Plane & Origin; xy xz & yz planes

Plane contains origin

The vector equation of plane $p$ in scalar-product form is given by $$ \boxed{ p : \textbf{r} \cdot \textbf{n} = 0 } $$

$\textbf{r}$ is the position vector of a point on plane $p$ (i.e. $\overrightarrow{OR}$)

$\textbf{n}$ is the normal vector of the plane

Origin not on plane & perpendicular distance from origin to plane

If the equation of plane $p$ is given by $ \textbf{r} \cdot \textbf{n} = d $, the shortest distance from the origin to plane $p$ is given by: $$ \boxed{ \text{Shortest distance from origin to plane} = { |d| \over | \textbf{n} | } } $$

The perpendicular from the origin to plane $p$ is also the shortest distance from the origin to plane $p$.

Proof: \begin{align} \text{Shortest distance from origin to plane} & = \text{Length of projection of } \overrightarrow{OA} \text{ onto } \textbf{n} \\ & = \left| \overrightarrow{OA} \cdot \hat{ \textbf{n} } \right| \\ & = \left| \textbf{a} \cdot { \textbf{n} \over | \textbf{n} | } \right| \\ & = { \left| \textbf{a} \cdot \textbf{n} \right| \over | \textbf{n} | } \\ & = { | d | \over | \textbf{n} | } \end{align}

xy, xz & yz planes

\begin{align} \text{Parametric form, } \phantom{0} \textbf{r} = \lambda \left( \begin{matrix} 1 \\ 0 \\ 0 \end{matrix} \right) & + \mu \left( \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \right), \phantom{0} \lambda, \mu \in \mathbb{R} \\ \\ \text{Scalar-product form, } \phantom{0} \textbf{r} \cdot \left( \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right) & = 0 \\ \\ \text{Cartesian equation, } z & = 0 \end{align}

Note for the parametric form, there are other possible direction vectors to use.

The xy-plane can be used to represent the horizontal plane or the ground in real-life problems.

\begin{align} \text{Parametric form, } \phantom{0} \textbf{r} = \lambda \left( \begin{matrix} 1 \\ 0 \\ 0 \end{matrix} \right) & + \mu \left( \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right), \phantom{0} \lambda, \mu \in \mathbb{R} \\ \\ \text{Scalar-product form, } \phantom{0} \textbf{r} \cdot \left( \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \right) & = 0 \\ \\ \text{Cartesian equation, } y & = 0 \end{align}

\begin{align} \text{Parametric form, } \phantom{0} \textbf{r} = \lambda \left( \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \right) & + \mu \left( \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right), \phantom{0} \lambda, \mu \in \mathbb{R} \\ \\ \text{Scalar-product form, } \phantom{0} \textbf{r} \cdot \left( \begin{matrix} 1 \\ 0 \\ 0 \end{matrix} \right) & = 0 \\ \\ \text{Cartesian equation, } x & = 0 \end{align}