A plane in

three-space can be defined by one

point in the plane and a

normal vector orthogonal to the plane.

** Vector Equation of a Plane:**
Given a point P_{0}(x_{0}, y_{0}, z_{0}) in the plane and a normal vector **n**, let P(x, y, z) be an arbitrary point in the plane. Let **r** and **r**_{0} be the position vectors of P and P_{0} respectively. Subtracting **r** from **r**_{0} gives us a vector inside the plane, which is orthogonal to **n**. Thus:

** n** . (**r** - **r**_{0}) = 0

*or*

**n** . **r** = **n** . **r**_{0}

** Scalar Equation of a Plane:**

Given a point P_{0}(x_{0}, y_{0}, z_{0}) in the plane and a normal vector **n** = , let P(x, y, z) be an arbitrary point in the plane. The vector equation then becomes:

a(x-x_{0}) + b(y-y_{0}) + c(z-z_{0}) = 0

**Linear Equation of a Plane:**

ax + by + cz = d

*where d = ax*_{0} + by_{0} + cz_{0}

This node made possible by *Calculus Concepts and Contexts* by James Stewart.