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