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science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Sun Jul 16, 2006 10:54 am    Post subject: Re: Rotations - why are they not vectors

"Narcoleptic Insomniac" <i_have_narcoleptic_insomnia@yahoo.com> wrote in
message
news:7171135.1153043965570.JavaMail.jakarta@nitrogen.mathforum.org...
 Quote: On Jul 16, 2006 2:21 AM, Terry Padden wrote: mariano.suarezalvarez@gmail.com> wrote in message news:1153025545.663758.95150@75g2000cwc.googlegroups.com... Terry Padden wrote: I am bothered by the mathematics of rotations. It is I believe mathematically acceptable for any physical reality to be defined on an abstract axiomatic basis. Then anything that fulfills a given defining set of axioms for a type of mathematical object is a mathematically valid example of the defined mathematical object. Since rotations are just transformations on R^3, when you ADD rotations you're just looking at the composition of two or more transformations. Moreover, since these transformations can be represented as matrices the composition of them is definately NOT VECTOR ADDITION!!!

What has any of that mumbo-jumbo got to do with the AXIOMS -

do with R3.
Narcoleptic Insomniac
science forum Guru

Joined: 02 May 2005
Posts: 323

Posted: Sun Jul 16, 2006 9:58 am    Post subject: Re: Rotations - why are they not vectors

On Jul 16, 2006 2:21 AM, Terry Padden wrote:

 Quote: mariano.suarezalvarez@gmail.com> wrote in message news:1153025545.663758.95150@75g2000cwc.googlegroups.com... Terry Padden wrote: I am bothered by the mathematics of rotations. It is I believe mathematically acceptable for any physical reality to be defined on an abstract axiomatic basis. Then anything that fulfills a given defining set of axioms for a type of mathematical object is a mathematically valid example of the defined mathematical object. A set does not make a vector space. You B***** Fool. Who mentioned sets !!

YOU mentioned sets implicitly when you brought up vector
spaces!

 Quote: You need to specify two operations, addition and multiplication by scalars. The obvious ones you moron. You can ADD rotations; and you can SCALE rotations by a suitable Field. Idiot !!

Since rotations are just transformations on R^3, when you
ADD rotations you're just looking at the composition of
two or more transformations.

Moreover, since these transformations can be represented
as matrices the composition of them is definately NOT

 Quote: [snip] NB I am aware that 2-D rotations do-not-commute, but it seems to me that that has nothing to do with axiomatics or my questions. Actually, the composition operation on rotations (with a fixed center) in the plane *is* commutative. Yes you fool - but that is in 1-D ; and I wrote do-not-commute in 2-D. Can't you even read a simple sentence ??? You may want to review Halmos' presentation of what a vector space is, if you think that commutativity is irrelevant, by the way... NO ! I want you to go AWAY !!!!!!

Stop being such a douchebag.
William Elliot
science forum Guru

Joined: 24 Mar 2005
Posts: 1906

Posted: Sun Jul 16, 2006 9:33 am    Post subject: Re: Rotations - why are they not vectors

On Sun, 16 Jul 2006, Terry Padden wrote:
 Quote: "William Elliot" wrote in message On Sun, 16 Jul 2006, Terry Padden wrote: Now consider simple (= 1-D) rotations of a spherical object about any given fixed axis. You mean 3D rotations? NO, you fool.

Get some manners.
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Sun Jul 16, 2006 7:23 am    Post subject: Re: Rotations - why are they not vectors

"William Elliot" <marsh@hevanet.remove.com> wrote in message
news:Pine.BSI.4.58.0607152245480.22224@vista.hevanet.com...
 Quote: On Sun, 16 Jul 2006, Terry Padden wrote: Now consider simple (= 1-D) rotations of a spherical object about any given fixed axis. You mean 3D rotations?

NO, you fool. The axis is fixed. Only 1-D rotations are allowed (so far)

 Quote: Could someone point out to me in what way such 1-D rotations do NOT meet the axiomatic criteria for a Vector Space. In 3D, rotations are represented as a vector in the direction of the axis of rotation with magnitude of angular displacement.

PLEASE GO AWAY and keep this irrelevant 3-D nonsense to yourself

 Quote: If 1-D rotations are axiomatically vectors, why cannot they be axiomatically compounded into multi-dimensional vector spaces ? Because the cross product doesn't?

You are showing gross stupidity by this irrelevance. I am asking about
AXIOMS - not mechanics. Please stop embarrassing yourself in public.

