Re: [Phys-l] Dot product (was nit vector notation)

[John Denker <jsd@av8n.com>]

On 08/18/2007 10:02 AM, Michael Edmiston wrote:
> Stimulated by the unit vector thread, I was reading John Denker's discussion 
> at http://www.av8n.com/physics/intro-vector.htm and would like to question 
> John (and others) about the dot product definition.
>
> I agree with and appreciate John's comments that the dot product of vectors 
> A and B should be defined as ABcos(theta).  What I question is John's 
> comment that A dot B should not appear in physics textbooks as AxBx + AyBy + 
> AzBz, unless what he specifically means by "not appear" is that it should 
> not appear as a definition.  I think it should appear as one way to evaluate 
> A dot B, and indeed can then be used along with the definition of A dot B to 
> find the angle between A and B.
>
> When I am discussing the dot product in my physics class defined as 
> ABcos(theta), the question inevitably crops up "how do we find theta?"  This 
> is especially a common question in three-dimension situations.
>
> It seems to me that showing there is a way to compute the dot product by 
> choosing a basis, expressing the vectors using that basis, then applying the 
> AxBx + AyBy + AzBz procedure is a worthwhile thing to teach and can 
> appropriately appear in physics books.  At the same time we can also discuss 
> why ABcos(theta) is a good definition but AxBx + AyBy + AzBz is really a 
> "computational procedure" (rather than a definition).
>
> Comments?

First comment:  This is a deep, incisive question.  It cuts 
close to the core of what "geometry" is, and hence what
"physics" is.

I can't give an answer deep enough to match the question,
so you'll have to settle for a partial answer.  In recent
years I've gradually learned more about this, and I have
a healthy respect for how much I still don't know.

Second comment:  This is another of those contrapuntal discussions
where some messages focus on how to introduce the topic to
naive students, while other messages focus on the ultimate
most-precise most-sophisticated formalism.  That's all OK
by me;  we just need to keep track of which is which.  Also,
knowing the ultimate answer often helps decide which of two
otherwise-comparable introductory approaches should be preferred.

In more detail:

1) One important, clear, quantitative line of argument says 
that we shouldn't *define* the dot product as 
    A·B := AxBx + AyBy +  AzBz              (1)
because that doesn't work for relativity.  Instead we have
    A·B := AxBx + AyBy +  AzBz - AtBt       (2)
with a _minus sign_ in front of the timelike term.

I have no objections to rewriting equation (1) as a simple
equation instead of a definition.  We can then see it as
a corollary of some deeper definition (as discussed below).
The corollary is valid only under certain restrictive 
conditions.

  I am quite aware that in computer science (as opposed
  to physics) they commonly define the dot product in
  accordance with equation (1).  This is a problem.  It
  is probably not fixable in the short term.

2) There is another line of argument that has to do with 
the role of Cartesian analytic geometry.  We all know 
that analytic geometry is very powerful.  Students get 
seduced by it, and they develop the misimpression that 
points are *defined* by their coordinates, and that 
analytic geometry "is" geometry.  They forget that 
geometry was a science for 2000 years before Descartes
was born.  Galileo did physics to the highest standards
without using analytic geometry.  His proofs are quite
formal and precise, in the Euclidean style.  There is no 
evidence that Galileo ever wrote an equals-sign in his 
entire life (and he may never have even /seen/ an equal
sign).

   _Discourse on Method_ was published in 1637, so
   Galileo can hardly be faulted for not knowing about
   it while preparing _Two New Sciences_ which came out
   in 1638.

I do not think points are *defined* by their coordinates,
but rather the reverse.  Given a point, and a reference
frame, you can find the coordinates.  Given the same
point and a different reference frame, you obtain a
different set of coordinates for the _same_ point.

This is not just philosophy or metaphysics;   this has
direct practical implications.  It is important to be
able to see points from more than one point of view,
and to quantify points using more than one reference
frame.  The idea that the laws of physics are frame-
independent idea goes back to Galileo.

  I'm not making a historical point here;  IMHO there
  is no pedagogical importance to the fact that Galileo
  came before Newton ... but IMHO Galileo's laws are
  more important than Newton's laws.  And they are more 
  appropriate to an introductory course.   
   -- Scaling laws
   -- Frame independence
   -- Conservation laws

This notion of frame independence is central to
thermodynamics.  Sometimes you know the energy as
a function of P and S, and sometimes you know it
as a function of P and T, or V and S, or whatever.
Now E is a function of state, but P and V and T
and S are also functions of state!  Here _state_
has the same meaning as _point_ did above.  If
you know the state, you can find E,F,G,H,S,V,P,T
et cetera as a function of _state_.

This becomes clear when you formulate thermodynamics
using even the most basic ideas of differential geometry, 
which allows you to detect /and cure/ a huge number of 
fallacies that are endemic in the usual presentations 
of thermodynamics.  I think it is good pedagogy when 
reformulating something forces fallacies to disappear, 
especially when it can be done without appreciable
extra effort.

It seems to me this is good mathematics (differential
topology) and good pedagogy ... so it is not mere
philosophical or metaphysical opinion.

As I have said before, students' understanding of vectors
is regrettably non-monotone.  

Students’ proficiency with geometrical vectors seems to be 
a regrettably non-monotonic function of their overall 
sophistication:
  a) They start out with a gut feeling about geometrical 
   vectors. In early grades they learn to draw vectors as 
   arrows. They add them graphically, tip-to-tail. 
  
  b) Later they learn about arrays of numbers, matrices, 
   and all that. They add components and array elements 
   numerically, component by component. (This assumes 
   the representations are all based on the same basis; 
   otherwise they shouldn’t be added at all.) So far, so 
   good.
 
   The problem is that all too commonly, their knowledge of 
   components eclipses their understanding of geometrical 
   vectors.

  c) Finally when they become truly sophisticated, they start 
   making use of geometrical vector concepts again.  They learn 
   to appreciate the distinction between a tensor and the array 
   elements of that tensor. 


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

The cleanest way I know to define the dot product is to 
/postulate/ the existence of three vectors in space (or
four vectors in spacetime) for which we know the dot 
products.  This is spelled out in a new section
  http://www.av8n.com/physics/intro-vector.htm#sec-dot-product

> the question inevitably crops up "how do we find theta?"  

That question should crop up!  :-)

One advantage of the axiomatic approach to the dot
product is that it allows the dot product to define for
us what we mean by angle.

Another huge advantage is that this approach generalizes
from space to spacetime, making relativity incomparably
less mysterious.