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

[Stefan Jeglinski <jeglin@4PI.COM>]

> 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*?

I don't know if this helps, but at least it gives me an opportunity
to recommend a delightful book I just finished reading: "An Imaginary
Tale: The Story of sqrt(-1)," by Paul J. Nahin.

The book is full of interesting history and also non-trivial but not
excessive math. I found many of the mathematical treatments more
concise and satisfying than what I was exposed to in grad school
texts! Although I have been proficient at -using- complex numbers for
quite a while, some concepts I have only now grasped for the first
time, but I sha'n't repeat them here due to embarrassment :-)

At any rate, the question has long been posed: how to geometrically
interpret sqrt(-1)? The answer is so simple as to be still somewhat
unsatisfying in a way: that the sqrt(-1) is a rotation operation. The
book describes it as a 90 degree rotation off the real axis, but
saying a rotation to a direction orthogonal to the real axis might be
a better way to put it IMHO.

That it is a rotation seems trivial to anyone who uses complex
numbers in the r*exp(i*theta) form, but the important point is that
the real and imaginary axes form a legitimate coordinate system, in
which the common notion of direction does not make complete sense but
which still is the best way to try to describe it (IOW, the direction
of the axis of real numbers and the direction of the axis of
imaginary numbers). Here, sqrt(-1) is used as a *bookkeeping*
variable when writing vectors in this coordinate system. I think it
is a very useful exercise to consider the real and imaginary axes as
a coordinate system with directions. It helps free the mind from
demanding that directions be strictly "worldly."

So I might suggest that i,j,k in the current thread are really
bookkeeping variables that describe nothing more than how one
direction is orthogonally rotated with respect to another. The
directions themselves can be NSEW or xyz based, but these are only
special cases. Some directions have to be "imagined."

I know this does not still answer the question "What does pointing in
a direction *mean*?" but stressing the bookkeeping viewpoint may help
avoid the -necessity- of trying to understand the 'unit' nature among
other things. Bookkeeping allows you to draw the parallel that you
cannot add feet and seconds, for example. Feet and seconds are
nothing more than dimensional bookkeeping variables to keep the user
honest. Not to get too nutty here, but the minus sign is also a
bookkeeping variable, determining the direction one moves on a number
axis.

Laying all this on a student, especially if they are only in high
school and are having a hard time grasping the concept, may be a
mistake, but perhaps someone here can create something new out of it.
Oh well. Perhaps you can tell students we are really just
accountants. With attitudes.




Stefan Jeglinski