A unit vector coplanar with \(\hat{i}+\hat{j}+\hat{k}\) and \(2 \hat{i}+\hat{j}+\hat{k}\) and perpendicular to \(\hat{i}+\hat{j}-\hat{k}\) is

Let \(\overline{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \overline{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{c}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\).
Then, required unit vectors are given by
\(\bar{\alpha}= \pm \frac{\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \cdot}{|\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|}\)
Now, \(\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}) \overline{\mathrm{c}}\)
\(\begin{array}{ll}
\therefore & =2(\hat{i}+\hat{j}+\hat{k})-1(2 \hat{i}+\hat{j}+\hat{k})=\hat{j}+\hat{k} \\
\therefore & |\bar{a} \times(\bar{b} \times \bar{c})|=\sqrt{1+1}=\sqrt{2}
\end{array}\)
Hence, required unit vectors are
\(\bar{\alpha}= \pm \frac{\hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{2}}\)