Beispiel Nr: 05
$ \text{Gegeben:} \\ \text{Ebene: } \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{Gesucht:} \text{Ebene in Koordinatenform: } n_1 x_1+n_2 x_2+n_3 x_3+k=0 \\ \\ \text{Determinante}\\ \textbf{Gegeben:} \\ \vec{x} =\left( \begin{array}{c} 1 \\ 2 \\ -7 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 1 \\ -1 \\ 0 \\ \end{array} \right)+ \sigma \left( \begin{array}{c} -1 \\ 0 \\ 1 \\ \end{array} \right) \\ \\ \\ \textbf{Rechnung:} \\ \vec{x} =\left( \begin{array}{c} 1 \\ 2 \\ -7 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 1 \\ -1 \\ 0 \\ \end{array} \right) + \sigma \left( \begin{array}{c} -1 \\ 0 \\ 1 \\ \end{array} \right) \\ D=\begin{array}{|ccc|} x_1-1\ & 1 & -1\\ x_2-2&-1 & 0\\ x_3+7& 0 & 1 \\ \end{array} \begin{array}{cc} x_1-1\ & 1 \\ x_2-2&-1 \\ x_3+7& 0 \end{array} =0 \\ (x_1-1) \cdot \left(-1\right) \cdot 1+ 1 \cdot 0 \cdot (x_3+7) + \left(-1\right) \cdot (x_2-2) \cdot 0 \\ - \left(-1\right) \cdot \left(-1\right) \cdot (x_3+7) - (x_1-1) \cdot 0 \cdot 0 - 1 \cdot (x_2-2) \cdot 1=0 \\ -1 x_1-1 x_2-1 x_3-4=0 \\ -1 x_1 -1 x_2 -1 x_3 -4 = 0 \\ \text{Koordinatenform in Hessesche Normalenform HNF} \\ -1 x_1-1 x_2-1 x_3-4=0 \\ \vec{n} = \left( \begin{array}{c} -1 \\ -1 \\ -1 \\ \end{array} \right) \\ \text{Länge des Normalenvektors} \\ \left|\vec{n}\right| =\sqrt{n_1^2+n_2^2+n_3^2} \\ \left|\vec{n}\right| =\sqrt{\left(-1\right)^2+\left(-1\right)^2+\left(-1\right)^2} \\ \left|\vec{n}\right| =1,73 \\ \text{HNF:} \dfrac{-1 x_1-1 x_2-1 x_3-4}{1,73}=0 \\ \\ $