$\begin{array}{l} \text{Gegeben:} \\ \text{Ebene: } \vec{x} =\left( \begin{array}{c} a1 \\ a2 \\ a3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} b1 \\ b2 \\ b3 \\ \end{array} \right) + \sigma \left( \begin{array}{c} c1 \\ c2 \\ c3 \\ \end{array} \right) \\ \text{Gesucht:} \text{Ebene in Koordinatenform: } n_1 x_1+n_2 x_2+n_3 x_3+k=0 \\ \\ \text{Determinante}\\ \textbf{Gegeben:} \\ \vec{x} =\left( \begin{array}{c} 1 \\ -4 \\ 2 \\ \end{array} \right) + \lambda \left( \begin{array}{c} -4 \\ 3 \\ -2 \\ \end{array} \right)+ \sigma \left( \begin{array}{c} 3 \\ 2 \\ 2 \\ \end{array} \right) \\ \\ \\ \textbf{Rechnung:} \\ \vec{x} =\left( \begin{array}{c} 1 \\ -4 \\ 2 \\ \end{array} \right) + \lambda \left( \begin{array}{c} -4 \\ 3 \\ -2 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 3 \\ 2 \\ 2 \\ \end{array} \right) \\ D=\begin{array}{|ccc|} x_1-1\ & -4 & 3\\ x_2+4&3 & 2\\ x_3-2& -2 & 2 \\ \end{array} \begin{array}{cc} x_1-1\ & -4 \\ x_2+4&3 \\ x_3-2& -2 \end{array} =0 \\ (x_1-1) \cdot 3 \cdot 2+ \left(-4\right) \cdot 2 \cdot (x_3-2) + 3 \cdot (x_2+4) \cdot \left(-2\right) \\ - 3 \cdot 3 \cdot (x_3-2) - (x_1-1) \cdot 2 \cdot \left(-2\right) - \left(-4\right) \cdot (x_2+4) \cdot 2=0 \\ 10 x_1+2 x_2-17 x_3+32=0 \\ 10 x_1 +2 x_2 -17 x_3 +32 = 0 \\ \text{Koordinatenform in Hessesche Normalenform HNF} \\ 10 x_1+2 x_2-17 x_3+32=0 \\ \vec{n} = \left( \begin{array}{c} 10 \\ 2 \\ -17 \\ \end{array} \right) \\ \text{Länge des Normalenvektors} \\ \left|\vec{n}\right| =\sqrt{n_1^2+n_2^2+n_3^2} \\ \left|\vec{n}\right| =\sqrt{10^2+2^2+\left(-17\right)^2} \\ \left|\vec{n}\right| =19,8 \\ \text{HNF:} \dfrac{10 x_1+2 x_2-17 x_3+32}{-19,8}=0 \\ \\ \end{array}$