Re: vector products

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

"Robert J. Beichner" wrote:
> Chapter 0 will be a series of ³tools² and techniques that physics students
> should find useful.
...
> Vectors including components, adding/subtracting, and scalar & vector
> products

Ahhh, vector products.

Here's a suggestion that will frighten some people and delight
others:  If we're going to multiply vectors, let's do it right,
using
 ++ dot product and wedge product
as opposed to
 -- dot product and cross product.

As far as I can tell, the cross product is a mistake.  It's
like trying to eat soup with a fork.  The wedge product is
like eating soup with a spoon.  It works better.  Lots better.
It is not one whit more complicated.  Indeed it is easier for
the students in every way.

The only drawback is that some teachers will be scared off by
the wedge product, if they already know about cross products
and are afraid to learn anything new.

Note that in the first chapters of the book, it is not
necessary to introduce the full Clifford Algebra formalism.
Instead, just go through and replace every instance of
v cross r with v wedge r.  It works the same.  It is
anti-commutative just like the cross product
        v wedge r = - r wedge v
and its magnitude is |v| |r| sin(theta).

In these early chapters, only the diagram changes:  v wedge r
is diagrammed as an area in the plane, not as a vector that
sticks up perpendicular to the plane.

In later chapters, we can trot out the full Geometric Product
        A B = A dot B + A wedge B
for applications such as rotation operators and for allowing
an astonishing simplification of the Maxwell equations.

See
  http://www.monmouth.com/~jsd/physics/maxwell-ga.htm#sec-ga-general
and references therein.