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Re: i,j,k -- & please sign your postings

I enjoyed the thoughtful response to David Bowman's question, but would like
to know from who it came. Not everybody's mail system ferrets out who
submitted the postings to Phys-L; my system only tells me that the message
came from Phys-L @ NAU. Therefore I would appreciate the courtesy of folks
signing their messages.

My personal answers to David's question is close to being along the lines of
below. I don't try to talk about basis vectors as differential operators to
my undergraduates; and not even that often to my 1st year graduate classes.
I save it for differential geometry or relativity courses. {which means that
I never get a chance to teach these days :-( , but that is another story}.
I remember when I was trying to make sense of the differential operator
approach to vectors; and finally only being able to understand when I was
able to make 1-1 correspondances of the algebra of combining the
differential operators to the "usual" algebra of combining vectors that I
had learned as an undergraduate. Which probably supports the idea written
below that "deep" understanding of vectors doesn't require differential
geometry.

I add, that in my actual teaching practice; I'm not nearly so axiomatic as
below: I very quickly introduce components and unit vectors (basis vectors
to the cognoscenti), so early, in fact, to my introductory students, that
they may not realize how arbitrary they are.

Joel Rauber





At 11:39 AM 9/17/99 -0400, David Bowman wrote:
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*?

Compared to the other questions you pose, that one seems
relatively easy.
As for "pointing" I would take a pencil or some such pointy
object and say
"Look, this is pointing at you. Now... It's not pointing at
you." An even
more graphic demonstration would require a wind-up toy:
"Look, it's walking
toward you. Now it's walking away from you."

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.

I'll believe that! I personally think differential geometry
is beautiful
and more fun that group theory --- but I'm not representative of your
students. And I think that coordinate bases are not the most
general, most
elegant, or most pedagogical way to think about vectors.

This also is related to the question Ludwik posed on Sat, 18 Sep 1999
02:32:02 -0400.

I first learned about vectors from a group-theory guy, so possibly I'm
prejudiced in that direction, but it also possible that
that's the right
direction. So let me describe the "axiomatic" approach:

-- We know that ordinary numbers can be added, subtracted,
and multiplied
by three.
-- But there are other things that can be subjected to the same
operations. The modular number systems are one example. Nowadays kids
learn about that in elementary school.
-- There are these things called vectors which can be added,
subtracted,
and multiplied by scalars. Vector addition can be represented by the
following picture.....
-- Beyond those operations, there is another fun thing we can do with
vectors: dot product. Usual axioms (bilinear, positive,
....). Triangle
inequality.
-- The dot product introduces the concept of angle between
vectors. A dot
B over norm A times norm B *defines* cosine to my way of thinking.
-- Given a vector you can make a unit vector by dividing by its norm.
-- We can write down projection operators that give the
projection of A
along B, and vice versa.
-- In some cases it is possible to write the identity
operator as a sum of
projection operators. Gram-Schmidt, blah, blah, blah.
!!!! Note that we can get this far without saying a single word about
coordinates or components or bases or axes !!!!
-- At this point you can point out that projecting onto a
basis (assuming
it exists) is equivalent to multiplying by an identity
operator, so it's
always allowed and sometimes convenient. The choice of basis
is arbitrary.

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?

It needs to be explained.

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?

Wow. Let's take that sentence apart.

Or do you just say that it is something which just
must be intuitively grasped

Kip Thorne likes to say that education is the process of
cultivating your
intuition. So yes, it needs to be intuitively grasped -- but
not "just"
intuitively grasped, and that's the goal of (not the opposite of) the
explanatory process.

you can
just think about vectors in terms of their properties under addition,
scalar multiplication, etc.,

Yes, that's the viewpoint I take, as sketched above. 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. And again, the
axiomatic view is not
at all inconsistent with intuitive understanding.

and save any deep understanding of them for
graduate school when they take a differential geometry or general
relativity course?

I say again a broad and deep understanding of vectors does not require
differential geometry. And it shouldn't wait until grad school.