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*From*: John Denker <jsd@av8n.com>*Date*: Mon, 06 Oct 2014 12:24:15 -0700

On 10/06/2014 11:45 AM, Jeffrey Schnick wrote:

It would be better to say that being closed under multiplications

means that the cross product of any two vectors, each of which is an

element of the set, is also an element of the set.

Right.

I would add that contrary to the wording of the subject

line, no vector is "closed". It is the /set/ that is

closed (with respect to a given operation).

Closure is a concept that comes up a lot in /group theory/.

However, in the context of the cross-product operator, a

closed set of vectors does not form a group. The group

axioms require an identity element, which is lacking here.

==============

Whenever you see a cross product, you should ask yourself

if the physics would be better expressed in terms of a

/wedge/ product. The answer is almost always "yes".

If you do that, you can set up a system that is not only

closed but forms a group. In three dimensions, a basis

for the group will contain

-- one scalar grade 0

-- three vectors grade 1

-- three pseudovectors grade 2

-- one pseudoscalar grade 3

for a total of 2^3 blades, which is the right answer.

To this way of looking at it, a set consisting of grade=1

vectors is /never/ closed under the wedge product operation,

because the wedge product of two vectors is a grade=2

pseudovector. On the other side of the same coin, the

set of grade=1 vectors is never closed under the dot

product operation either, because the dot product of two

vectors is a grade=0 scalar.

If you include all 8 blades, the set is closed, and forms

a group. Forsooth it's more than a group, it's an /algebra/

i.e. Clifford algebra.

===========

Very unlike the cross product, the wedge product generalizes

to any number of dimensions, not just 3 dimensions. There

are tremendous applications in d=2 and d=4. For a discussion

of why this is useful, see

https://www.av8n.com/physics/clifford-intro.htm

**References**:**[Phys-L] carbon wars***From:*John Denker <jsd@av8n.com>

**Re: [Phys-L] carbon wars***From:*"Folkerts, Timothy J" <FolkertsT@bartonccc.edu>

**Re: [Phys-L] carbon wars***From:*John Denker <jsd@av8n.com>

**Re: [Phys-L] carbon wars***From:*Ze'ev Wurman <zeev@ieee.org>

**Re: [Phys-L] carbon wars***From:*"Folkerts, Timothy J" <FolkertsT@bartonccc.edu>

**Re: [Phys-L] carbon wars***From:*John Denker <jsd@av8n.com>

**Re: [Phys-L] carbon wars***From:*"Folkerts, Timothy J" <FolkertsT@bartonccc.edu>

**Re: [Phys-L] carbon wars***From:*"Craig Lucanus" <lucanus@iinet.net.au>

**[Phys-L] closed vectors***From:*Paul Lulai <plulai@stanthony.k12.mn.us>

**Re: [Phys-L] closed vectors***From:*Jeffrey Schnick <JSchnick@Anselm.Edu>

**Re: [Phys-L] closed vectors***From:*Jeffrey Schnick <JSchnick@Anselm.Edu>

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