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.
Tips
- Sketch the points first – a quick drawing shows whether a slope should be positive or negative.
- Keep the order of the points consistent in every formula.
- Vertical lines have an undefined slope; horizontal lines have slope zero.
- Use the same units on both axes when you measure distances or angles.
Common mistakes to avoid
- Swapping x and y coordinates.
- Subtracting the coordinates in a different order in the numerator and denominator of the slope.
- Forgetting that the distance is always positive.
Formula
Direction r = n₁ × n₂, line: P₀ + λ·rWorked 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.
Frequently asked questions
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.
What formula does the Line of Intersection of Two Planes Calculator use?
Direction r = n₁ × n₂, line: P₀ + λ·r.