# Equation of a Plane

## Table of Contents

## Scalar Equation #

The scalar equation of a plane is:

$$ ax + by + cz = d $$

Where

- \( \small a, b, c \) are the components of the normal vector to the plane, and
- \( \small \small d \) is the smallest distance from the origin to the plane.

## Vector Equation #

The vector equation of a plane is:

$$ (r - r_0 )\cdot n = 0 $$

Where

- \( \small r \) represents a point on the plane,
- \( \small r_0 \) is a specific point on the plane (the position vector of a known point),
- \( \small n \) is the normal vector to the plane.

What this equation says is that the plane is defined by all the position vectors that start on the plane that are perpendicular to the normal vector.

The vector equation for the plane can be converted to the scalar equation by expanding the vectors \( \small n = (a, b, c)\), \( r = (x, y, z) \), and \( r_0 = (x_0, y_0, z_0) \) and then taking the dot product and grouping the constant terms into \( \small d \).