Beispiel Nr: 08
$ \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 \\ 2 \\ 7 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 2 \\ 6 \\ 8 \\ \end{array} \right) \\ \text{Gerade 2: } \vec{x} =\left( \begin{array}{c} 9 \\ 7 \\ 8 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 5 \\ 3 \\ 3 \\ \end{array} \right) \\ \\ \\ \textbf{Rechnung:} \\ \text{Gerade 1: } \vec{x} =\left( \begin{array}{c} 1 \\ 2 \\ 7 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 2 \\ 6 \\ 8 \\ \end{array} \right) \\ \text{Gerade 2: } \vec{x} =\left( \begin{array}{c} 9 \\ 7 \\ 8 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 5 \\ 3 \\ 3 \\ \end{array} \right) \\ \text{Richtungsvektoren: } \\ \left( \begin{array}{c} 2 \\ 6 \\ 8 \\ \end{array} \right) =k \cdot \left( \begin{array}{c} 5 \\ 3 \\ 3 \\ \end{array} \right) \\ \begin{array}{cccc} 2&=&+5 k& \quad /:5 \quad \Rightarrow k=\frac{2}{5} \\ 6&=&+3 k & \quad /:3 \quad \Rightarrow k=2 \\ 8&=&+3 k & \quad /:3 \quad \Rightarrow k=2\frac{2}{3} \\ \end{array} \\ \\ \Rightarrow \text{Geraden sind nicht parallel} \\ \left( \begin{array}{c} 1 \\ 2 \\ 7 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 2 \\ 6 \\ 8 \\ \end{array} \right) = \left( \begin{array}{c} 9 \\ 7 \\ 8 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 5 \\ 3 \\ 3 \\ \end{array} \right) \\ \begin{array}{cccccc} 1& +2\lambda &=& 9& +5\sigma& \quad /-1 \quad /-5 \sigma\\ 2& +6\lambda &=& 7& +3 \sigma& \quad /-2 \quad /-3 \sigma\\ 7& +8\lambda &=& 8& +3 \sigma& \quad /-7 \quad /-3 \sigma\\ \end{array} \\ \\I \qquad 2 \lambda -5 \sigma =8\\ II \qquad 6 \lambda -3 \sigma = 5 \\ III \qquad 8 \lambda +3 \sigma = 1 \\ \\ \text{Aus 2 Gleichungen }\lambda \text{ und } \sigma \text{ berechnen } \\ I \qquad 2 \lambda -5 \sigma =8 \qquad / \cdot3\\ II \qquad 6 \lambda -3 \sigma = 5 \qquad / \cdot\left(-1\right)\\ I \qquad 6 \lambda -15 \sigma =24\\ II \qquad -6 \lambda +3 \sigma = -5 \\ \text{I + II}\\ I \qquad 6 \lambda -6 \lambda-15 \sigma +3 \sigma =24 -5\\ -12 \sigma = 19 \qquad /:\left(-12\right) \\ \sigma = \frac{19}{-12} \\ \sigma=-1\frac{7}{12} \\ \sigma \text{ in I}\\ I \qquad 6 \lambda -15 \cdot \left(-1\frac{7}{12}\right) =24 \\ 6 \lambda +23\frac{3}{4} =24 \qquad / -23\frac{3}{4} \\ 6 \lambda =24 -23\frac{3}{4} \\ 6 \lambda =\frac{1}{4} \qquad / :6 \\ \lambda = \frac{\frac{1}{4}}{6} \\ \lambda=\frac{1}{24} \\ \lambda \text{ und } \sigma \text{ in die verbleibende Gleichung einsetzen} \\ III \quad 7+\frac{1}{24}\cdot8=8-1\frac{7}{12}\cdot3 \\ 7\frac{1}{3}=3\frac{1}{4} \\ \text{Geraden sind windschief} \\ $