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Rotations - why are they not vectors
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Sun Jul 16, 2006 10:54 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"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 -

PLEASE GO AWAY and stop parading your stupidity. Rotations have nothing to
do with R3.
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Narcoleptic Insomniac
science forum Guru


Joined: 02 May 2005
Posts: 323

PostPosted: Sun Jul 16, 2006 9:58 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

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
VECTOR ADDITION!!!

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.
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Sun Jul 16, 2006 9:33 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

On Sun, 16 Jul 2006, Terry Padden wrote:
Quote:
"William Elliot" <marsh@hevanet.remove.com> 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.
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Sun Jul 16, 2006 7:23 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"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.
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Sun Jul 16, 2006 7:23 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

<mariano.suarezalvarez@gmail.com> wrote in message
news:1153025545.663758.95150@75g2000cwc.googlegroups.com...
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 !!!!!!
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Sun Jul 16, 2006 5:51 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

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.
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mariano.suarezalvarez@gma
science forum addict


Joined: 28 Apr 2006
Posts: 58

PostPosted: Sun Jul 16, 2006 4:52 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

Terry Padden wrote:
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
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Sun Jul 16, 2006 4:42 am    Post subject: Rotations - why are they not vectors Reply with 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.

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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