The equation of line passing through the point $(1,2,3)$ and perpendicular to the lines…
The equation of line passing through the point $(1,2,3)$ and perpendicular to the lines $\frac{x-2}{3}=\frac{y-1}{2}=\frac{z+1}{-2}$ and $\frac{x}{2}=\frac{y}{-3}=\frac{z}{1}$ is
Required line is perpendicular to the lines $\frac{x-2}{3}=\frac{y-1}{2}=\frac{z+1}{-2}$ and $\frac{x}{2}=\frac{y}{-3}=\frac{z}{1}$
$\therefore \quad$ Required line is parallel to vector
$\overline{\mathrm{b}}=\left|\begin{array}{ccc}
\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\
3 & 2 & -2 \\
2 & -3 & 1
\end{array}\right|=-4 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}-13 \hat{\mathrm{k}}$
$\therefore \quad$ The equation of the required line is
$(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(-4 \hat{i}-7 \hat{j}-13 \hat{k})$