Equation of a Plane Calculator
Find the standard equation of a 3D plane Ax + By + Cz + D = 0 passing through three points or through a point with a given normal vector.
Enter Calculation Inputs
Fill in the fields below. Instant client-side computation runs automatically in your browser as you type.
Common Searches & Natural Queries
What is the Plane Equation?
In three-dimensional Euclidean space, any three non-collinear points uniquely define a flat two-dimensional plane. By forming two displacement vectors on the plane and computing their cross product, we derive the normal vector (A, B, C) perpendicular to the surface.
Mathematical Formula & Methodology
The mathematical model underlying this tool operates on the following verified equation:
Normal Vector N = (P₂ - P₁) × (P₃ - P₁) = (A, B, C); Ax + By + Cz + D = 0, where D = -(Ax₁ + By₁ + Cz₁)
Understanding the Parameters:
- Point 1: x₁ — Primary input variable.
- Point 1: y₁ — Primary input variable.
- Point 1: z₁ — Primary input variable.
- Point 2: x₂ — Primary input variable.
- Point 2: y₂ — Primary input variable.
- Point 2: z₂ — Primary input variable.
- Point 3: x₃ — Primary input variable.
- Point 3: y₃ — Primary input variable.
- Point 3: z₃ — Primary input variable.
How to Use This Calculator
Input the 3D coordinates (x, y, z) for three distinct non-collinear points. Click Calculate to generate the normal vector components and the Cartesian plane equation.
Worked Numerical Example
Given P₁(1, 2, -1), P₂(2, 3, 1), P₃(3, -1, 2): Vector v = P₂ - P₁ = (1, 1, 2). Vector w = P₃ - P₁ = (2, -3, 3). Cross product N = v × w = ((1)(3) - (2)(-3), (2)(2) - (1)(3), (1)(-3) - (1)(2)) = (9, 1, -5). Thus A = 9, B = 1, C = -5. D = -(9(1) + 1(2) - 5(-1)) = -(9 + 2 + 5) = -16. General equation: 9x + y - 5z - 16 = 0.
Key Considerations & Standards
If the three points are collinear (lie on a single straight line), the cross product equals (0, 0, 0) and no unique plane can be determined.
Frequently Asked Questions
What if the points chosen are on the same line?
If points are collinear, their cross product is zero. Infinitely many planes pass through a single line; non-collinear points are required for a unique plane.
What is the significance of coefficients A, B, and C?
Vector [A, B, C] is the orthogonal (normal) vector pointing straight perpendicular out of the plane surface.