Re: another way of corrupting the youth

[Arnulfo Castellanos Moreno <acastell@FISICA.USON.MX>]

An option to introduce GA if an student knows:
algebra, trigonometry, vectors and analytic geometry.

Given two vectors a, b in two dimensions with an angle
theta between them, you identify the paralelogram
between a and b. You can show that its area is
(norm of a)*(norm of b)*Sin(theta).

You define a geometric product such that any vector v is
operated as

(v)(v)=(v)^2 0 =v dot v
with "dot" the inner well known product

Doing v = a + b
and comparing (a+b)(a+b) with (a+b) dot (a+b)
a few of algebra will give you the condition

(a)(b) + (b)(a) = 2 a dot b

Watching this relation you will obtain two conclusions
1) a and b anticommute if a dot b = 0,
2) if a and b have the same direction, then a and b commute.

You can separate
(a)(b) = (1/2)[ (a)(b) + (b)(a)] + (1/2)[ (a)(b) - (b)(a)]
to identify a symmetric part and an antisymmetric one.

so (a)(b) = a dot b + a edge b, with a edge b the antisymmetric part.

Taking  i = (1,0) and j = (0,1) as your basis in two dimensions
and representing a= i*a_1 + j*a_2 and b= i*b_1 + j*b_2
where * is the usual multiplication between reals, you can do several
things:

A)
first: identify (i) edge (j) as a square with a direction
and (j) edge (i) as another square with the contrary direction.
The figures cited by John S. Denker are useful at this point.

B)
You can define I =  (i)(j) and show that (I)(I) = -1, without going
to complex numbers yet.

C)
You can show that operating (I)(a) you get a rotated by -Pi/2
so, you can define Exp(I*theta) to rotate the vector a by any angle.
Each one of the calculus is as easy as a multiplication in algebra, but
taking into account that (i)(j) = - (j)(i), (i)(i) = (j)(j) = 1.
You do not need a matrix to do it.

D)
You can see that the norm of a edge b is the norm of the paralelogram.

E)
After that, you can study angular momentum by defining
the bivector L = r edge p and the torque = r edge F.
Given the properties cited above. This can be seen in two dimensions
and associated to rotations.

Arnulfo Castellanos Moreno




----- Original Message -----
From: "John S. Denker" <jsd@MONMOUTH.COM>
To: <PHYS-L@lists.nau.edu>
Sent: Saturday, September 07, 2002 3:07 PM
Subject: another way of corrupting the youth


> As the saying goes, learning proceeds from the
> known to the unknown.
>
> So I wrote up a discussion of complex numbers in
> one column, compared to Clifford Algebra in the
> other column.
>
> I'm imagining a sequence where people learn in
> the following sequence:
>  -- plain old real numbers
>  -- vectors
>  -- complex numbers
>  -- Clifford Algebra.
> or perhaps
>  -- plain old real numbers
>  -- complex numbers
>  -- vectors
>  -- Clifford Algebra.
>
> The point being that most of the ideas in
> Clifford Algebra can be seen as non-shocking
> generalizations of ideas that are already
> known from complex numbers, with a little
> bit of vector technology thrown in.
>
> http://www.monmouth.com/~jsd/physics/complex-clifford.htm