Cross product in exterior algebra...?

Higher-dimensional geometry (previously "Polyshapes").

Cross product in exterior algebra...?

Postby Paul » Tue Nov 16, 2004 9:46 pm

I tried posting this to the Math Message Board, but I'm not getting any responses there... :(

I don't use the Clifford, or Geometric product, here, so when there's no operator I mean apply ordinary multiplication of scalars.

Image

I found this rather easy way to resolve wedge products here... wedge-product (This is why I sometimes drop explicitly stating the basis vectors) The author's method appears to work, and he doesn't use the basis vectors explicitly... however, if the subscripts of the vector components themselves stand for the basis vectors, then having e<sub>31</sub> be the <i>default</i>? basis bivector would mess his system up,... or at least, his system loses it's correspondence to the basis vectors...?

I thought Hamilton choose the basis bivectors for the quaternions because he decided he wanted a left-handed basis... which makes ijk = -1, as he desired. However, he also has the negative signs, which is equivalent to one additional transposition...?

I'm confused... how do I justify using e<sub>31</sub>, instead of e<sub>13</sub>...?

Can someone go over this is some detail... it seems like this could be of critical importance. How do I know how to choose the permutation ordering of basis bivectors, trivectors, etc. if I go to 4, or more, dimensions?
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Postby Paul » Tue Nov 16, 2004 9:57 pm

Also... it just occured to me that the circular permutation rule for the cross product, i.e. 123, 231, 312, also implies that the difference of vector components should be (a<sub>3</sub>b<sub>1</sub> - a<sub>1</sub>b<sub>3</sub>)e<sub>31</sub> ...

But, if I understand the rules for resolving wedge products on that webpage correctly, i.e. wedge-product, shouldn't a<sub>3</sub>b<sub>1</sub> be the term negated?
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Re: Cross product in exterior algebra...?

Postby pat » Wed Nov 17, 2004 5:25 am

Paul wrote:I tried posting this to the Math Message Board, but I'm not getting any responses there... :(


Well, I'm not sure what you're asking. I'll have to re-read... but this strikes me:

I don't use the Clifford, or Geometric product, here, so when there's no operator I mean apply ordinary multiplication of scalars.


But, right around the middle (and again after that) you write:
( e<sub>12</sub>e<sub>123</sub> + e<sub>13</sub>e<sub>123</sub> + e<sub>23</sub>e<sub>123</sub> )

Those aren't scalars. It looks like the multiplication you're using there *is* the Clifford product. e<sub>j</sub>e<sub>k</sub> = -e<sub>k</sub>e<sub>j</sub> for j not equal to k and e<sub>j</sub>e<sub>j</sub> = 1.
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Re: Cross product in exterior algebra...?

Postby pat » Wed Nov 17, 2004 5:33 am

Paul wrote:How do I know how to choose the permutation ordering of basis bivectors, trivectors, etc. if I go to 4, or more, dimensions?


I think the answer here is that you don't. Take a look at the ordering identities for the Octonions. They're a very odd sort of cylic. I'd have to turn the light on in my son's room to track down my copy of On Quaternions and Octonions by Conway and Smith. It's an excellent book which summarizes the relationships of the basis vectors of the Octonions more simply than I've seen anywhere else. But, the thing about them are that they're not cyclic. There are four separate cycles (IIRC).
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Postby Paul » Wed Nov 17, 2004 11:38 am

Hello Pat,

Thanks for your response.

Yes, Conway and Smith's book On Quaternions and Octionions is an excellent book... I, too, have that book. I will reference it in regards to this issue.
(e<sub>12</sub>e<sub>123</sub> + e<sub>13</sub>e<sub>123</sub> + e<sub>23</sub>e<sub>123</sub>)

Yes, the products between the basis vectors are geometric products. I made a mistake.


Now, to respond to what may be the emotive portion of your response.

I appreciate all your responses. I don't work at a job where I do anything to do with mathematics, nor programming... although I do have a BS in Computer Science. I really don't have anyone in real life these days to correspond with about my mathematical interests.

For me, at least for now, it's just a hobby. For many of you, it's also part of your work. I'd think you're more likely to get tired of it,... and someone like me is likely to sometimes make you feel worn-out. I too, however, get tired of it sometimes.

Anyway, perhaps it helps to understand my situation that may be correlated with my seemingly child-like interest here. Right now, all this is basically just a hobby for me.
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Postby Paul » Fri Nov 26, 2004 7:38 pm

Hello Pat and all,

I apologize to everyone for my denseness when I tried doing this several days again,... and apparently confused myself pretty well,... and perhaps some of you as well.

I believe I probably made a permutation transposition error. Using the modification Pat suggested, i.e. using a negative sign to indicate transposition changes, probably eliminated the error, and the resulting confusion. Pat's modification just makes it easier to do each change on each row, regardless of whether it's a transposition change... and still clearly track what's going on.

As you can see here, it appears all the rules I've learned for the 3D cross and wedge products do appear to work as I was taught:


Image
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