This discussion seems to allow me an opportunity to ask phys-l list
members how they would explain to an inquisitive student just what the
precise meaning of a unit vector *actually* is. I'm not talking here
about how does one *represent* a vector by drawing a picture of it, but
of its *meaning*. What do you tell an inquisitive student *what it
means* that a vector points in the x-direction with magnitude 1? Or even
(forget about the magnitude for now) just what *it means* that a vector
points in a given direction? What does pointing in a direction *mean*?
When I explained what I *really* mean by a basis vector such as i,j, or
k for a Cartesian coordinate basis (as it is understood in differential
geometry) in terms of partial differential operators acting as
directional derivatives (acting on the space of scalar fields defined on
the coordinated manifold) along a given coordinate direction when the
other coordinates on that manifold are held fixed, I seemed to get blank
stares rather than bright "light bulb" expressions of an "aha" insight.
How do others explain this so the students actually have some
understanding of the *real meaning* of the phrase that a given vector
"points in a given direction with a given magnitude"? Do you actually
try to explain it? Or do you just say that it is something which just
must be intuitively grasped and that short of any such intuition you can
just think about vectors in terms of their properties under addition,
scalar multiplication, etc., and save any deep understanding of them for
graduate school when they take a differential geometry or general
relativity course?