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

[John Denker <jsd@MONMOUTH.COM>]

At 06:04 AM 10/1/99 -0400, David Bowman wrote in part:
> As far as the cleanness and elegance factor goes, I have to agree with
> ... the superiority of the axiomatic abstract algebraic approach

OK, cool.

> 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.

I think there are two questions here:
 * what is a vector, and
 * what role do vectors play in physics.

The same can be said of complex numbers:
  * what is a complex number, and
  * what role do complex numbers play in physics.

Indeed complex numbers were known to mathematicians for decades before they
played any significant role in physics.

For vectors the history is the other way around -- the applications
preceded the abstraction -- but conceptually what is true for complex
numbers is also true for vectors:  if vectors had no physical applications
they would still be vectors.

> Nothing in the [axiomatic]
> development seems to explain the *physical meaning* of the directional
> properties of vectors in *physical* space.

True.  The physics of vector quantities needs to be explained separately
from the definition of vectors.

There are many types of physics-vectors, and still the physics-vectors are
a small subset of all the vectors that we know about.

> 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 tangent vectors are a subset of the physics-vectors, and therefore a
tiny sub-subset of all the vectors that we know about.  They must be taught
as *examples* of vectors, not to be confused with the *definition* of vectors.

(As a specific illustration:  Isospin is a vector, but it is not
particularly intuitive to think of it as a tangent, or a motion, or even an
impending motion.)

It's also worth noting that in many cases vectors, as axiomatically
defined, only *approximately* describe real physics.  Consider the
geometric addition of distance vectors using the parallelogram rule.
Geometry was invented, as the name suggests, to measure the fields in
Egypt.  If you send real surveyors to lay out a large figure with four
equal-length sides and four interior angles of 90 degrees, the figure won't
exactly close!  That's because the "horizontal" land in Egypt is curved.

It is only in certain limits (e.g. small figures) and/or by the use of
mathematical abstractions (e.g. tangent spaces) that we can make contact
between vectors and real physics.

==============

And since the question asked *especially* about the physics of i, j, and k
in particular, there is a simple answer:
        THERE IS NO PHYSICS of i, j, and k.
The physics does not care about the choice of basis.  Any equation that
purports to predict otherwise is just wrong.

______________________________________________________________
copyright (C) 1999 John S. Denker             jsd@monmouth.com