The cartesian equationof the plane whose vector equation is
$\overline{r}=(1+\lambda -\mu )\overline{i}+(2-\lambda )\overline{j}+(3-2\lambda +2\mu )\overline{k}$ whare $\lambda ,\mu$ are scalars is

$$\begin{gathered} \overline{\mathrm{r}}=\mathrm{xi}+\mathrm{yj}+\mathrm{zk} \\ \mathrm{compare} \mathrm{i}, \mathrm{j}, \mathrm{k} \mathrm{coff} \\ 1+\mathrm{\lambda } -\mathrm{M}=\mathrm{x}, 2-\mathrm{\lambda }=\mathrm{y} \overline{) \mathrm{z}=3-2\mathrm{\lambda }+2\mathrm{M}} \\ \mathrm{\lambda }=2-\mathrm{y} \overline{) \mathrm{z}=3-4+2\mathrm{y}+6-2\mathrm{x}-2\mathrm{y}} \\ \mathrm{M}=1+2-\mathrm{y}-\mathrm{x} \\ =3-\mathrm{x}-\mathrm{y} \overline{) 2\mathrm{x}+\mathrm{t}=5} \end{gathered}$$