Parameterform von Ebenen

$$\begin{array}{rlr}E& :\vec{x}=\vec{a}+\lambda \vec{u}+\mu \vec{v};& \lambda ,\mu \in \mathbb{R}\end{array}$$ $$\begin{array}{rl}\vec{a}& :Verankerungsvektor\\ \vec{u},\vec{v}& :Richtungsvektoren\end{array}$$

Beispiel: Umrechnung zur Normalenform

$$\begin{array}{rl}E& :\vec{x}=\left(\begin{array}{c}2\\ 3\\ -4\end{array}\right)+\lambda \left(\begin{array}{c}1\\ 2\\ -4\end{array}\right)+\mu \left(\begin{array}{c}-2\\ 3\\ 0\end{array}\right)|\ast \vec{n}=\left(\begin{array}{c}1\\ 2\\ -4\end{array}\right)\times \left(\begin{array}{c}-2\\ 3\\ 0\end{array}\right)=\left(\begin{array}{c}12\\ 8\\ 7\end{array}\right)\\ & ⟺\left(\begin{array}{c}12\\ 8\\ 7\end{array}\right)\ast \vec{x}=\left(\begin{array}{c}12\\ 8\\ 7\end{array}\right)\left[\left(\begin{array}{c}2\\ 3\\ -4\end{array}\right)+\lambda \left(\begin{array}{c}1\\ 2\\ -4\end{array}\right)+\mu \left(\begin{array}{c}-2\\ 3\\ 0\end{array}\right)\right]\\ & ⟺\left(\begin{array}{c}12\\ 8\\ 7\end{array}\right)\ast \vec{x}=20\end{array}$$