Given two lines.

$$\begin{gathered} l_1:(3+t) \hat{i}+(-1+2 t) \hat{j}+(4+2 t) \hat{k} \text { and } \\ l_2:(3+2 s) \hat{i}+(3+2 s) \hat{j}+(2+s) \hat{k} \\ \Rightarrow l_1: \frac{x-3}{1}=\frac{y+1}{2}=\frac{z-4}{2}=t \text { and } \\ \quad l_2: \frac{x-3}{2}=\frac{y-3}{2}=\frac{z-2}{1}=s \end{gathered}$$

Given, a line $$l$$ is perpendicular to $$l_1$$ and $$l_2$$

$$\therefore \quad l$$ is parallel to $$l_1 \times l_2$$

$$\begin{aligned} & \Rightarrow \vec{l}_1 \times \vec{l}_2=\left|\begin{array}{lll} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 2 \\ 2 & 2 & 1 \end{array}\right| \\ & \Rightarrow \vec{l}_1 \times \vec{l}_2=-2 \hat{i}+3 \hat{j}-2 \hat{k} \end{aligned}$$

The equation of line $$l$$ passing through origin and parallel to $$-2 \hat{i}+3 \hat{j}-2 \hat{k}$$ is

$$l: \frac{x}{-2}=\frac{y}{3}=\frac{z}{-2}=\lambda$$

Let P is the point of intersection of $$l$$ and $$l_1$$

$$\begin{aligned} & \Rightarrow \mathrm{P}=(-2 \lambda, 3 \lambda,-2 \lambda)=(3+t,-1+2 t, 4+2 t) \\ & \Rightarrow \lambda=\frac{3+t}{-2}=\frac{-1+2 t}{3}=\frac{4+2 t}{-2} \\ & \Rightarrow t=-1, \lambda=-1 \end{aligned}$$

$$\text { So, } \mathrm{P}=(2,-3,2)$$

Let Q is a point on the line $$l_2$$

$$Q=(3+2 s, 3+2 s, 2+s)$$

Apply $$|\mathrm{PQ}|=\sqrt{17}$$

$$\begin{aligned} & \Rightarrow \sqrt{(3+2 s-2)^2+(3+2 s+3)^2+(2+s-2)^2} \\ & =\sqrt{17} \\ & \Rightarrow \sqrt{9 s^2+28 s+37}=\sqrt{17} \\ & \Rightarrow \quad 9 s^2+28 s+37=17 \\ & \Rightarrow \quad 9 s^2+28 s+20=0 \\ & \Rightarrow \mathrm{s}=\frac{-28 \pm \sqrt{28^2-4 \times 9 \times 20}}{2.9} \\ & \Rightarrow \mathrm{s}=\frac{-28 \pm 8}{18} \\ & \Rightarrow \mathrm{s}=\frac{-10}{9},-2 \\ \end{aligned}$$

If $$s=-2$$, then $$Q=(-1,-1,0)$$

$$\text { If } s=-\frac{10}{9} \text {, then } Q=\left(\frac{7}{9}, \frac{7}{9}, \frac{8}{9}\right)$$

Hints :

(i) If $$l$$ is perpendicular to $$\vec{l}_1$$ and $$\vec{l}_2$$, then $$l$$ is parallel to $$\vec{l}_1 \times \vec{l}_2$$.

(ii) A general point on the line

$$\begin{gathered} \quad \frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}=\lambda \text { is } \\ \left(x_1+\lambda a, y_1+\lambda b, z_1+\lambda c\right) . \end{gathered}$$