Math Planes Parametric equation

# Parametric equation of a plane

Planes play a major role in analytical geometry. Similar to lines, there is also a parametric equation for planes. However, there is a position vector and two direction vectors.

$\text{E: } \vec{x} = \vec{a} + r \cdot \vec{u} + s \cdot \vec{v}$
• $\vec{x}$ is the general vector of the plane
• $\vec{a}$ is the position vector
• $\vec{u}, \vec{v}$ are the direction vectors
• $r, s$ are parameters
!

### Remember

A plane is clearly defined by three points.

## Parametric equation using 3 points

If 3 points $A$, $B$, $C$ are given, you can easily set up a parametric equation of the plane.

$\text{E: } \vec{x} = \vec{OA} + r \cdot \vec{AB} + s \cdot \vec{AC}$
i

### Method

1. Position vector of a point as position vector of the plane
2. Direction vectors: any two connecting vectors of the given points
3. Insert position vector and direction vector

### Example

Determine the parametric equation of plane $E$ using points $A(2|1|1)$, $B(3|2|1)$ and $C(3|6|3)$.

1. #### Position vector

$\vec{OA}=\begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$
2. #### Connecting vector

$\vec{AB}$ $=\begin{pmatrix} 3-2 \\ 2-1 \\ 1-1 \end{pmatrix}$ $=\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$

$\vec{AC}$ $=\begin{pmatrix} 3-2 \\ 6-1 \\ 3-1 \end{pmatrix}$ $=\begin{pmatrix} 1 \\ 5 \\ 2 \end{pmatrix}$

3. #### Insert

$\text{E: } \vec{x} = \vec{OA} + r \cdot \vec{AB} + s \cdot \vec{AC}$

$\text{E: } \vec{x} = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} + r \cdot \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ $+ s \cdot \begin{pmatrix} 1 \\ 5 \\ 2 \end{pmatrix}$