Let the foot point of the perpendicular drawn
from the point $P(3,-1,1)$ on the straight line be $L$.
$\therefore$ Hence, $L$ lies on the straight line
$\therefore L(2 t, 2+3 t, 3+4 t)$ [where, $t$ is arbitrary
constant]
$\therefore$ The direction ratios of $P L$ are $(2 t-3,2+3 t+1$,
Again, the direction ratios of the straight line.
$\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ are $(2,3,4)$.
Since, $P L$ is perpendicular on the straight line.
Then, $(2 t-3) \cdot 2+(3 t+3) \cdot 3+(4 t-8) \cdot 4=0$
$\begin{aligned} & 4 t-6+9 t+9+16 t-32=0 \\ & 29 t=29\end{aligned}$
$\because \quad t=1$
Hence, $L(2,5,7)$
$\therefore \quad(P L)=\sqrt{(2-3)^2+(5+1)^2+(7-11)^2}=\sqrt{53}$