Equations of Planes

In three-space, planes can be represented using vector, parametric, and scalar equations.

Vector Equation of a Plane

Planes can be defined using a point and two non-parallel direction vectors:

r = r0 + ta + sb

Example

Consider a plane passing through P(2,3,4) with direction vectors a = [1,2,3] and b = [4,5,6].

The vector equation is:

[x, y, z] = [2, 3, 4] + t[1,2,3] + s[4,5,6]

Parametric Equations

Expanding the vector equation gives:

• x = 2 + t + 4s
• y = 3 + 2t + 5s
• z = 4 + 3t + 6s

Scalar Equation of a Plane

A plane's equation can be written in the form:

Ax + By + Cz + D = 0

Example

Given three points A(1,1,1), B(2,3,4), and C(3,5,6), find the scalar equation.

Compute vectors AB and AC, take their cross product to get the normal vector, then substitute a point to find D.

The final equation is:

Ax + By + Cz + D = 0