Point-Normal

Definition

Point-normal is a form of an equation used to uniquely determine a line in $R^2$ or a plane in $R^3$ and so forth. It is represented by the vector equation

$$\mathbf{n}\cdot \overrightarrow{P_{0}P}=0$$

Remark: Geometry

One learns in analytic geometric that a line in $R^2$ is determined uniquley by its slope and one of its points, and that a plane in $R^3$ is determined uniquely by its “inclination” and one of its points.

One way of specifying scope and inclination using this fact is to use a nonzero vector $\mathbf{n}$, called a normal, that is orthogonal to the plane line or plane in question.

For a line through point $P_0(x_0,y_0)$ that has normal $\mathbf{n}=(a,b)$,

$$a(x-x_0)+b(y-y_0)=0$$

For a plane through point $P_0(x_0,y_0,z_0)$ that has normal $\mathbf{n}=(a,b,c)$,

$$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$$