Beispiel Nr: 02
$\text{Gegeben:} \\ \text{Ebene1: } \vec{x} =\left( \begin{array}{c} a1 \\ a2 \\ a3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} b1 \\ b2 \\ b3 \\ \end{array} \right) + \sigma \left( \begin{array}{c} c1 \\ c2 \\ c3 \\ \end{array} \right) \\ \text{Ebene2: } n1 x_1+n2 x_2+n3 x_3+k1=0 \\ \\ \text{Gesucht:} \\\text{Lage der Ebenen zueinander} \\ \\ \textbf{Gegeben:} \\ \text{Ebene1: } \vec{x} =\left( \begin{array}{c} 4 \\ 4 \\ 9 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 5 \\ 4 \\ 5 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 8 \\ 9 \\ 7 \\ \end{array} \right) \\ \text{Ebene2: } 2 x_1+3 x_2+7 x_3+7=0 \\ \\ \\ \textbf{Rechnung:} \\ \text{Ebene: } \vec{x} =\left( \begin{array}{c} 4 \\ 4 \\ 9 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 5 \\ 4 \\ 5 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 8 \\ 9 \\ 7 \\ \end{array} \right) \\ \text{Ebene: } 2 x_1+3 x_2+7 x_3+7=0 \\ \begin{array}{cccc} x_1=& 4 &+5\lambda &+8\sigma \\ x_2=&4 &+4\lambda &+9\sigma \\ x_3=&9 &+5\lambda &+9\sigma\\ \end{array} \\ 2( 4+5\lambda+8\sigma) +3(4+4\lambda+9\sigma) +7 (9+5\lambda+7\sigma)+7=0 \\ 57\lambda+92\sigma+90=0 \\ \\ \sigma=\frac{-57 \lambda -90}{92} \\ \sigma= -\frac{57}{92} \lambda -\frac{45}{46} \\ \vec{x} = \left( \begin{array}{c} 4 \\ 4 \\ 9 \\ \end{array} \right) +\lambda \cdot \left( \begin{array}{c} 5 \\ 4 \\ 5 \\ \end{array} \right) +(-\frac{57}{92}\lambda-\frac{45}{46}) \cdot \left( \begin{array}{c} 8 \\ 9 \\ 7 \\ \end{array} \right) \\ \text{Schnittgerade: } \vec{x} =\left( \begin{array}{c} -3\frac{19}{23} \\ -4\frac{37}{46} \\ 2\frac{7}{46} \\ \end{array} \right) + \lambda \left( \begin{array}{c} \frac{1}{23} \\ -1\frac{53}{92} \\ \frac{61}{92} \\ \end{array} \right) \\ \\ $