Wähle den Richtungsvektor der Geraden $\overrightarrow{\mathrm{u}}=\left(\begin{array}{c}-1\\ 3\\ 1\end{array}\right)\backslash overrightarrow\{\backslash mathrm\; u\}=\backslash begin\{pmatrix\}-1\backslash \backslash 3\backslash \backslash 1\backslash end\{pmatrix\}$ als ersten Richtungsvektor und den Vektor zwischen dem Punkt $\mathrm{P}\backslash mathrm\; P$  und dem Aufpunkt $\mathrm{A}\backslash mathrm\; A$  der Geraden als zweiten Richtungsvektor der Ebene $\mathrm{E}\backslash mathrm\; E$ .

Berechne  $\overrightarrow{\mathrm{P}\mathrm{A}}\backslash overrightarrow\{\backslash mathrm\{PA\}\}$  .

$\overrightarrow{\mathrm{P}\mathrm{A}}=\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{P}}=\left(\begin{array}{c}2\\ 1\\ -3\end{array}\right)-\left(\begin{array}{c}1\\ -3\\ -3\end{array}\right)=\left(\begin{array}{c}1\\ 4\\ 0\end{array}\right)\backslash overrightarrow\{\backslash mathrm\{PA\}\}=\backslash overrightarrow\{\backslash mathrm\; A\}-\backslash overrightarrow\{\backslash mathrm\; P\}=\backslash begin\{pmatrix\}2\backslash \backslash 1\backslash \backslash -3\backslash end\{pmatrix\}-\backslash begin\{pmatrix\}1\backslash \backslash -3\backslash \backslash -3\backslash end\{pmatrix\}=\backslash begin\{pmatrix\}1\backslash \backslash 4\backslash \backslash 0\backslash end\{pmatrix\}$

Wähle  $\mathrm{P}\backslash mathrm\; P$  als Aufpunkt der Ebene $\mathrm{E}\backslash mathrm\; E$  .

$\Rightarrow \mathrm{E}:\overrightarrow{\mathrm{x}}=\left(\begin{array}{c}1\\ -3\\ -3\end{array}\right)+\lambda \cdot \left(\begin{array}{c}-1\\ 3\\ 1\end{array}\right)+\mu \cdot \left(\begin{array}{c}1\\ 4\\ 0\end{array}\right)\backslash Rightarrow\backslash ;\backslash mathrm\; E:\backslash ;\backslash overrightarrow\{\backslash mathrm\; x\}=\backslash begin\{pmatrix\}1\backslash \backslash -3\backslash \backslash -3\backslash end\{pmatrix\}+\backslash mathrm\backslash lambda\backslash cdot\backslash begin\{pmatrix\}-1\backslash \backslash 3\backslash \backslash 1\backslash end\{pmatrix\}+\backslash mathrm\backslash mu\backslash cdot\backslash begin\{pmatrix\}1\backslash \backslash 4\backslash \backslash 0\backslash end\{pmatrix\}$