Re: i,j,k -- OK, how do the students get it?

["Prof. John P. Ertel (wizard)" <jpe@NADN.NAVY.MIL>]

I've stayed out of the fray on this one just to see what would come
forward.  The question below, however, gives me cause to step up to the
plate.  I really don't think I want to complicate the concepts of "vector"
and "unit vector" in students minds.  In all my courses, I simply remind
my students that "true vectors" (as opposed to "pseudo vectors") are
things that have magnitude and direction.  The magnitude contains both the
"amount" of the vector and its units.  The direction is contained by a
unit vector which has magnitude of ONE (hence its name).  As an example,
the instantaneous velocity is a vector while the instantaneous speed is
its magnitude.  The direction of the velocity is contained in a unit
vector that is instantaneously tangent to the trajectory and, in fact,
defines the trajectory at any instant:
        V(vector) = v(scalar) * ev^(v-unit vector)
If I ever go on too long about the meaning of vectors and unit vectors, I
find I weaken (rather than strengthen) their understanding.  I find the
preceding simple explanation together with (at most) one other example
        R(vector) = r(scalar) * er^(r-unit vector)
and some quiet time for the student to think and absorb the concept to be
the best solution.

This definition also does away with any need to talk about specific axes
when dealing with the concept of unit vectors.  The i,j,k-axes may be
helpful specific examples, but I find  e1^, e2^, e3^ to be more helpful to
the student.  They are not taught at the very beginning to artificially
limit their definition of unit-vectors.

BTW: As a very young man, I was taught to call r-unit vector by the name
"away-hat".  This name has always seemed to convey the full meaning of the
r-unit vector with no other explanation required!


On Fri, 17 Sep 1999, 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*?
> >
> > 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?
> >
> > David Bowman
> > dbowman@georgetowncollege.edu

+=================================+=================================+
ERTEL SENDS.
                               /^\-/^\
                              /       \
                             |    *    |
                             |#########|
                     ===========================
                       / Prof. John P. Ertel \
                      / USNA-NESA Faculty Rep \
                     /   veteran Eagle Scout   \
 +==============================================================+
 |     JPE@nadn.navy.mil         or         JPErtel@BigFoot.com |
 |   Physics Department, 9C               Office : 410-293-6657 |
 |     Michelson  Rm-338                     DSN :     281-6657 |
 |      572 Holloway RD                      FAX : 410-293-3729 |
 |    U. S. Naval Academy                                       |
 |   Annapolis, MD 21402-5026          HomeVoice : 410-757-6618 |
 | {Call HomeVoice to activate FAX}      homeFAX : 410-757-6619 |
 +==============================================================+
     http://physics.usna.edu/physics/Faculty/Ertel/ertel.html
            http://physics.usna.edu/ECAs/NESA/nesa.htm