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
Step-by-Step Solution
- 1.Line 1: y = 0x + 0
- 2.Line 2: y = 0x + 0
- 3.Slopes are equal: 0 = 0
- 4.Intercepts are equal: 0 = 0
- 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.