I'm sorry about the delayed response on this thread, however, ...
Thanks to all those who responded to my request for a student-
understandable explanation for the 'real meaning' of unit/basis
vectors in particular, and the 'real meaning' of "pointing in a
direction" in general for vectors in physical space. The advice offered
included John Denker's advocacy (endorsed by Joel Rauber) of the
axiomatic abstract algebraic approach (A^4?), Stefan Jeglinski's
suggestion, based on an analogy with complex numbers, of considering
i,j,k as mere placeholding or "bookkeeping variables that describe
nothing more than how one direction is orthogonally rotated with respect
to another", and John Ertel's suggestion of breaking up a vector--
illustrated by a *velocity* vector--into a product of a scalar magnitude
and an an associated unit vector whose direction is defined by the local
tangent to the 'motion'. Unfortunately, but probably understandably, the
whole body of advice is mutually conflicting. For instance, John D's
comment about the A^4 approach, "I consider it more general, more
elegant, and less trouble than other approaches, including the
differential-geometry approach, and the very troublesome
vector-is-a-list-of-numbers approach" seems to be at odds with (and calls
"very troublesome") the 'bookkeeping philosophy' as illustrated by
Stefan's suggestion: "Perhaps you can tell students we are really just
accountants. With attitudes.".
As far as the cleanness and elegance factor goes, I have to agree with
John D. about the superiority of the A^4 way of doing things.
Unfortunately, it doesn't seem to do much for actually answering the
student's question about the "real meaning" of the vectors (esp. i,j,&k)
and their directions we use in physics. A^4 is is *so* abstract (and
this is where its beauty and power come from) that is applies equally to
any abstract vector space defined over the reals that happens to be
endowed with a bilinear norm-inducing inner product. Nothing in the
development seems to explain the *physical meaning* of the directional
properties of vectors in *physical* space. The abstract approach is
just as well at home describing vectors in physical space as in
describing the abstract vectors in any Hilbert space. This approach is
useful when trying to get students to see the common algebraic properties
of vectors and their operations in physical space with the corresponding
properties and operations some other space such as a mathematical space
of functions or the Hilbert spaces of quantum theory, or some other state
space used in fields ranging from economics, to control theory. As
elegant as the A^4 method is, it doesn't seem to get at the *meaning* of
physical vectors and their directional properties in the *physical space*
of physics (which sort of acts as the spatial scaffolding or substrate
on/in which physical events happen).
Similarly, the 'bookkeeping' approach runs into a similar problem in
answering the original question. It seems to me that the main difference
between the 'bookkeeping' approach and A^4 is that A^4 defines the space
*including* its endowed inner product structure, and the 'bookkeeping'
method constructs and imposes that structure later *after* the space is
defined in terms of its place-holding basis elements. This gives the
latter approach an inelegant dependence on the original basis used to
define the space.
I kind of like John E.'s explanation in terms of the local tangent of
the motion for a velocity, and, presumably, the local tangent to an
'impending motion' of for some other kinds of vectors. Such an idea
does seem to sort of translate the differential geometry ideas about the
meaning of vectors on a geometric manifold into something understandable
to a student.