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
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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.
$\text{E: } \vec{x} = \vec{OA} + r \cdot \vec{AB} + s \cdot \vec{AC}$
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Method
- Position vector of a point as position vector of the plane
- Direction vectors: any two connecting vectors of the given points
- 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 $C(3|6|3)$.
Position vector
$\vec{OA}=\begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$Connecting vector
$\vec{AB}$ $=\begin{pmatrix} 3-2 \\ 2-1 \\ 1-1 \end{pmatrix}$ $=\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$$\vec{AC}$ $=\begin{pmatrix} 3-2 \\ 6-1 \\ 3-1 \end{pmatrix}$ $=\begin{pmatrix} 1 \\ 5 \\ 2 \end{pmatrix}$
Insert
$\text{E: } \vec{x} = \vec{OA} + r \cdot \vec{AB} + s \cdot \vec{AC}$$\text{E: } \vec{x} = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} + r \cdot \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ $+ s \cdot \begin{pmatrix} 1 \\ 5 \\ 2 \end{pmatrix}$