$\begin{array}{l} \text{Gegeben:} \\ \text{Ebene: } \vec{x} =\left( \begin{array}{c} a1 \\ a2 \\ a3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} b1 \\ b2 \\ b3 \\ \end{array} \right) + \sigma \left( \begin{array}{c} c1 \\ c2 \\ c3 \\ \end{array} \right) \\ \text{Gesucht:} \text{Ebene in Koordinatenform: } n_1 x_1+n_2 x_2+n_3 x_3+k=0 \\ \\ \text{Determinante}\\ \textbf{Gegeben:} \\ \vec{x} =\left( \begin{array}{c} 5 \\ -1 \\ 2 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 3 \\ 0 \\ 2 \\ \end{array} \right)+ \sigma \left( \begin{array}{c} 0 \\ 3 \\ -1 \\ \end{array} \right) \\ \\ \\ \textbf{Rechnung:} \\ \vec{x} =\left( \begin{array}{c} 5 \\ -1 \\ 2 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 3 \\ 0 \\ 2 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 0 \\ 3 \\ -1 \\ \end{array} \right) \\ D=\begin{array}{|ccc|} x_1-5\ & 3 & 0\\ x_2+1&0 & 3\\ x_3-2& 2 & -1 \\ \end{array} \begin{array}{cc} x_1-5\ & 3 \\ x_2+1&0 \\ x_3-2& 2 \end{array} =0 \\ (x_1-5) \cdot 0 \cdot \left(-1\right)+ 3 \cdot 3 \cdot (x_3-2) + 0 \cdot (x_2+1) \cdot 2 \\ - 0 \cdot 0 \cdot (x_3-2) - (x_1-5) \cdot 3 \cdot 2 - 3 \cdot (x_2+1) \cdot \left(-1\right)=0 \\ -6 x_1+3 x_2+9 x_3+15=0 \\ -6 x_1 +3 x_2 +9 x_3 +15 = 0 \\ \text{Koordinatenform in Hessesche Normalenform HNF} \\ -6 x_1+3 x_2+9 x_3+15=0 \\ \vec{n} = \left( \begin{array}{c} -6 \\ 3 \\ 9 \\ \end{array} \right) \\ \text{Länge des Normalenvektors} \\ \left|\vec{n}\right| =\sqrt{n_1^2+n_2^2+n_3^2} \\ \left|\vec{n}\right| =\sqrt{\left(-6\right)^2+3^2+9^2} \\ \left|\vec{n}\right| =11,2 \\ \text{HNF:} \dfrac{-6 x_1+3 x_2+9 x_3+15}{-11,2}=0 \\ \\ \end{array}$