$\begin{array}{l} \text{Gegeben:} \text{Gerade 1: } \vec{x} =\left( \begin{array}{c} a1 \\ a2 \\ a3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} b1 \\ b2 \\ b3 \\ \end{array} \right) \\ \text{Gerade 2: } \vec{x} =\left( \begin{array}{c} c1 \\ c2 \\ c3 \\ \end{array} \right) + \sigma \left( \begin{array}{c} d1 \\ d2 \\ d3 \\ \end{array} \right) \\ \text{Gesucht:} \text{Die Lage der Geraden zueinander.} \\ \\ \textbf{Gegeben:} \\ \text{Gerade 1: } \vec{x} =\left( \begin{array}{c} -3 \\ 1 \\ 5 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 1 \\ 2 \\ -3 \\ \end{array} \right) \\ \text{Gerade 2: } \vec{x} =\left( \begin{array}{c} 1 \\ 5 \\ -3 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 5 \\ 4 \\ -1 \\ \end{array} \right) \\ \\ \\ \textbf{Rechnung:} \\ \text{Gerade 1: } \vec{x} =\left( \begin{array}{c} -3 \\ 1 \\ 5 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 1 \\ 2 \\ -3 \\ \end{array} \right) \\ \text{Gerade 2: } \vec{x} =\left( \begin{array}{c} 1 \\ 5 \\ -3 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 5 \\ 4 \\ -1 \\ \end{array} \right) \\ \text{Richtungsvektoren: } \\ \left( \begin{array}{c} 1 \\ 2 \\ -3 \\ \end{array} \right) =k \cdot \left( \begin{array}{c} 5 \\ 4 \\ -1 \\ \end{array} \right) \\ \begin{array}{cccc} 1&=&+5 k& \quad /:5 \quad \Rightarrow k=\frac{1}{5} \\ 2&=&+4 k & \quad /:4 \quad \Rightarrow k=\frac{1}{2} \\ -3&=&-1 k & \quad /:-1 \quad \Rightarrow k=3 \\ \end{array} \\ \\ \Rightarrow \text{Geraden sind nicht parallel} \\ \left( \begin{array}{c} -3 \\ 1 \\ 5 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 1 \\ 2 \\ -3 \\ \end{array} \right) = \left( \begin{array}{c} 1 \\ 5 \\ -3 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 5 \\ 4 \\ -1 \\ \end{array} \right) \\ \begin{array}{cccccc} -3& +1\lambda &=& 1& +5\sigma& \quad /+3 \quad /-5 \sigma\\ 1& +2\lambda &=& 5& +4 \sigma& \quad /-1 \quad /-4 \sigma\\ 5& -3\lambda &=& -3& -1 \sigma& \quad /-5 \quad /+1 \sigma\\ \end{array} \\ \\I \qquad 1 \lambda -5 \sigma =4\\ II \qquad 2 \lambda -4 \sigma = 4 \\ III \qquad -3 \lambda -1 \sigma = -8 \\ \\ \text{Aus 2 Gleichungen }\lambda \text{ und } \sigma \text{ berechnen } \\ I \qquad 1 \lambda -5 \sigma =4 \qquad / \cdot2\\ II \qquad 2 \lambda -4 \sigma = 4 \qquad / \cdot\left(-1\right)\\ I \qquad 2 \lambda -10 \sigma =8\\ II \qquad -2 \lambda +4 \sigma = -4 \\ \text{I + II}\\ I \qquad 2 \lambda -2 \lambda-10 \sigma +4 \sigma =8 -4\\ -6 \sigma = 4 \qquad /:\left(-6\right) \\ \sigma = \frac{4}{-6} \\ \sigma=-\frac{2}{3} \\ \sigma \text{ in I}\\ I \qquad 2 \lambda -10 \cdot \left(-\frac{2}{3}\right) =8 \\ 2 \lambda +6\frac{2}{3} =8 \qquad / -6\frac{2}{3} \\ 2 \lambda =8 -6\frac{2}{3} \\ 2 \lambda =1\frac{1}{3} \qquad / :2 \\ \lambda = \frac{1\frac{1}{3}}{2} \\ \lambda=\frac{2}{3} \\ \lambda \text{ und } \sigma \text{ in die verbleibende Gleichung einsetzen} \\ III \quad 5+\frac{2}{3}\cdot\left(-3\right)=-3-\frac{2}{3}\cdot\left(-1\right) \\ 3=-2\frac{1}{3} \\ \text{Geraden sind windschief} \\ \end{array}$