$\begin{array}{l} \text{Gegeben:} \\ \text{Ebene: } \vec{x} =\left( \begin{array}{c} a1 \\ a2 \\ a3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} b1 \\ b2 \\ b3 \\ \end{array} \right) + \sigma \left( \begin{array}{c} c1 \\ c2 \\ c3 \\ \end{array} \right) \\ \text{Gesucht:} \text{Ebene in Koordinatenform: } n_1 x_1+n_2 x_2+n_3 x_3+k=0 \\ \\ \text{Determinante}\\ \textbf{Gegeben:} \\ \vec{x} =\left( \begin{array}{c} 2 \\ -1 \\ -3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 0 \\ 1 \\ 0 \\ \end{array} \right)+ \sigma \left( \begin{array}{c} -4 \\ 5 \\ -2 \\ \end{array} \right) \\ \\ \\ \textbf{Rechnung:} \\ \vec{x} =\left( \begin{array}{c} 2 \\ -1 \\ -3 \\ \end{array} \right) + \lambda \left( \begin{array}{c} 0 \\ 1 \\ 0 \\ \end{array} \right) + \sigma \left( \begin{array}{c} -4 \\ 5 \\ -2 \\ \end{array} \right) \\ D=\begin{array}{|ccc|} x_1-2\ & 0 & -4\\ x_2+1&1 & 5\\ x_3+3& 0 & -2 \\ \end{array} \begin{array}{cc} x_1-2\ & 0 \\ x_2+1&1 \\ x_3+3& 0 \end{array} =0 \\ (x_1-2) \cdot 1 \cdot \left(-2\right)+ 0 \cdot 5 \cdot (x_3+3) + \left(-4\right) \cdot (x_2+1) \cdot 0 \\ - \left(-4\right) \cdot 1 \cdot (x_3+3) - (x_1-2) \cdot 5 \cdot 0 - 0 \cdot (x_2+1) \cdot \left(-2\right)=0 \\ -2 x_1+0 x_2+4 x_3+16=0 \\ -2 x_1 +4 x_3 +16 = 0 \\ \text{Koordinatenform in Hessesche Normalenform HNF} \\ -2 x_1+0 x_2+4 x_3+16=0 \\ \vec{n} = \left( \begin{array}{c} -2 \\ 0 \\ 4 \\ \end{array} \right) \\ \text{Länge des Normalenvektors} \\ \left|\vec{n}\right| =\sqrt{n_1^2+n_2^2+n_3^2} \\ \left|\vec{n}\right| =\sqrt{\left(-2\right)^2+0^2+4^2} \\ \left|\vec{n}\right| =4,47 \\ \text{HNF:} \dfrac{-2 x_1+0 x_2+4 x_3+16}{-4,47}=0 \\ \\ \end{array}$