If \(\bar{a}=2 \hat{i}-\hat{j}+\hat{k}, \bar{b}=\hat{i}+\hat{j}-2 \hat{k}\) and \(\bar{c}=4 \hat{i}-2 \hat{j}+\hat{k}\), then the unit vector in the direction of \(3 \bar{a}+\bar{b}-2 \bar{c}\) is

\(\begin{aligned}
& 3 \overline{\mathrm{a}}+\overline{\mathrm{b}}-2 \overline{\mathrm{c}} \\
& =3(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})+(\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}})-2(4 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \\
& =-(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}
\end{aligned}\)
\(\therefore \quad\) The unit vector in the direction of \(3 \overline{\mathrm{a}}+\overline{\mathrm{b}}-2 \overline{\mathrm{c}}\) is
\(\begin{aligned}
& \frac{-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}}{|\hat{-\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}|} \\
& =\frac{-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}}{\sqrt{(-1)^2+2^2+(-1)^2}} \\
& =\frac{1}{\sqrt{6}}(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})
\end{aligned}\)