Wähle  $\overrightarrow{\mathrm{v}}\backslash overrightarrow\{\backslash mathrm\; v\}$  als ersten Richtungsvektor und  $\overrightarrow{\mathrm{A}\mathrm{B}}\backslash overrightarrow\{\backslash mathrm\{AB\}\}$  als zweiten Richtungsvektor der Ebene.

Berechne $\overrightarrow{\mathrm{A}\mathrm{B}}\backslash overrightarrow\{\backslash mathrm\{AB\}\}$ .

$\overrightarrow{\mathrm{A}\mathrm{B}}=\overrightarrow{\mathrm{B}}-\overrightarrow{\mathrm{A}}=\left(\begin{array}{c}3\\ 7\\ 5\end{array}\right)-\left(\begin{array}{c}1\\ 3\\ -2\end{array}\right)=\left(\begin{array}{c}2\\ 4\\ 7\end{array}\right)\backslash overrightarrow\{\backslash mathrm\{AB\}\}=\backslash overrightarrow\{\backslash mathrm\; B\}-\backslash overrightarrow\{\backslash mathrm\; A\}=\backslash begin\{pmatrix\}3\backslash \backslash 7\backslash \backslash 5\backslash end\{pmatrix\}-\backslash begin\{pmatrix\}1\backslash \backslash 3\backslash \backslash -2\backslash end\{pmatrix\}=\backslash begin\{pmatrix\}2\backslash \backslash 4\backslash \backslash 7\backslash end\{pmatrix\}$

Wähle  $\overrightarrow{\mathrm{O}\mathrm{A}}\backslash overrightarrow\{\backslash mathrm\{OA\}\}$  als Aufpunkt der Ebene und setze in die Ebenengleichung  $\mathrm{E}:\overrightarrow{\mathrm{x}}=\overrightarrow{\mathrm{O}\mathrm{A}}+\lambda \cdot \overrightarrow{\mathrm{v}}+\mu \cdot \overrightarrow{\mathrm{A}\mathrm{B}}\backslash mathrm\; E:\backslash ;\backslash overrightarrow\{\backslash mathrm\; x\}=\backslash overrightarrow\{\backslash mathrm\{OA\}\}+\backslash mathrm\backslash lambda\backslash cdot\backslash overrightarrow\{\backslash mathrm\; v\}+\backslash mathrm\backslash mu\backslash cdot\backslash overrightarrow\{\backslash mathrm\{AB\}\}$  ein.

$\mathrm{E}:\overrightarrow{\mathrm{x}}=\left(\begin{array}{c}1\\ 3\\ -2\end{array}\right)+\lambda \cdot \left(\begin{array}{c}-2\\ 10\\ -7\end{array}\right)+\mu \cdot \left(\begin{array}{c}2\\ 4\\ 7\end{array}\right)\backslash mathrm\; E:\backslash ;\backslash overrightarrow\{\backslash mathrm\; x\}=\backslash begin\{pmatrix\}1\backslash \backslash 3\backslash \backslash -2\backslash end\{pmatrix\}+\backslash mathrm\backslash lambda\backslash cdot\backslash begin\{pmatrix\}-2\backslash \backslash 10\backslash \backslash -7\backslash end\{pmatrix\}+\backslash mathrm\backslash mu\backslash cdot\backslash begin\{pmatrix\}2\backslash \backslash 4\backslash \backslash 7\backslash end\{pmatrix\}$

Wähle  $\overrightarrow{\mathrm{v}}\backslash overrightarrow\{\backslash mathrm\; v\}$  als ersten Richtungsvektor und  $\overrightarrow{\mathrm{A}\mathrm{B}}\backslash overrightarrow\{\backslash mathrm\{AB\}\}$  als zweiten Richtungsvektor der Ebene.

Berechne $\overrightarrow{\mathrm{A}\mathrm{B}}\backslash overrightarrow\{\backslash mathrm\{AB\}\}$ .

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

Wähle  $\overrightarrow{\mathrm{O}\mathrm{A}}\backslash overrightarrow\{\backslash mathrm\{OA\}\}$  als Aufpunkt der Ebene und setze in die Ebenengleichung $\mathrm{E}:\overrightarrow{\mathrm{x}}=\overrightarrow{\mathrm{O}\mathrm{A}}+\lambda \cdot \overrightarrow{\mathrm{v}}+\mu \cdot \overrightarrow{\mathrm{A}\mathrm{B}}\backslash mathrm\; E:\backslash ;\backslash overrightarrow\{\backslash mathrm\; x\}=\backslash overrightarrow\{\backslash mathrm\{OA\}\}+\backslash mathrm\backslash lambda\backslash cdot\backslash overrightarrow\{\backslash mathrm\; v\}+\backslash mathrm\backslash mu\backslash cdot\backslash overrightarrow\{\backslash mathrm\{AB\}\}$  ein.

