Consider the lines L_1 and L_2 given by L_1: (x-1)/2=(y-3)/1=(z-2)/2; L_2: (x-2)/1=(y-2)/2=(z-3)/3. A line L_3 having direction ratios 1,-1,-2,…

Consider the lines $\displaystyle L_1$ and $\displaystyle L_2$ given by $$\begin{aligned} & L_1: \frac{x-1}{2}=\frac{y-3}{1}=\frac{z-2}{2} \\ & L_2: \frac{x-2}{1}=\frac{y-2}{2}=\frac{z-3}{3} . \end{aligned} $$ A line $\displaystyle L_3$ having direction ratios $\displaystyle 1,-1,-2$, intersects $\displaystyle L_1$ and $\displaystyle L_2$ at the points $\displaystyle P$ and $\displaystyle Q$ respectively. Then the length of line segment $\displaystyle P Q$ is