$\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} 1 \\ -1 \\ 2 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 3 \\ -4 \\ 2 \\ \end{array} \right) \\ \text{Gerade 2: } \vec{x} =\left( \begin{array}{c} 1 \\ -1 \\ 2 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 3 \\ -7 \\ 1 \\ \end{array} \right) \\ \\ \\ \textbf{Rechnung:} \\ \text{Gerade 1: } \vec{x} =\left( \begin{array}{c} 1 \\ -1 \\ 2 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 3 \\ -4 \\ 2 \\ \end{array} \right) \\ \text{Gerade 2: } \vec{x} =\left( \begin{array}{c} 1 \\ -1 \\ 2 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 3 \\ -7 \\ 1 \\ \end{array} \right) \\ \text{Richtungsvektoren: } \\ \left( \begin{array}{c} 3 \\ -4 \\ 2 \\ \end{array} \right) =k \cdot \left( \begin{array}{c} 3 \\ -7 \\ 1 \\ \end{array} \right) \\ \begin{array}{cccc} 3&=&+3 k& \quad /:3 \quad \Rightarrow k=1 \\ -4&=&-7 k & \quad /:-7 \quad \Rightarrow k=\frac{4}{7} \\ 2&=&+1 k & \quad /:1 \quad \Rightarrow k=2 \\ \end{array} \\ \\ \Rightarrow \text{Geraden sind nicht parallel} \\ \left( \begin{array}{c} 1 \\ -1 \\ 2 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 3 \\ -4 \\ 2 \\ \end{array} \right) = \left( \begin{array}{c} 1 \\ -1 \\ 2 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 3 \\ -7 \\ 1 \\ \end{array} \right) \\ \begin{array}{cccccc} 1& +3\lambda &=& 1& +3\sigma& \quad /-1 \quad /-3 \sigma\\ -1& -4\lambda &=& -1& -7 \sigma& \quad /+1 \quad /+7 \sigma\\ 2& +2\lambda &=& 2& +1 \sigma& \quad /-2 \quad /-1 \sigma\\ \end{array} \\ \\I \qquad 3 \lambda -3 \sigma =0\\ II \qquad -4 \lambda +7 \sigma = 0 \\ III \qquad 2 \lambda +1 \sigma = 0 \\ \\ \text{Aus 2 Gleichungen }\lambda \text{ und } \sigma \text{ berechnen } \\ I \qquad 3 \lambda -3 \sigma =0 \qquad / \cdot\left(-4\right)\\ II \qquad -4 \lambda +7 \sigma = 0 \qquad / \cdot\left(-3\right)\\ I \qquad -12 \lambda +12 \sigma =0\\ II \qquad 12 \lambda -21 \sigma = 0 \\ \text{I + II}\\ I \qquad -12 \lambda +12 \lambda+12 \sigma -21 \sigma =0 +0\\ -9 \sigma = 0 \qquad /:\left(-9\right) \\ \sigma = \frac{0}{-9} \\ \sigma=0 \\ \sigma \text{ in I}\\ I \qquad -12 \lambda +12 \cdot 0 =0 \\ -12 \lambda +0 =0 \qquad / -0 \\ -12 \lambda =0 -0 \\ -12 \lambda =0 \qquad / :\left(-12\right) \\ \lambda = \frac{0}{-12} \\ \lambda=0 \\ \lambda \text{ und } \sigma \text{ in die verbleibende Gleichung einsetzen} \\ III \quad 2+0\cdot2=2+0\cdot1 \\ 2=2 \\ \lambda \text{ oder } \sigma \text{ in die Geradengleichung einsetzen} \\ \\ \vec{x} = \left( \begin{array}{c} 1 \\ -1 \\ 2 \\ \end{array} \right) +0 \cdot \left( \begin{array}{c} 3 \\ -4 \\ 2 \\ \end{array} \right) \\ \text{Schnittpunkt: }S(1,-1,2) \\ \\ \end{array}$