How do you find the equation of a plane through three points?

Build two vectors from one point to the other two points, take their cross product to get a normal vector, then substitute one point to solve for D.

What is the normal vector of a plane?

It is a vector perpendicular to the plane. In Ax + By + Cz +, the normal vector is (A, B, C).

Can three collinear points define a plane?

No. Infinitely many planes pass through a single line, so the calculator rejects collinear or nearly collinear points.

Why can one plane have multiple equations?

Multiplying every coefficient by the same nonzero number gives an equivalent equation for the same plane.

Plane Equation Calculator

Find the equation of a plane in 3D from three points, a point and normal vector, or a point with two direction vectors, with normal vector and intercepts.

Plane definition

Point M1 (on the plane)

Point M2

Point M3

Define the plane by three points, by a point and a normal vector, or by a point and two in-plane direction vectors.

Plane equation from points or a normal vector

English search results for plane equation calculators focus on turning 3D coordinate data into the standard plane equation Ax + By + Cz +. The most common modes are three non-collinear points, a point and normal vector, or a point with two non-parallel directions in the plane.

A x + B y + C z + D = 0

A, B, and C are the plane normal components; D shifts the plane through the given point.

n = (P_{2} - P_{1}) \times (P_{3} - P_{1})

For three points, the cross product of two in-plane vectors gives the normal vector.

What the calculator returns

The standard Cartesian plane equation.
The normal vector and its length.
A simplified integer coefficient form when possible.
Axis-intercept status for x, y, and z axes.
Validation for collinear points, zero normals, and parallel direction vectors.

Three points define a unique plane only when they are not collinear.

Examples

Input	Plane equation
(1, 0, 0), (0, 1, 0), (0, 0, 1)	x + y + z -
point (0, 0, 2), normal (0, 0, 1)	z -
point (0, 0, 0), vectors (1, 0, 0) and (0, 1, 0)

Frequently Asked Questions

Sources and References

Calculations are based on the listed reference sources. Links open in a new tab.

Updated: June 2, 2026

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