Beispiel Nr: 05
$\text{Gegeben:} \text{Zwei parallele Geraden}\\ \text{Gerade 1: } \vec{x} =\left( \begin{array}{c} a_1 \\ a_2 \\ a_3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} b_1 \\ b_2 \\ b_3 \\ \end{array} \right) \\ \text{Gerade 2: } \vec{x} =\left( \begin{array}{c} c_1 \\ c_2 \\ c_3 \\ \end{array} \right) + \sigma \left( \begin{array}{c} d_1 \\ d_2 \\ d_3 \\ \end{array} \right) \\ \text{Gesucht:} \text{Ebene aus zwei Geraden} \\ \text{Parallele Geraden} \\ \textbf{Gegeben:} \\ \text{Gerade1: }\\ \vec{x} =\left( \begin{array}{c} -5 \\ 8 \\ 0 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 4 \\ -6 \\ 0 \\ \end{array} \right) \\ \text{Gerade2: }\\ \vec{x} =\left( \begin{array}{c} 5 \\ 3 \\ -3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 2 \\ 3 \\ 1 \\ \end{array} \right) \\ \\ \\ \textbf{Rechnung:} \\ \text{Gerade1: }\\ \vec{x} =\left( \begin{array}{c} -5 \\ 8 \\ 0 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 4 \\ -6 \\ 0 \\ \end{array} \right) \\ \text{Gerade2: }\\ \vec{x} =\left( \begin{array}{c} 5 \\ 3 \\ -3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 2 \\ 3 \\ 1 \\ \end{array} \right) \\ \vec{AC} =\left( \begin{array}{c} 5+5 \\ 3-8 \\ -3-0 \\ \end{array} \right) = \left( \begin{array}{c} 10 \\ -5 \\ -3 \\ \end{array} \right) \\ \vec{x} =\left( \begin{array}{c} -5 \\ 8 \\ 0 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 4 \\ -6 \\ 0 \\ \end{array} \right)+ \sigma \left( \begin{array}{c} 10 \\ -5 \\ -3 \\ \end{array} \right) \\ $