Plane

A plane can be represented in 2 forms: scalar form and normal form.

Scalar Equation of Plane in ${\mathbb{R}}^3$

$\pi :ax+by+cz=d$ The normal vector $\vec{n}=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$

Vector Form of the Plane

$\pi :\vec{x}=\left[\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right]=\overrightarrow{OP}+s\vec{u}+t\vec{v}s,t\in \mathbb{R}$ where $\vec{u},\vec{v}$ are non-zero non-parallel vectors. Intuitively, there is a unique plane that goes across two lines that are not parallel.

The normal vector $\vec{n}=\vec{u}\times \vec{v}$