When I was teaching introductory physics laboratory, I would tell my
students that there is a difference between _vector_quantities and
_vectors_. Vectors are mathematical entities with a definite "rules" of
combination while vector quantities are physical quantities.
It amazes me that these mathematical entities (vectors) can be used to
represent certain physical quantities (vector quantities) and such that
the mathematics of vectors can also be applied in the analysis of these
physical quantities!
Let me illustrate:
* in some quantities, a "pure number" (scalar) describes the "amount" of
units in a quantity: for an object having mass of M1 = (10) kg, the (10)
describes the number of kilograms in the object. The same is true for
another mass M2 = (20) kg.
* in other quantities, a vector is used to describes the units:
- for an upward force expressed as F1= (3j)newtons, (5j) is the
mathematical entity (the vector) that describes the newton units in the
force.
- if another force is also acting (horizontal this time), we may express
it as F2 = (4i) newtons, where (4i) is the mathematical entity describing
the newton units in this second force.
* in doing arithmetic, it is the mathematical entities that are
considered.
for summations: (as long as they have the same units)
(1) Msum = M1 + M2 = (10)kg + (20)kg = (10+20)kg = (30)kg
where usual algebra is done on the mathematical entities
(2) Fsum = F1+F2 = (3j)N + (4i)N = (3j+4i) N
In analyzing the summation of forces (eqn-2), the physical problem is
therefore reduced to a vector algebra problem : analyzing the vector
(3j +4i) that describes the newtons in the resultant force.
Therefore, I am supportive of the idea of representing the unit vector j
as having (1/4) the length of the vector used to _represent_ F1. I am of
the impression that once we use an arrow to represent the force, we are
_not_ drawing the "force" but the vector used to represent the force.
Afterall, an arrow is not a force but a representation of the vector. We
draw arrows to represent the vector algebra problem to which the physical
problem was reduced to.
In this context, there is no confusion for the unit vectors having no
physical units. They are not supposed to have units, being purely
mathematical entities. Forces have units; the vectors used to represent
the forces don't. The vectors describe the newtons in the force.
I am not sure if I am making sense, but I hope I do. Otherwise, pity the
students that i have taught in this manner.