Let me summarize what I learned in this thread. Please correct or
improve the description.
Symbols i,j,k, for example when used to describe a force F=2i-3j,
become "unit vectors" in that expression. All by themselves they
are only place holders. A coordinate system to represent F (a set of
three axes) and a unit (to quantify forces) must be chosen to turn
i,j,k into unit vectors. Without this they are only symbols.
In that sense they are like numbers. A pure number means nothing
in practice unless we say what it represents. In one problem it may
be a number of apples, in another a number of volts, or the length
of an arrow. In the same way i,j,k mean nothing unless a frame of
reference is chosen (or implied) and a physical unit is chosen to
represent a physical quantity. We call i,j,k unit vectors because
they become vectors, not because they are vectors by themselves.
In other words, we may say that by themselves i,j,k are variables
rather than specific vectors. There is nothing profound in this
observation. We never deal with i,j,k "by themselves", only with
them being components of expressions which represent vectors.
That is why calling i,j,k "unit vectors" is not confusing.
In one problem 1i (written as i) will stand for the horizontal
velocity (in m/s), in another for the force pulling an object along
an inclined plane, etc. This is not related to the choice of a scale
(for example one inch on paper for 0.25 N, 1 cm for 5 m/s or
1 mm for 1 a.u.). Scales become important only when problems
are solved graphically rather than algebraically.
Ludwik Kowalski