How can vectors with different units (position, speed, ...) "share" the same space? My question is too stupid to be googled, so I'll ask it here (because I didn't get any answers from google).

Context: 
I have a three dimensional space and the units for $x,y,z$ are given in meters. On the plane $xy$ there is an area $A$ with fluid coming at out of it at velocity $V$ ($3\,\mathrm m/\mathrm s$). My textbook just draws this velocity vector with a magnitude of $3$ on top of area $A$. 

My question is: 
Why is it valid to draw this vector in a system with no reference to time (because velocity is m/s)? Why is it just assumed that its magnitude is $3$ on this coordinate system?

Thanks.
You might say that you have several vector spaces overlapping, one space $A$ for spacial locations (with an arbitrarily picked origin and units in meters, say), one space $B$ for velocities with units in m/s, say (and add spaces for acceleration, impulse, force, ... if needed). Principally, these are totally distinct spaces and it makes no sense e.g. to add a vector $v\in B$  to a vector $r\in A$. However, for any given timespan $t$, we have a map $v\mapsto tv$ from $B$ to $A$. It is convenient to "draw" both spaces in the same 3d space - actually this is not much different from the convention to "draw" a 3d sapce on the falt page of a textbook in the first place. Some means of distinction should be used, maybe different colours; or one agrees upon a standard timespan to map the velocities space into the dislpacement space.

You may have noticed that I mentioned the arbitrariness of choice of origin for $A$, but said nothing about the origin of $B$. Well, actually the most correct way of handling the situation would be to attach a copy of $B$ to every single point in $A$; one speaks of tangential space and hs to concider topological peoperties to relate velocities attached to differente (nearby) points. But I'm afraid that would be even more confusing than the illustration you already complain about ...
Because size of the velocity vector is the value of the velocity without units, and direction is the angle it creates with the determined beginning of axis