 Quote: NB I am aware that 2-D rotations do-not-commute, Planar 2D rotations around a point are basically scalars.

That's your own little secret. Keep it that way. Now please go away and
think about axioms for a few years.
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Sun Jul 16, 2006 7:23 am    Post subject: Re: Rotations - why are they not vectors

<mariano.suarezalvarez@gmail.com> wrote in message
 Quote: Terry Padden wrote: I am bothered by the mathematics of rotations. It is I believe mathematically acceptable for any physical reality to be defined on an abstract axiomatic basis. Then anything that fulfills a given defining set of axioms for a type of mathematical object is a mathematically valid example of the defined mathematical object. A set does not make a vector space.

You B***** Fool. Who mentioned sets !!

 Quote: You need to specify two operations, addition and multiplication by scalars.

The obvious ones you moron. You can ADD rotations; and you can SCALE
rotations by a suitable Field. Idiot !!

 Quote: [snip] NB I am aware that 2-D rotations do-not-commute, but it seems to me that that has nothing to do with axiomatics or my questions. Actually, the composition operation on rotations (with a fixed center) in the plane *is* commutative.

Yes you fool - but that is in 1-D ; and I wrote do-not-commute in 2-D.
Can't
you even read a simple sentence ???

 Quote: You may want to review Halmos' presentation of what a vector space is, if you think that commutativity is irrelevant, by the way... NO ! I want you to go AWAY !!!!!!
William Elliot
science forum Guru

Joined: 24 Mar 2005
Posts: 1906

Posted: Sun Jul 16, 2006 5:51 am    Post subject: Re: Rotations - why are they not vectors

On Sun, 16 Jul 2006, Terry Padden wrote:

 Quote: Now consider simple (= 1-D) rotations of a spherical object about any given fixed axis. You mean 3D rotations?

 Quote: Could someone point out to me in what way such 1-D rotations do NOT meet the axiomatic criteria for a Vector Space. In 3D, rotations are represented as a vector in the direction of the axis

of rotation with magnitude of angular displacement.

 Quote: If 1-D rotations are axiomatically vectors, why cannot they be axiomatically compounded into multi-dimensional vector spaces ? Because the cross product doesn't?

 Quote: NB I am aware that 2-D rotations do-not-commute,

Planar 2D rotations around a point are basically scalars.
mariano.suarezalvarez@gma

Joined: 28 Apr 2006
Posts: 58

Posted: Sun Jul 16, 2006 4:52 am    Post subject: Re: Rotations - why are they not vectors

 Quote: I am bothered by the mathematics of rotations. It is I believe mathematically acceptable for any physical reality to be defined on an abstract axiomatic basis. Then anything that fulfills a given defining set of axioms for a type of mathematical object is a mathematically valid example of the defined mathematical object.

A set does not make a vector space. You need to specify
two operations, addition and multiplication by scalars. Moreover,
these need to satisfy certain relations, which you have seen in Halmos'
book.

Now, you want to view the set of rotations as a vector space,
somehow. Well. You need to decide what those two operations will be.
Before you do that, the question "are rotations a vector space?"
is meaningless.

 Quote: [snip] NB I am aware that 2-D rotations do-not-commute, but it seems to me that that has nothing to do with axiomatics or my questions.

Actually, the composition operation on rotations (with a fixed center)
in the plane *is* commutative.

You may want to review Halmos' presentation of what a vector space
is, if you think that commutativity is irrelevant, by the way...

-- m
science forum beginner

Joined: 17 Jun 2005
Posts: 28

 Posted: Sun Jul 16, 2006 4:42 am    Post subject: Rotations - why are they not vectors I am bothered by the mathematics of rotations. It is I believe mathematically acceptable for any physical reality to be defined on an abstract axiomatic basis. Then anything that fulfills a given defining set of axioms for a type of mathematical object is a mathematically valid example of the defined mathematical object. Now consider simple (= 1-D) rotations of a spherical object about any given fixed axis. Superficially, to me (not a mathematician), such "angular displacements" meet all of the formal axioms for a Vector Space (as given in e.g. Halmos) as well as 1-D linear displacements do. Could someone point out to me in what way such 1-D rotations do NOT meet the axiomatic criteria for a Vector Space. If 1-D rotations are axiomatically vectors, why cannot they be axiomatically compounded into multi-dimensional vector spaces ? NB I am aware that 2-D rotations do-not-commute, but it seems to me that that has nothing to do with axiomatics or my questions. I am not suggesting that rotations ought to be physically vectors. I am just trying to get clarification of the math picture for vectors.

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