The coplanar points $$A,\,B,\,C,\,D$$   are $$\left( {2 - x,\,2,\,2} \right),\,\left( {2,\,2 - y,\,2} \right),\,\left( {2,\,2,\,2 - z} \right)$$         and $$\left( {1,\,1,\,1} \right)$$   respectively. Then :

Solution :
$$\eqalign{ & \overrightarrow {AB} = \overrightarrow {OB} - \overrightarrow {OA} \cr & = \left\{ {2\overrightarrow i + \left( {2 - y} \right)\overrightarrow j + 2\overrightarrow k } \right\} - \left\{ {\left( {2 - x} \right)\overrightarrow i + 2\overrightarrow j + 2\overrightarrow k } \right\} \cr & = x\overrightarrow i - y\overrightarrow j \cr & \overrightarrow {AC} = \overrightarrow {OC} - \overrightarrow {OA} \cr & = \left\{ {2\overrightarrow i + 2\overrightarrow j + \left( {2 - z} \right)\overrightarrow k } \right\} - \left\{ {\left( {2 - x} \right)\overrightarrow i + 2\overrightarrow j + 2\overrightarrow k } \right\} \cr & = x\overrightarrow i - z\overrightarrow k \cr & \overrightarrow {AD} = \overrightarrow {OD} - \overrightarrow {OA} \cr & = \left\{ {\overrightarrow i + \overrightarrow j + \overrightarrow k } \right\} - \left\{ {\left( {2 - x} \right)\overrightarrow i + 2\overrightarrow j + 2\overrightarrow k } \right\} \cr & = \left( {x - 1} \right)\overrightarrow i - \overrightarrow j - \overrightarrow k \cr} $$
As these vectors are coplanar, \[\left| \begin{array}{l} \,\,\,\,x\,\,\,\,\,\,\,\,\,\, - y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\\ \,\,\,\,x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\, - z\\ x - 1\,\,\,\,\, - 1\,\,\,\,\, - 1 \end{array} \right| = 0\]
On simplification, $$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$$