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Equation of a Plane from a Normal and Point — result sheet
Build the Cartesian equation of a plane from a nonzero normal vector and one point on the plane.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: a x + b y + c z + d = 0, where d = -(a x0 + b y0 + c z0) and (a,b,c) is nonzero.
A plane is determined by a nonzero normal vector and one point on the plane. The point-normal form expands to ax + by + cz + d = 0 by solving d from the entered point.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Normal a — The x coefficient of the plane normal.; minimum -1000000; maximum 1000000
- Normal b — The y coefficient of the plane normal.; minimum -1000000; maximum 1000000
- Normal c — The z coefficient of the plane normal.; minimum -1000000; maximum 1000000
- Point x0 — The x coordinate of a point on the plane.; minimum -1000000; maximum 1000000
- Point y0 — The y coordinate of a point on the plane.; minimum -1000000; maximum 1000000
- Point z0 — The z coordinate of a point on the plane.; minimum -1000000; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Normal a | 1 |
| Normal b | 2 |
| Normal c | 3 |
| Point x0 | 1 |
| Point y0 | 1 |
| Point z0 | 1 |
d = -6 and the plane equation is x + 2y + 3z - 6 = 0.
Assumptions and limits
- The normal coefficients and point coordinates use one Cartesian 3D coordinate system.
- The normal vector (a,b,c) is nonzero; the returned equation is an algebraic plane description without unit conversion or symbolic simplification beyond its displayed form.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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