Equation of a plane

Summary

A plane in $\mathbb{R}^3$ is the set of the solutions of one linear equation in the coordinates. Let $p_0 = (x_0, y_0, z_0)$ be a point and let $n = (a, b, c)$ be a nonzero vector. The plane through $p_0$ with the normal vector $n$ is the set of the points $p$ with $n \cdot (p - p_0) = 0$. In coordinates this is the point-normal form $$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0.$$ Expanded, it is the general form $a x + b y + c z = d$ with $d = a x_0 + b y_0 + c z_0$. Conversely, for each real $d$ and each $(a, b, c) \neq (0, 0, 0)$, this equation defines a plane with the normal vector $(a, b, c)$.

Remarks

Two equations define the same plane if and only if one is a nonzero multiple of the other. For a plane through the point $p_0$ that contains the linearly independent vectors $u$ and $v$, the cross product $u \times v$ is a normal vector. Two planes are parallel or equal if and only if their normal vectors are multiples of each other.

Explanations

L1 human

The normal vector of a plane points straight out of it. A point $p$ is on the plane exactly when the arrow from $p_0$ to $p$ lies flat in the plane. Then this arrow is perpendicular to the normal vector. The dot product test turns this into one equation. For example, $z = 0$ is the floor plane with the normal vector $(0, 0, 1)$.