If we agree that in physics the term "unit" refers to things
such as kg, m/s and N, then the term "unit vector" is
probably not appropriate for i, j and k.
Let me elaborate. First, i, j and k are dimensionless,
otherwise they could not be used for so many different
physical quantities. (In F=3i+4j the unit, N, goes with
the scalars 3 and 4, not with i and j.) Second, the
dimensionless quantities, such as v/c, are scalars;
the operation of division does not apply to vectors
pointing in different directions.
So what does the word "unit vector" mean? One unit
of what? F=1j is a vector whose length is 1 N, and
which is directed along the y axis. Likewise F=1i is
a vector. But i and j alone are not vectors. A vector
quantity may have many different directions; j is
always pointing along the +y axis.
I am not proposing to change anything. Only to
recognize that the word "unit" means a different
thing here. Would you agree? We multiply scalars
by i, j and k in order to create vectors. What would
be a better name for i, j and k?
Ludwik Kowalski