[Phys-l] migrating cross product --> wedge product

[John Denker <jsd@av8n.com>]

On 09/06/2010 09:19 AM, Stefan Jeglinski wrote:

> Question: on your clifford page and its discussion of bivectors, why 
> do you (apparently) maintain what I would call a conventional view of 
> the dot product but eschew the term "cross product" in lieu of wedge 
> product (if I read it correctly)? 

An excellent question.  I am kicking myself for not having
already addressed this point on the web page.  I just now added
a couple of subsections:
  http://www.av8n.com/physics/clifford-intro.htm#sec-visualize
  http://www.av8n.com/physics/clifford-intro.htm#sec-peda-symmetry

Thanks for the question.

> It appears to be only because of 
> the (not unimportant) extension to higher dimensions (that is, all 
> cross products are wedge, but not all wedge are cross) not to mention 
> symmetry considerations. 

Not "only" for that reason.

> Fine, but I think the engineering proponents 
> of free and bound vectors are firmly grounded in 3 dimensions at 
> most.

OK.....

> The discussion of bivector, for them, is an unnecessary 
> complication, just as the distinction between free and bound has also 
> been in this discussion.

Maybe the higher-dimensional stuff is unnecessary, but there
is a lot more to the story than that.  As a minor point,
the bivector approach also works in *lower* dimensions, i.e.
two dimensions (unlike the cross product) so that is already
useful, but this is just the flea on the penguin on the tip
of the iceberg.

I prefer the geometric approach for the simplest of practical 
pedagogical reasons: I can get good results using wedge products. 
The more elementary the context, and the more unprepared the 
students, the more helpful wedge products are. I can visualize 
the wedge product, and I can get the students to visualize it. 
(This stands in stark contrast to the cross product, which 
tends to be very mysterious to students.)

Specifically: Consider gyroscopic precession. I have done the 
pedagogical experiment more times than I care to count. I 
have tried it both ways, using cross products (pseudovectors) 
and/or using wedge products (bivectors). Precession can be 
understood using little more than the addition of bivectors, 
adding them edge-to-edge.  For the picture on this, see
  http://www.av8n.com/physics/clifford-intro.htm#sec-addition
I can use my hands to represent the bivectors to be added, or 
(even better) I can show up with simple cardboard props.

Getting students to visualize angular momentum as a bivector 
in the plane of rotation takes no time at all. In contrast, 
getting them to visualize it as a pseudovector along the 
axis of rotation is a big production; even a bright student 
is going to struggle with this, and the not-so-bright 
students are never going to get it.

   Maybe this just means I’m doing a lousy job of explaining 
   cross products, but even if that’s true, I’ll bet there are 
   plenty of teachers out there who find themselves in the 
   same situation and would benefit from taking the bivector 
   approach. 

========

Another whole category of reasons has to do with symmetry.
See
  http://www.av8n.com/physics/pierre-puzzle.htm

I take very seriously the symmetry of the laws of physics.
IMHO this is the flesh and bone of physics.