$\begin{array}{l} \text{Gegeben:} \\ \text{Ebene1: } \vec{x} =\left( \begin{array}{c} a1 \\ a2 \\ a3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} b1 \\ b2 \\ b3 \\ \end{array} \right) + \sigma \left( \begin{array}{c} c1 \\ c2 \\ c3 \\ \end{array} \right) \\ \text{Ebene2: } n1 x_1+n2 x_2+n3 x_3+k1=0 \\ \\ \text{Gesucht:} \\\text{Lage der Ebenen zueinander} \\ \\ \textbf{Gegeben:} \\ \text{Ebene1: } \vec{x} =\left( \begin{array}{c} 1 \\ 9 \\ 5 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 1 \\ 4 \\ 5 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 5 \\ 9 \\ 9 \\ \end{array} \right) \\ \text{Ebene2: } 4 x_1+3 x_2+4 x_3+1=0 \\ \\ \\ \textbf{Rechnung:} \\ \text{Ebene: } \vec{x} =\left( \begin{array}{c} 1 \\ 9 \\ 5 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 1 \\ 4 \\ 5 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 5 \\ 9 \\ 9 \\ \end{array} \right) \\ \text{Ebene: } 4 x_1+3 x_2+4 x_3+1=0 \\ \begin{array}{cccc} x_1=& 1 &+1\lambda &+5\sigma \\ x_2=&9 &+4\lambda &+9\sigma \\ x_3=&5 &+5\lambda &+9\sigma\\ \end{array} \\ 4( 1+1\lambda+5\sigma) +3(9+4\lambda+9\sigma) +4 (5+5\lambda+9\sigma)+1=0 \\ 36\lambda+83\sigma+52=0 \\ \\ \sigma=\frac{-36 \lambda -52}{83} \\ \sigma= -\frac{36}{83} \lambda -\frac{52}{83} \\ \vec{x} = \left( \begin{array}{c} 1 \\ 9 \\ 5 \\ \end{array} \right) +\lambda \cdot \left( \begin{array}{c} 1 \\ 4 \\ 5 \\ \end{array} \right) +(-\frac{36}{83}\lambda-\frac{52}{83}) \cdot \left( \begin{array}{c} 5 \\ 9 \\ 9 \\ \end{array} \right) \\ \text{Schnittgerade: } \vec{x} =\left( \begin{array}{c} -2\frac{11}{83} \\ 3\frac{30}{83} \\ -\frac{53}{83} \\ \end{array} \right) + \lambda \left( \begin{array}{c} -1\frac{14}{83} \\ \frac{8}{83} \\ 1\frac{8}{83} \\ \end{array} \right) \\ \\ \end{array}$