# Line of Intersection of Two Planes Calculator

URL: https://speedycalculator.com/calculators/math/line-of-intersection-of-two-planes/
Last updated: 2026-10-30

> The line of intersection calculator finds the line where two planes a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂ meet, as a point plus a direction vector, or reports parallel or coincident planes.

## Inputs

- Plane 1: a₁
- b₁
- c₁
- d₁
- Plane 2: a₂
- b₂
- c₂
- d₂

## Results

- Line of intersection
- Direction vector r
- Point P₀ on the line

## Formula

```
Direction r = n₁ × n₂,  line: P₀ + λ·r
```

## Explanation

Two non-parallel planes always meet in a line – think of the crease where a wall meets the floor. Finding it is a standard step in 3D geometry, CAD and computer graphics.

### How to use the Line of Intersection of Two Planes Calculator

- **Plane 1: a₁**
- **b₁**
- **c₁**
- **d₁**
- **Plane 2: a₂**
- **b₂**
- **c₂**
- **d₂**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The direction is the cross product of the two normal vectors. A point on the line is found by setting the coordinate with the largest direction component to zero and solving the remaining 2 × 2 system.

## Worked example

Inputs — Plane 1: a₁: 1; b₁: 1; c₁: 1; d₁: 1; Plane 2: a₂: 1; b₂: -1; c₂: 2; d₂: 3.

Results — **Line of intersection = (x, y, z) = (0, −0.33333333, 1.3333333) + t·(3, −1, −2)**; Direction vector r = (3, −1, −2); Point P₀ on the line = (0, −0.33333333, 1.3333333).

## Key terms

- **Parametric equation**: x = x₀ + λr_x, y = y₀ + λr_y, z = z₀ + λr_z.

## FAQ

**When do two planes not intersect?**
When they are parallel and distinct (normal vectors parallel, different d).

**Is the answer unique?**
The line is; its parametric description is not – any point and any multiple of r work.

**How do I check the result?**
Substitute the point and the point plus r into both plane equations.

**Can I find the angle between the planes?**
Yes, from the angle between their normals.


## Sources

- [OpenStax – Calculus Volume 3](https://openstax.org/details/books/calculus-volume-3)
- [Wolfram MathWorld – Plane-plane intersection](https://mathworld.wolfram.com/Plane-PlaneIntersection.html)