Re: Displacement and position (was: displacement and graphs)

["John S. Denker" <jsd@MONMOUTH.COM>]

At 12:30 PM 10/10/01 -0500, Rick Tarara wrote:
> I suspect
> most people (including physicists) don't really think of position formally
> as a vector quantity.  We do realize that our position it is relative to
> some reference point and deep down that does imply a direction but that is
> not the way we really consider position (and almost certainly not the way
> students do).  As I sit here in South Bend, Indiana, I don't consider my
> location in terms of a direction (other than perhaps that this is Northern
> Indiana).  Late next week I will be in Chicago.  Again I have a rough
> feeling for the position of Chicago, but I really only put a strong
> directional notion to this when I consider going FROM South Bend TO Chicago.

The foregoing viewpoint is 100% correct.  It is actually more sophisticated
than I would have expected from the typical student, but anybody who makes
that argument gets 100% respect from me.

To reiterate:  Points without an origin are just points.  If/when you pick
an origin, there is an isomorphism between the space of points and the
space of vectors connecting the origin to those points.  This is quite a
sophisticated distinction, even trickier than the distinction between
"numbers" and "numerals".  We should not expect students to be able to
handle this distinction, but if they want to, more power to 'em.

> In other words, right or wrong, there is a strong tendency to
> consider location or position as scalar quantities and displacement as the
> vector quantity

That's going a bit too far.  The points are neither scalars nor
vectors.  They're just points.

If they were scalars, you would be able to add them and multiply them et
cetera.  But you can't.  Not by a long shot.

They are not scalars.  They are abstractions.

> In intro courses where students have no background in vectors
> and where one is NOT going to really teach vectors, this seems like a useful
> approach towards the concept of velocity.

I'm not sure about that.

Perhaps we can compromise on something like this:  The laws of physics are
structured in such a way that we never care about an abstract point, or a
point all by itself.  The only things that ever matter are things like
 -- the distance along the line AB from one point and another, and
 -- the angle between two such lines AB and AC.
Therefore we shall duck all questions about what a "point" is, and consider
only the separation vectors.  We will sometimes find it convenient to
choose an origin+basis, but this is never mandatory.