Calculator Inputs
Enter both plane equations in the form ax + by + cz + d = 0. The result appears above this form after submission.
Example Data Table
This worked example matches the default values loaded into the calculator.
| Plane 1 | Plane 2 | Direction Vector | Point on Line | Parametric Form |
|---|---|---|---|---|
| x + y + z - 6 = 0 | x - y + 2z - 5 = 0 | <3, -1, -2> | (0, 7/3, 11/3) |
x = 3t y = 7/3 - t z = 11/3 - 2t |
Formula Used
Let the planes be:
Plane 2: a₂x + b₂y + c₂z + d₂ = 0
The normal vectors are:
n₂ = <a₂, b₂, c₂>
The line direction vector is the cross product of both normals:
If v = <0, 0, 0>, the planes are parallel or coincident. Otherwise, the planes intersect in a unique line.
A point on the line is found by fixing one variable and solving the remaining 2×2 system from both plane equations.
The final line forms are:
Parametric form:
x = x₀ + vₓt
y = y₀ + vᵧt
z = z₀ + v_z t
Symmetric form:
(x - x₀)/vₓ = (y - y₀)/vᵧ = (z - z₀)/v_z
How to Use This Calculator
- Enter coefficients for the first plane equation.
- Enter coefficients for the second plane equation.
- Set the parameter range for the graph.
- Choose how many plot points you want.
- Press Calculate Intersection Line.
- Read the status message above the form.
- Review the point, direction, and line forms.
- Use the CSV or PDF buttons to export results.
Frequently Asked Questions
1. What does this calculator compute?
It checks whether two planes intersect, are parallel, or coincide. When an intersection exists, it returns a point on the line, the direction vector, vector form, parametric form, symmetric form, and a graph.
2. Why is the cross product used?
Each plane has a normal vector. The intersection line must be perpendicular to both normals. The cross product of the normals gives a direction vector that satisfies that geometric requirement.
3. What happens if the planes are parallel?
If the normal vectors are parallel, their cross product becomes zero. The calculator then tests whether the planes are distinct parallel planes or the exact same plane.
4. What does coincident mean here?
Coincident planes are identical planes written in equivalent forms. Their intersection is not a single line. Instead, every point on that plane belongs to both equations.
5. Can I enter decimal coefficients?
Yes. The calculator accepts integers and decimals. It formats the final results clearly and tries to simplify integer direction vectors when possible.
6. Why might the symmetric form look incomplete?
If one direction component is zero, that coordinate stays constant along the line. In that case, the calculator shows a fixed coordinate statement beside the remaining ratio terms.
7. What does the graph represent?
The graph plots points generated from the line equation over your chosen parameter interval. It helps you inspect direction, position, and scale in three-dimensional space.
8. When should I change the parameter range?
Change it when the visible segment looks too short, too long, or poorly centered. A wider range extends the plotted line, while a narrower range zooms in.