Wähle den Vektor zwischen den Aufpunkten  ${\mathrm{A}}_{\mathrm{g}}\{\backslash mathrm\; A\}\_\backslash mathrm\; g$  und  ${\mathrm{A}}_{\mathrm{h}}\{\backslash mathrm\; A\}\_\backslash mathrm\; h$  der

beiden Geraden als ersten Richtungsvektor der Ebene.

Berechne  $\overrightarrow{{\mathrm{A}}_{\mathrm{g}}{\mathrm{A}}_{\mathrm{h}}}\backslash overrightarrow\{\{\backslash mathrm\; A\}\_\backslash mathrm\; g\{\backslash mathrm\; A\}\_\backslash mathrm\; h\}$  .

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

Wähle einen der Aufpunkte der beiden Geraden, z.b.  ${\mathrm{A}}_{\mathrm{g}}\{\backslash mathrm\; A\}\_\backslash mathrm\; g$  ,

als Aufpunkt der Ebene.

Wähle einen der Richtungsvektoren der beiden Geraden

z.B.  $\overrightarrow{{\mathrm{v}}_{\mathrm{g}}}\backslash overrightarrow\{\{\backslash mathrm\; v\}\_\backslash mathrm\; g\}$  , als zweiten Richtungsvektor der Ebene.

Stelle nun die Ebenengleichung auf.

$\mathrm{E}:\overrightarrow{\mathrm{x}}=\left(\begin{array}{c}2\\ -1\\ 1\end{array}\right)+\lambda \cdot \left(\begin{array}{c}-1\\ 4\\ -2\end{array}\right)+\mu \cdot \left(\begin{array}{c}1\\ -2\\ 1\end{array}\right)\backslash mathrm\; E:\backslash ;\backslash overrightarrow\{\backslash mathrm\; x\}=\backslash begin\{pmatrix\}2\backslash \backslash -1\backslash \backslash 1\backslash end\{pmatrix\}+\backslash mathrm\backslash lambda\backslash cdot\backslash begin\{pmatrix\}-1\backslash \backslash 4\backslash \backslash -2\backslash end\{pmatrix\}+\backslash mathrm\backslash mu\backslash cdot\backslash begin\{pmatrix\}1\backslash \backslash -2\backslash \backslash 1\backslash end\{pmatrix\}$