Let $\overline{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$,
$\begin{aligned}
& \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}} \\
& \overline{\mathrm{c}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}
\end{aligned}$
$\overline{\mathrm{b}} \times \overline{\mathrm{c}}$ is perpendicular to both $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$.
$\therefore \quad \overline{\mathrm{b}} \times \overline{\mathrm{c}}=\left|\begin{array}{ccc}
\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\
2 & 1 & -1 \\
1 & 3 & 2
\end{array}\right|=5 \hat{i}-5 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ The direction ratios of the required line are $5,-5,5$ i.e. $1,-1,1$
$\therefore \quad$ The required equation of line is
$\overline{\mathrm{r}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$