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
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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

  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

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

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