The plane determined by the points $\left(-1,2,-2\right), \left(0, 1, 1\right)$ and $\left(1, 1, 2\right)$ passes through

$\begin{array}{rcl}\left|\begin{array}{ccc}\mathrm{x}-{\mathrm{x}}_1& \mathrm{y}-{\mathrm{y}}_1& \mathrm{z}-{\mathrm{z}}_1\\ {\mathrm{x}}_2-{\mathrm{x}}_1& {\mathrm{y}}_2-{\mathrm{y}}_1& {\mathrm{z}}_2-{\mathrm{z}}_1\\ {\mathrm{x}}_3-{\mathrm{x}}_1& {\mathrm{y}}_3-{\mathrm{y}}_1& {\mathrm{z}}_3-{\mathrm{z}}_1\end{array}\right|& =& 0\\ \left|\begin{array}{ccc}\mathrm{x}+1& \mathrm{y}-2& \mathrm{z}+2\\ 1& -1& 3\\ 2& -1& 4\end{array}\right|& =& 0\\ (\mathrm{x}+1) (-4+3)-(\mathrm{y}-2) (-2)+(\mathrm{z}+2) \left(1\right)& =& 0\\ -\mathrm{x}-1+2\mathrm{y}-4+\mathrm{z}+2& =& 0\\ \mathrm{x}-2\mathrm{y}-\mathrm{z}+3& =& 0 \mathrm{and} \mathrm{verigy}.\end{array}$