General Equation of a Plane

Every first-degree equation in x, y, z represents a plane in 3D. The normal to the plane has direction ratios (a, b, c). The vector form is r·n = −d where n = (a, b, c).

A plane in 3D is determined by a point $A(\mathbf{a})$ on it and a normal vector $\mathbf{n}$ perpendicular to it.

For any point $P(\mathbf{r})$ on the plane, $\overrightarrow{AP} = \mathbf{r}-\mathbf{a}$ is perpendicular to $\mathbf{n}$:

$$(\mathbf{r}-\mathbf{a})\cdot\mathbf{n} = 0 \implies \mathbf{r}\cdot\mathbf{n} = \mathbf{a}\cdot\mathbf{n} = d$$

Writing $\mathbf{n} = (a, b, c)$ and $\mathbf{r} = (x, y, z)$:

$$ax+by+cz = d \quad \text{or} \quad ax+by+cz+d' = 0$$

The normal direction is $(a, b, c)$: Any plane perpendicular to a given direction has its equation determined by that direction and one point.

Special planes:

How many conditions determine a plane? One plane through three non-collinear points (three conditions). One plane through a line with a given normal (one condition beyond the line). The general equation has 4 parameters $(a:b:c:d)$ — one is a scaling factor, so 3 independent parameters → 3 conditions needed.