$\mathrm{E}:\overrightarrow{\mathrm{x}}=\left(\begin{array}{c}2\\ 1\\ -3\end{array}\right)+\lambda \cdot \left(\begin{array}{c}-1\\ 3\\ 1\end{array}\right)+\mu \cdot \left(\begin{array}{c}-1\\ -4\\ 0\end{array}\right)\backslash mathrm\; E:\backslash ;\backslash overrightarrow\{\backslash mathrm\; x\}=\backslash begin\{pmatrix\}2\backslash \backslash 1\backslash \backslash -3\backslash end\{pmatrix\}+\backslash mathrm\backslash lambda\backslash cdot\backslash begin\{pmatrix\}-1\backslash \backslash 3\backslash \backslash 1\backslash end\{pmatrix\}+\backslash mathrm\backslash mu\backslash cdot\backslash begin\{pmatrix\}-1\backslash \backslash -4\backslash \backslash 0\backslash end\{pmatrix\}$

Wähle  $\overrightarrow{\mathrm{v}}\backslash overrightarrow\{\backslash mathrm\; v\}$  als ersten Richtungsvektor und  $\overrightarrow{\mathrm{A}\mathrm{B}}\backslash overrightarrow\{\backslash mathrm\{AB\}\}$  als

zweiten Richtungsvektor der Ebene.

Berechne  $\overrightarrow{\mathrm{A}\mathrm{B}}\backslash overrightarrow\{\backslash mathrm\{AB\}\}$  .

$\overrightarrow{\mathrm{A}\mathrm{B}}=\overrightarrow{\mathrm{B}}-\overrightarrow{\mathrm{A}}=\left(\begin{array}{c}4\\ -3\\ -1\end{array}\right)-\left(\begin{array}{c}8\\ 13\\ 9\end{array}\right)=\left(\begin{array}{c}-4\\ -16\\ -10\end{array}\right)\backslash overrightarrow\{\backslash mathrm\{AB\}\}=\backslash overrightarrow\{\backslash mathrm\; B\}-\backslash overrightarrow\{\backslash mathrm\; A\}=\backslash begin\{pmatrix\}4\backslash \backslash -3\backslash \backslash -1\backslash end\{pmatrix\}-\backslash begin\{pmatrix\}8\backslash \backslash 13\backslash \backslash 9\backslash end\{pmatrix\}=\backslash begin\{pmatrix\}-4\backslash \backslash -16\backslash \backslash -10\backslash end\{pmatrix\}$

Um einen einfacheren Vektor zu erhalten, kann man gegebenenfalls noch $-2-2$  ausklammern.

$\Rightarrow \overrightarrow{\mathrm{A}\mathrm{B}}=\left(\begin{array}{c}2\\ 8\\ 5\end{array}\right)\backslash Rightarrow\backslash ;\backslash overrightarrow\{\backslash mathrm\{AB\}\}=\backslash begin\{pmatrix\}2\backslash \backslash 8\backslash \backslash 5\backslash end\{pmatrix\}$

Wähle  $\overrightarrow{\mathrm{O}\mathrm{A}}\backslash overrightarrow\{\backslash mathrm\{OA\}\}$  als Aufpunkt der Ebene und setze in die Ebenengleichung $\mathrm{E}:\overrightarrow{\mathrm{x}}=\overrightarrow{\mathrm{O}\mathrm{A}}+\lambda \cdot \overrightarrow{\mathrm{v}}+\mu \cdot \overrightarrow{\mathrm{A}\mathrm{B}}\backslash mathrm\; E:\backslash ;\backslash overrightarrow\{\backslash mathrm\; x\}=\backslash overrightarrow\{\backslash mathrm\{OA\}\}+\backslash mathrm\backslash lambda\backslash cdot\backslash overrightarrow\{\backslash mathrm\; v\}+\backslash mathrm\backslash mu\backslash cdot\backslash overrightarrow\{\backslash mathrm\{AB\}\}$  ein.

$\mathrm{E}:\overrightarrow{\mathrm{x}}=\left(\begin{array}{c}8\\ 13\\ 9\end{array}\right)+\lambda \cdot \left(\begin{array}{c}-6\\ 1\\ 1\end{array}\right)+\mu \cdot \left(\begin{array}{c}2\\ 8\\ 5\end{array}\right)\backslash mathrm\; E:\backslash ;\backslash overrightarrow\{\backslash mathrm\; x\}=\backslash begin\{pmatrix\}8\backslash \backslash 13\backslash \backslash 9\backslash end\{pmatrix\}+\backslash mathrm\backslash lambda\backslash cdot\backslash begin\{pmatrix\}-6\backslash \backslash 1\backslash \backslash 1\backslash end\{pmatrix\}+\backslash mathrm\backslash mu\backslash cdot\backslash begin\{pmatrix\}2\backslash \backslash 8\backslash \backslash 5\backslash end\{pmatrix\}$