I am working on a problem that states:
"A swimmer capable of swimming at a speed $c$ in still water is swimming in a stream in which the current is $u$ (which we assume to be less than $c$). Suppose the swimmer swims upstream a distance $L$ and then returns downstream to the starting point. Find the time necessary to make the round trip, and compare it with the time to swim across the stream a distance $L$ and return "
I was able to get to the point where I got the value of the velocity of the swimmer relative to the stream to be $v = -c$ Hence relative to the observer, the velocity of the swimmer would be $v = -c + u $
In the solutions manual they state
"As expected, the velocity relative to the ground has magnitude smaller than $c$ ; it is also negative, since the swimmer is swimming in the negative $x$ direction, so $|v| = c − u$.) "
I understand why the magnitude of $v$ would be $c - u$, but does this mean everytime I see $u - c = v$, I should change it to $c - u$ because mathematically, $|v| = \sqrt{(u-c)^2} = c - u$?