Beispiel Nr: 03
$\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} 3 \\ 6 \\ 0 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 4 \\ 0 \\ 6 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 5 \\ 0 \\ 0 \\ \end{array} \right) \\ \text{Ebene2: } 5 x_1+0 x_2+6 x_3+5=0 \\ \\ \\ \textbf{Rechnung:} \\ \text{Ebene: } \vec{x} =\left( \begin{array}{c} 3 \\ 6 \\ 0 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 4 \\ 0 \\ 6 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 5 \\ 0 \\ 0 \\ \end{array} \right) \\ \text{Ebene: } 5 x_1+0 x_2+6 x_3+5=0 \\ \begin{array}{cccc} x_1=& 3 &+4\lambda &+5\sigma \\ x_2=&6 &+0\lambda &+0\sigma \\ x_3=&0 &+6\lambda &+0\sigma\\ \end{array} \\ 5( 3+4\lambda+5\sigma) +0(6+0\lambda+0\sigma) +6 (0+6\lambda+0\sigma)+5=0 \\ 56\lambda+25\sigma+20=0 \\ \\ \sigma=\frac{-56 \lambda -20}{25} \\ \sigma= -2\frac{6}{25} \lambda -\frac{4}{5} \\ \vec{x} = \left( \begin{array}{c} 3 \\ 6 \\ 0 \\ \end{array} \right) +\lambda \cdot \left( \begin{array}{c} 4 \\ 0 \\ 6 \\ \end{array} \right) +(-2\frac{6}{25}\lambda-\frac{4}{5}) \cdot \left( \begin{array}{c} 5 \\ 0 \\ 0 \\ \end{array} \right) \\ \text{Schnittgerade: } \vec{x} =\left( \begin{array}{c} -1 \\ 6 \\ 0 \\ \end{array} \right) + \lambda \left( \begin{array}{c} -7\frac{1}{5} \\ 0 \\ 6 \\ \end{array} \right) \\ \\ $