engineering and construction
  • Physics simulations
  • Navigation systems

Solution Methods

2D Lines

Set equations equal and solve for coordinates using algebraic substitution or Cramer's rule.

3D Lines

Set up system of parametric equations and solve for parameter values, then calculate intersection point.

Note: Numerical precision may affect results for nearly parallel lines. Use appropriate tolerance values for real-world applications.