Equation of a Plane

In $3$-dimensional space, a plane is uniquely determined by a point contained by the plane and a vector perpendicular to the plane, called the normal vector.

Suppose that we have a plane $P$ containing a point $P_0=(x_0, y_0, z_0)$ and with normal vector $\vec{v}=\left<a,b,c\right>$.

Let $P_1=(x, y, z)\in P$ be any point in our given plane. Since $\vec{v}$ is the normal vector to the plane $P$, we have: $$\vec{P_0P_1}\perp\vec{v}$$ $$\left<x-x_0, y-y_0, z-z_0\right>\perp\left<a, b, c\right>$$ Two vectors are perpendicular if and only if their dot product equals $0$. Therefore: $$\left<x-x_0, y-y_0, z-z_0\right>\cdot\left<a, b, c\right>=0$$ $$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$$ This is indeed the equation of the plane. This form of describing the plane using an equation is called the point-normal form, because it describes a plane uniquely using a point in the plane and a normal vector.