Equation of a Plane Through 3 Points Calculator

How it works

n = PQ × PR = (a, b, c) · a(x − x₁) + b(y − y₁) + c(z − z₁) = 0 ⇔ ax + by + cz = d with d = a·x₁ + b·y₁ + c·z₁ · distance from the origin = |d| / √(a² + b² + c²)

Two vectors lying in the plane — PQ and PR — have a cross product perpendicular to both, which is the plane’s normal vector (a, b, c). Any point X of the plane satisfies n · (X − P) = 0, which expands to ax + by + cz = d. The result is scaled to the smallest whole-number coefficients. If the cross product is the zero vector the points lie on one line and infinitely many planes pass through them.

Worked example

P(1, 2, −1), Q(2, 3, 1), R(3, −1, 2): PQ = (1, 1, 2) and PR = (2, −3, 3), so n = PQ × PR = (1·3 − 2·(−3), 2·2 − 1·3, 1·(−3) − 1·2) = (9, 1, −5). Then d = 9·1 + 1·2 − 5·(−1) = 16, so the plane is 9x + y − 5z = 16. Check with R: 27 − 1 − 10 = 16.

Sources: OpenStax Calculus Volume 3 — 2.5 Equations of Lines and Planes in Space · Paul’s Online Math Notes — Equations of Planes

Frequently asked questions