The equation for a line segment is very similar to that for a line, it is:
$(x,y,z) = tQ + (1-t)P,\ 0 \leq t \leq 1$ (this one goes FROM $P$ TO $Q$, as $t$ goes from 0 to 1, if we switch $P$ and $Q$ it goes the other way).
In our case, this is $(2t+1,2-t,t-1)$ with $0 \leq t \leq 1$.
Suppose we choose $0 < t_0 < 1$. Let's use the standard distance formula, to calculate the distance of the point $B$ corresponding to $t_0$.
We have $d(P,B) = \sqrt{(1-(2t_0+1))^2 + (2-(2-t_0))^2 + (-1-(t_0-1))^2}$
$= \sqrt{4t_0^2 + t_0^2 + t_0^2} = \sqrt{6}t_0$
and $d(Q,B) = \sqrt{(3-(2t_0+1))^2 + (1-(2-t_0))^2 + (0-(t_0 - 1))^2}$
$= \sqrt{(2-2t_0)^2 + (-1+t_0)^2 + (t_0-1)^2} = \sqrt{6}(1-t_0)$ (we use $1-t_0$ since we want the positive square root).
Can you continue?
(Intuitively, how far along the interval from 0 to 1 do you have to be, to where how far you have to go is half of how far you've come? Use this to check your answer).