Can anyone please explain how option (A) is correct. I have tried both dot product and cross product methods but option (A) is not matched.

1) using dot product :-

$\vec{a} = 2i - 2j + k$ and $\vec{b} = i + j -2k$ and let, option (A) = $\vec{c} = \frac{(i + j +k) }{\sqrt{3}}$

Now , $\vec{a}. \vec{c} $ = $\frac{2-2+1}{\sqrt{3}} = \frac{1}{\sqrt{3}} \neq 0$

2) by using cross product of $\vec{a}\;\;and\;\; \vec{b} $ to get the a vector which is perpendicular to the plane in which vectors $\vec{a}\;\; and\;\; \vec{b}$ lies :-

$\vec{n}$= $\begin{vmatrix} \hat{i} & \hat{j} &\hat{k} \\ 2&-2 &1 \\ 1&1 &-2 \end{vmatrix} = \hat{i} \begin{vmatrix} -2 &1 \\ 1& -2 \end{vmatrix} -\hat{j}\begin{vmatrix} 2 &1 \\ 1& -2 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 &-2 \\ 1& 1 \end{vmatrix} = 3\hat{i} + 5\hat{j} + 4\hat{k}$

Now , unit length vector in the direction of vector $\vec{n}$ is :-

$\hat{n}$ = $\frac{\vec{n}}{|\vec{n}|}$ = $\frac{3\hat{i} + 5\hat{j} + 4\hat{k}}{\sqrt{9 +25+16}}$ = $\frac{3\hat{i} + 5\hat{j} + 4\hat{k}}{5\sqrt{2}}$

It is not matching with any given options. So , Please tell me , Am I missing something or given options are wrong ?