Cross product of 3 1-vectors in 4-space...

Higher-dimensional geometry (previously "Polyshapes").

Cross product of 3 1-vectors in 4-space...

Postby Paul » Thu Nov 18, 2004 6:01 am

Hello Pat and all,

Actually, I only stumbled on that confusing issue concerning the bivector index ordering with the 3D cross product because I was having trouble finding the component formula for the cross product of 3 1-vectors in 4-space... and so, I thought I'd look at the relation between the wedge and cross products in 3D to see if that might give me some hints about what I might be doing wrong in deriving the component formula for the cross product of 3 1-vectors in 4-space.

At this time, I still don't think I've resolved the issue concerning the bivector index ordering with the 3D cross product, but I do think I've found the correct component formula for the cross product of 3 1-vectors in 4-space.

Also, in regards to various products... It occurred to me that I might be able to say that all implied products (those without explicit operators) are geometric products. It seems the issue might be somewhat confused, but the geometric product of two scalars may just be their ordinary scalar product.

If I read correctly, it appears that the operation table for the geometric product on this webpage... Multivector Methods might support the argument that the geometric product of two scalars is just their ordinary scalar product.

However, the operation table for the wedge product (on the same webpage) would seem to indicate that the wedge product of scalars also gives the ordinary scalar product. If one is utilizing an inner product that gives 0 for all scalar operands, then this scheme would seem to work.

On the other hand, if one chooses an inner product returning the scalar product for scalar operands, then one could use the "thin" outer product.

Of course, later on here, I also use the dot to indicate both the inner product of vectors and the scalar product.

It might better if I approached the problem from another perspective. How about if I say to assume all operands are scalars, unless indicated otherwise. Everyone knows basis multivectors aren't scalars, so... maybe this'll work.

Anyway... As I said, I really set out to derive the component formula for the cross product of 3 1-vectors in 4-space,... which, if I understand correctly, will yield a 1-vector normal to the 3D-realm of the 3 1-vectors.

I think I've now accomplished that. However, it was a real pain to get all the formalism correct.

Image

I tested it on these 3 1-vectors, and it seems to work...

Image

Of course, even if I have derived the component formula correctly, it'd probably be more useful to have a more general formula for finding the normal to a 3D-realm from 4 arbitrary points of 4-space.

Also... I'm not certain how this component formula will behave if the 3 1-vectors happen to be corealmic. Does anyone know how this component formula will behave if the 3 1-vectors are corealmic...?


Also... one other question has been on my mind. Can the norm of the wedge product of two blades be described geometrically as a 'base' equal to the hypervolume of one blade multiplied by a 'height' equal to the hypervolume of the other blade...?

Of course, even if this geometric interpretation might have some validity, I'm dubious it could be extended to general multivectors...?
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Postby Paul » Thu Nov 18, 2004 12:25 pm

Hello Pat and all,

Does anyone know how this component formula will behave if the 3 1-vectors are corealmic...?


Sorry to be so dense again... especially after all you've been doing with the wedge product, Pat. Correct me if I'm wrong, but we know the 3 1-vectors are corealmic being that their wedge product is zero...
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Postby pat » Thu Nov 18, 2004 10:02 pm

Paul wrote:Correct me if I'm wrong, but we know the 3 1-vectors are corealmic being that their wedge product is zero...


I haven't caught up on all of the above yet... but this statement is quite correct.
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Postby pat » Thu Nov 18, 2004 10:10 pm

I read the page you linked to about wedge products for stringed instrument tuning. Intriguing....

For my purposes, I find it easier to just keep track of the sign as I go along through the canceling, reordering process.
1341234
-1314234
1134234
34234
-43234
42334
424
-442
-2

more later,
Patrick
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Re: Cross product of 3 1-vectors in 4-space...

Postby pat » Thu Nov 18, 2004 10:24 pm

Paul wrote:It occurred to me that I might be able to say that all implied products (those without explicit operators) are geometric products. It seems the issue might be somewhat confused, but the geometric product of two scalars may just be their ordinary scalar product.


I'd agree with that. The geometric product of two scalars is just the product of those two scalars: a1 * b1 = (ab)1*1 = (ab)1.

Of course, even if I have derived the component formula correctly, it'd probably be more useful to have a more general formula for finding the normal to a 3D-realm from 4 arbitrary points of 4-space.


This is a normal to a plane which goes through the origin and each of these three points. It also happens to be normal to all parallel realms. If you want to start with four arbitrary points: a, b, c, and d, then you can just calculate a' = a - d, b' = b - d, c' = c - d and use the above with a', b', and c'.

Or, if you really want to go nutty, you can just take your above forumula, and where you have x<sub>j</sub> where x is one of { a, b, c } and j is one of { 1, 2, 3, 4 }, you can substitute in ( x<sub>j</sub> - d<sub>j</sub> ). The very magic of linear operators!

Also... one other question has been on my mind. Can the norm of the wedge product of two blades be described geometrically as a 'base' equal to the hypervolume of one blade multiplied by a 'height' equal to the hypervolume of the other blade...?


I'm gonna have to read that a few more times before taking a stab.
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Postby Paul » Thu Nov 18, 2004 10:40 pm

Hello Pat,

This is a normal to a plane which goes through the origin and each of these three points.


I'm pretty sure you meant realm here, not plane... correct?
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Postby pat » Thu Nov 18, 2004 10:43 pm

Paul wrote:I'm pretty sure you meant realm here, not plane... correct?


Ah, yes.... in my mind... planes are always of dimension (n-1). I try to say hyperplane to avoid confusion.... but.... alas....
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Re: Cross product of 3 1-vectors in 4-space...

Postby pat » Thu Nov 18, 2004 10:48 pm

Paul wrote:Also... one other question has been on my mind. Can the norm of the wedge product of two blades be described geometrically as a 'base' equal to the hypervolume of one blade multiplied by a 'height' equal to the hypervolume of the other blade...?


Only if the two blades don't share component vectors, right? That is... only if the two blades are orthogonal. Or, do your 'base' and 'height' in quotes sort-of imply that each is two-dimensional... and that your hypervolume is four-dimensional... and so you'd expect zero if your 'base' and 'height' didn't fan out to a full four-dimensions?

I ask, because you could do this same thing even if the 1-vectors in your space were more than four-dimensional.
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