$Equation of a Plane$

Equation of a Plane

Find the equation of a plane through three points, from a point and a normal vector or two direction vectors: general and intercept forms, the normal vector.

What is known:

Three points of the plane

Point and normal vector

Point and two direction vectors

General equation Ax + By + Cz + D = 0

Space: in the plane (x, y) in space (x, y, z)

Coordinates are whole numbers, decimals or fractions: 3, −1.5, 2/3.

Test yourself

The plane equation calculator writes the general equation of a plane Ax + By + Cz + D = 0 through three points, from a point and a normal vector, or from a point and two direction vectors, and from the general equation it finds the normal vector, the intercept form and the distance from the origin.

Equations of a plane

What is known	Equation
Point M₀ and normal (A, B, C)	A(x − x₀) + B(y − y₀) + C(z − z₀) = 0
Three points A, B, C	normal n = AB × AC, then as above
Point and two vectors a, b	normal n = a × b
General equation	Ax + By + Cz + D = 0
Intercept form	x/a + y/b + z/c = 1
Distance from the origin	d = \|D\| / √(A² + B² + C²)

Example. The plane through A(1, 0, 0), B(0, 2, 0), C(0, 0, 3).
AB = (−1, 2, 0), AC = (−1, 0, 3), n = AB × AC = (6, 3, 2)
6(x − 1) + 3y + 2z = 0 ⇒ 6x + 3y + 2z − 6 = 0, intercept form: x/1 + y/2 + z/3 = 1.

If the three points lie on one line, the cross product is zero and infinitely many planes pass through them — the calculator will tell you.