How Does the Cross Product Work in Geometric Algebra?

[krcmd1]

geometric algebra cross product

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Homework Statement [/b]

my text (Geometric Algebra for Physicists, by Doran and Lasenby), p. 69, deals with rotating frame {fsubk} (I assume in 3D)

d/dt (fsubk) = omega X fsubk omega being angular velocity

then

omega X fsubk = (-I omega) dot fsubk = fsubk dot (I omega), where I is pseudoscalar and (I omega) is I think the geometric product of I and omega.

I don't understand these algebraic steps. Can someone explain?

I did follow earlier explanations of why with vectors a,b that a X b = -I(a wedge b)

If I have posted this to the wrong forum, would the moderators kindly forward it to a more appropriate one?


Thank you all very much!

Ken Cohen

 

[pkleinod]

$$\omega \times f = -I (\omega \wedge f) = -(I\omega)\cdot f$$

The first equality is just the definition of the cross product as the dual of
a wedge product (outer product). The next
step invokes a duality relationship between the dot product and the outer
product: the dual of the outer product of a blade A with blade B is
just the dot product of the dual of A with B (Do you know about this?).
On the left of the equals
sign we see the dual of an outer product and on the right we have the
dual of one of the factors dotted with the second factor. For the final
step, use the general relationship

$$A_r \cdot B_s = (-)^{r(s-1)} B_s \cdot A_r$$ for $$r \leq s$$

Identify the vector f with A_r and the other factor with
B_s, so that s=2 (bivector!) and r=1 The sign is then -1:

$$-(I\omega)\cdot f = f \cdot (I\omega)$$.

 

[krcmd1]

Thank you very, very much.

I am struggling to teach my self math to understand physics, and it's a lonely stumbling effort.

what program do you use to enter notation?

the equality of the dual of outer product equalling the dot of the dual was not presented as far as I can tell up to that point in the text I'm using.

do you think I'm barking up the wrong tree with geometric algebra entirely?

thanks, again.

 

[pkleinod]

what program do you use to enter notation?

This forum allows an implementation of TeX. You just sandwich the TeX expression
between two markup symbols: (tex) ab = a \cdot b + a \wedge b (/tex) to get

$$ab = a \cdot b + a \wedge b$$.
But use square brackets [] instead of round brackets (). You can also click on an equation and a window pops up
showing you how it is done. For more info: https://www.physicsforums.com/misc/howtolatex.pdf

the equality of the dual of outer product equaling the dot of the dual was not presented as far as I can tell up to that point in the text I'm using.

Look for section 4.1.4 on pages 96 to 97 of your book. Indeed, skip to chapter 4 and
read it right away. I think it will answer a lot of your questions. You can always come
back to where you are. Chapter 4 is pretty well self-contained.

do you think I'm barking up the wrong tree with geometric algebra entirely?

Not at all! For all of physics that has to do with vectors (mechanics, spacetime,
special relativity, electrodynamics) and for other topics (point groups, lie algebra,
conformal geometry, computer graphics, robotics), geometric algebra offers
a better understanding and provides better tools for attacking problems. Anything to do with rotations is best done with GA (quaternions are just the even-grade sub-algebra
of 3D space). Once you have the GA perspective on something like electrodynamics,
you will never look back. There is also a lot of baggage that you can forget about: e.g.
the vector cross product, axial as opposed to polar vectors. On the other hand, some
topics, such as quantum mechanics, although they can be approached by GA, would be
very difficult to learn entirely from the GA point of view, in my opinion. Also, although
GA is very valuable, you will not readily find others willing to talk physics in this language.
Which brings me to your remark:

I am struggling to teach my self math to understand physics, and it's a lonely stumbling effort.

Essentially that is what I am doing, and I find it lonely as well. There are three other
very good books that I can recommend but they are very expensive; however, I don't know what I would do without them:

"Clifford Algebra to GA" by David Hestenes and Garret Sobczyk
(Kluwer, 1984, reprinted in 2002)
This has it all but is not an easy read.

"New Foundations for Classical Mechanics" (second edition)
David Hestenes, (Kluwer, 1999, reprinted in 2003)
A great deal on GA in addition to a wonderful treatment of mechanics.

"GA for Computer Science" by
Leo Dorst, Daniel Fontijne, and Stephen Mann
(Morgan Kaufmann, Elsevier, 2007)
A very good introduction with many things spelled out. It
includes many exercises and an internet web-site with software
for viewing many of the figures dynamically and interactively:
http://www.geometricalgebra.net/

Good luck in your endeavours. If you have specific questions to me, why not
send me a private communication within physics forums---I think this is possible.

 

[krcmd1]

You are very generous. I will!