$\begin{array}{l} \text{Gegeben:} \\ \text{Ebene: } \vec{x} =\left( \begin{array}{c} a1 \\ a2 \\ a3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} b1 \\ b2 \\ b3 \\ \end{array} \right) + \sigma \left( \begin{array}{c} c1 \\ c2 \\ c3 \\ \end{array} \right) \\ \text{Gesucht:} \text{Ebene in Koordinatenform: } n_1 x_1+n_2 x_2+n_3 x_3+k=0 \\ \\ \text{Determinante}\\ \textbf{Gegeben:} \\ \vec{x} =\left( \begin{array}{c} 3 \\ 0 \\ 2 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 5 \\ -2 \\ 8 \\ \end{array} \right)+ \sigma \left( \begin{array}{c} 2 \\ 0 \\ 4 \\ \end{array} \right) \\ \\ \\ \textbf{Rechnung:} \\ \vec{x} =\left( \begin{array}{c} 3 \\ 0 \\ 2 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 5 \\ -2 \\ 8 \\ \end{array} \right) + \sigma \left( \begin{array}{c} 2 \\ 0 \\ 4 \\ \end{array} \right) \\ D=\begin{array}{|ccc|} x_1-3\ & 5 & 2\\ x_2-0&-2 & 0\\ x_3-2& 8 & 4 \\ \end{array} \begin{array}{cc} x_1-3\ & 5 \\ x_2-0&-2 \\ x_3-2& 8 \end{array} =0 \\ (x_1-3) \cdot \left(-2\right) \cdot 4+ 5 \cdot 0 \cdot (x_3-2) + 2 \cdot (x_2-0) \cdot 8 \\ - 2 \cdot \left(-2\right) \cdot (x_3-2) - (x_1-3) \cdot 0 \cdot 8 - 5 \cdot (x_2-0) \cdot 4=0 \\ -8 x_1-4 x_2+4 x_3+16=0 \\ -8 x_1 -4 x_2 +4 x_3 +16 = 0 \\ \text{Koordinatenform in Hessesche Normalenform HNF} \\ -8 x_1-4 x_2+4 x_3+16=0 \\ \vec{n} = \left( \begin{array}{c} -8 \\ -4 \\ 4 \\ \end{array} \right) \\ \text{Länge des Normalenvektors} \\ \left|\vec{n}\right| =\sqrt{n_1^2+n_2^2+n_3^2} \\ \left|\vec{n}\right| =\sqrt{\left(-8\right)^2+\left(-4\right)^2+4^2} \\ \left|\vec{n}\right| =9,8 \\ \text{HNF:} \dfrac{-8 x_1-4 x_2+4 x_3+16}{-9,8}=0 \\ \\ \end{array}$