Let $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{c}=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$
Now, $\mathbf{b} \times \mathbf{c}=\left|\begin{array}{ccc}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 1 & 1 & -2 \\ -1 & 2 & 1\end{array}\right|=5 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$
Now, $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\left|\begin{array}{ccc}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 1 & -2 & 2 \\ 5 & 1 & 3\end{array}\right|=-8 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}$
Now, any vector perpendicular to a and lying in the plane containing $\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ is $\pm(a \times(b \times c))$
$$
\begin{aligned}
\therefore \text { Required unit vector } &=\pm \frac{(-8 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+11 \hat{\mathbf{k}})}{\sqrt{(-8)^{2}+(7)^{2}+(11)^{2}}} \\
&=\pm \frac{(-8 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+11 \hat{\mathbf{k}})}{\sqrt{234}}
\end{aligned}
$$