We know that a vector perpendicular to the plane containing the points
$ A , B , C$ is given by $A \times B + B \times C + C \times A$
We have, $A = \hat{i} - \hat{j} + 2\hat{k}, \vec{B} = 2\hat{i} + 0 \hat{j} - \hat{k}$
and $C= 0 \hat{i} + 2 \hat{j} + \hat{k}$
Now, $A \times B= \begin{vmatrix}\hat{i}& \hat{j}&\hat{k}\\ 1&-1&2\\ 2&0&-1\end{vmatrix} = \hat{i} + 5\hat{j} +2 \hat{k} $
$ B \times C = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ 2&0&-1\\ 0&2&1\end{vmatrix} = 2\hat{i} - 2\hat{j} + 4\hat{k} $
$C \times\vec{A} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ 0&2&1\\ 1&-1&2\end{vmatrix} = 5\hat{i} + \hat{j} - 2\hat{k}$
Thus, $A \times B + B \times C + C \times B =(\hat{ i }+5 \hat{ j }+2 \hat{ k }) $
$+(2 \hat{ i }-2 \hat{ j }+4 \hat{ k })+(5 \hat{ i }+\hat{ j }-2 \hat{ k }) $
$=8 \hat{ i }+4 \hat{ j }+4 \hat{ k }$