$\begin{array}{l} \text{Gegeben:} \\ \text{Ebene: } \vec{x} =\left( \begin{array}{c} a1 \\ a2 \\ a3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} b1 \\ b2 \\ b3 \\ \end{array} \right) + \sigma \left( \begin{array}{c} c1 \\ c2 \\ c3 \\ \end{array} \right) \\ \text{Gesucht:} \text{Ebene in Koordinatenform: } n_1 x_1+n_2 x_2+n_3 x_3+k=0 \\ \\ \text{Determinante}\\ \textbf{Gegeben:} \\ \vec{x} =\left( \begin{array}{c} 1 \\ -2 \\ 2 \\ \end{array} \right) + \lambda \left( \begin{array}{c} -3 \\ 4 \\ -5 \\ \end{array} \right)+ \sigma \left( \begin{array}{c} 2 \\ 3 \\ 6 \\ \end{array} \right) \\ \\ \\ \textbf{Rechnung:} \\ \vec{x} =\left( \begin{array}{c} 1 \\ -2 \\ 2 \\ \end{array} \right) + \lambda \left( \begin{array}{c} -3 \\ 4 \\ -5 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 2 \\ 3 \\ 6 \\ \end{array} \right) \\ D=\begin{array}{|ccc|} x_1-1\ & -3 & 2\\ x_2+2&4 & 3\\ x_3-2& -5 & 6 \\ \end{array} \begin{array}{cc} x_1-1\ & -3 \\ x_2+2&4 \\ x_3-2& -5 \end{array} =0 \\ (x_1-1) \cdot 4 \cdot 6+ \left(-3\right) \cdot 3 \cdot (x_3-2) + 2 \cdot (x_2+2) \cdot \left(-5\right) \\ - 2 \cdot 4 \cdot (x_3-2) - (x_1-1) \cdot 3 \cdot \left(-5\right) - \left(-3\right) \cdot (x_2+2) \cdot 6=0 \\ 39 x_1+8 x_2-17 x_3+11=0 \\ 39 x_1 +8 x_2 -17 x_3 +11 = 0 \\ \text{Koordinatenform in Hessesche Normalenform HNF} \\ 39 x_1+8 x_2-17 x_3+11=0 \\ \vec{n} = \left( \begin{array}{c} 39 \\ 8 \\ -17 \\ \end{array} \right) \\ \text{Länge des Normalenvektors} \\ \left|\vec{n}\right| =\sqrt{n_1^2+n_2^2+n_3^2} \\ \left|\vec{n}\right| =\sqrt{39^2+8^2+\left(-17\right)^2} \\ \left|\vec{n}\right| =43,3 \\ \text{HNF:} \dfrac{39 x_1+8 x_2-17 x_3+11}{-43,3}=0 \\ \\ \end{array}$