Plane inputs
Formula used
A plane needs one point and two independent directions. Their cross product creates a normal vector. The normal controls every Cartesian equation.
How to use
Select the input method matching your known information. Enter coordinates carefully and include negative signs. Then add optional comparison values.
Choose precision and parameter names before calculating. Review each equivalent equation in the results. Use the graph to inspect orientation.
Copy the merged solution or download another format. Saved calculations remain inside local browser storage. Clear them through browser settings.
Example data
| Method | Inputs | Normal vector | Cartesian result |
|---|---|---|---|
| Point and normal | P(1,2,3), n(2,−1,4) | ⟨2,−1,4⟩ | 2x − y + 4z = 12 |
| Three points | P₁(1,0,2), P₂(3,1,0), P₃(0,2,1) | Cross product result | Calculated automatically |
| Scalar | 2x − y + 3z = 7 | ⟨2,−1,3⟩ | Already supplied |
Frequently asked questions
What defines a unique plane?
Three non-collinear points define one plane. A point plus two independent directions also works.
Why can direction vectors fail?
Parallel directions have a zero cross product. They cannot span a plane.
Can the normal vector be reversed?
Yes. Multiplying every plane coefficient by minus one gives the same plane.
How is the Cartesian equation found?
Use the normal components as coefficients. Then calculate D using the known point.
How does point verification work?
Substitute the point into the scalar equation. Equal sides confirm plane membership.
What is the shortest point distance?
It is measured along the normal direction. The calculator also returns the closest point.
When are two planes parallel?
Their normal vectors are scalar multiples. Coincident planes also satisfy both constants.
When are two planes perpendicular?
The dot product of their normals equals zero. Numerical tolerance handles decimal rounding.
Can this calculator find intersection lines?
Yes. Non-parallel planes intersect in one line. Its direction is the normals’ cross product.