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Rotations - why are they not vectors
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Abraham Buckingham
science forum addict


Joined: 10 Mar 2005
Posts: 98

PostPosted: Sun Jul 16, 2006 3:09 pm    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.

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.

For clarification please explicitly demonstrate your rotations and how
they satisfy the the axioms of a vector space.
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Lynn Kurtz
science forum Guru


Joined: 02 May 2005
Posts: 603

PostPosted: Sun Jul 16, 2006 8:34 pm    Post subject: Re: Rotations - why are they not vectors Reply with quote

On Sun, 16 Jul 2006 12:57:25 GMT, "Terry Padden"
<TPadden@bigpond.net.au> wrote:


Quote:
For 45 years taxes from my hard earned income have been used to fund things
such as the internet and university education. It took me some time to
frame the question. For that it is not unreasonablke for me to expect
people to make a reasonable effort to understand a question before replying
to it . As a starter they should at least read it.

Well, Bully for you. This is usenet group where many people, who know
a lot more mathematics than you do, freely give of their time and
knowledge to help others. You have no right to "expect" anything from
anyone here. Given the tone of your responses, I am surprised anyone
is giving you a civil answer.

--Lynn
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 12:49 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"Stephen Montgomery-Smith" <stephen@math.missouri.edu> wrote in message
news:Mtsug.38362$FQ1.35840@attbi_s71...
Quote:
Terry Padden wrote:

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 ?


I think people might be confused about what you mean by "1-D rotations."
The convention normally used is to describe rotations in R^n as n
dimensional rotations. I don't think you are following this.

I am following the convention - for axiomatic vector spaces. The dimension
has nothing to do with Rn. It has to do with Basis / Unit vectors.
Rotations about a fixed axis require only one basis vector = an angle of any
size, say 1/2 an hour.

Quote:
Now the sophisticated way to describe your issue, I think, is to say that
there are two answers, depending upon whether you are describing the Lie
Group or the Lie Algebra.

Considering the question this is more sophist gobbledegook than
sophistication.

Quote:
Actual rotations are not described by vectors, but by matrices -

My question has nothing to do with how conventionally one does represent
rotation - but asks why they cannot be represented as vectors !
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 12:49 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

<abe.buckingham@gmail.com> wrote in message
news:1153062571.285535.132550@p79g2000cwp.googlegroups.com...
Quote:

Terry Padden wrote:

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.



Quote:
For clarification please explicitly demonstrate your rotations and how
they satisfy the the axioms of a vector space.


Consider the time of day = rotations of a sphere about a fixed axis =
continuous angular displacements.

You can add angles / times; conceptually time is reversible so you can have
negative rotations corresponding to any positive one; you can scale them
using your choice of number field; any unit of angular displacement
(minutes, seconds, hours) is a 1-D basis.
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 12:49 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"Jürgen Ren" <jurgenr@web.de> wrote in message
news:v8kkb2d23is638hejae7mfeqpjjgd7ldhg@4ax.com...
Quote:
On Sun, 16 Jul 2006 04:42:30 GMT, "Terry Padden"
TPadden@bigpond.net.au> wrote:

Could someone point out to me in what way such 1-D rotations do NOT meet
the
axiomatic criteria for a Vector Space.

They meet the requirements. It's a one-dimensional vector space.

Thank you. That is what worries me.


Quote:

If 1-D rotations are axiomatically vectors, why cannot they be
axiomatically
compounded into multi-dimensional vector spaces ?

I have no idea what you mean by "axiomatically compounded",

Neither do I really; I am struggling with the idea.

Quote:
but the
answer is that you can form the direct product of any number of such
spaces in the usual way to get higher-dimensional vector spaces.


I thought so; but hm ? So 2 x 1-D rotations is a vector space and 1-D
rotations are commutative. Where then from the axioms does the
non-commutaivity of 2-D rotations come from ? Puzzled I am.
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 1:21 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"Sylvain Croussette" <sylvaincroussette2@yahoo.ca> wrote in message
news:1153062256.844818.81550@b28g2000cwb.googlegroups.com...
Quote:
The problem is that he is not a mathematician (as he said
himself) and he is using "1-D" and "2-D" in a different context than
that of a mathematician.


NO! The problem is that you are ignorant about what a dimension is in the
theory of vector spaces. According to mathematicians, e.g. Halmos, the
dimension of a vector space is the number of basis vectors required to
specify any vector in the space. An LVS is 1-D if you need only 1 Basis /
Unit vector to specify any other e.g (as I already wrote) Rotations about a
fixed axis.

GO AWAY - until you understand the question.
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Stephen Montgomery-Smith1
science forum Guru


Joined: 01 May 2005
Posts: 487

PostPosted: Mon Jul 17, 2006 2:00 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

I am reminded of a story told to me by my driving instructor - he tried
to teach a student who, every time he tried to correct him, would cut
him off. The student repeatedly failed his driving test until, in
frustration, he hit one of the testers, ended up being charged with
assault, and no-one willing to take him for another driving test.

Halmos is a good introduction to the abstract approach to vector spaces,
but it is just that, an introduction. When you communicate your
problem, you have to expect that you don't fully know the language of
mathematics, and conversely, that we don't fully understand the words as
you mean them. I see this all the time, even with experts in different
disciplines (e.g. mathematics and engineering). The only way to
communicate is to be patient with each other, and slowly try to learn
the other person's language.

Unfortunately you are reacting in a very hostile fashion towards those
who are trying to help you. Quite possibly they are not understanding
where you are coming from, because you do not understand the common
language that has developed. But this is no-ones fault. Try to respond
in a nice fashion, because when we understand exactly what your question
is, someone here is going to have the answer. But if you turn everybody
off, no-one is going to want to try.

Best Stephen
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William Elliot
science forum Guru


Joined: 24 Mar 2005
Posts: 1906

PostPosted: Mon Jul 17, 2006 2:51 am    Post subject: [] Rotations - why are they not vectors Reply with quote

On Mon, 17 Jul 2006, Stephen Montgomery-Smith wrote:

Quote:
Unfortunately you are reacting in a very hostile fashion towards those
who are trying to help you. Quite possibly they are not understanding
where you are coming from, because you do not understand the common
language that has developed. But this is no-ones fault. Try to respond
in a nice fashion, because when we understand exactly what your question
is, someone here is going to have the answer. But if you turn everybody
off, no-one is going to want to try.

The patience of you respondents is amazing.
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jw12jw12jw12@yahoo.com
science forum beginner


Joined: 28 Oct 2005
Posts: 16

PostPosted: Mon Jul 17, 2006 4:31 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.
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.

Given the previous posts I'm hesitant about adding anything, but here
goes:

1-D rotations ARE a vector space...this space is isomorphic to R^1.For
example, one of the axioms of a vector space is: u+v=v+u , and in
the case of 1-D rotations a rotation of (for example) 30 degrees
followed by a rotation of 40 degrees is equivalent to a rotation of 40
degrees follwed by a rotation of 30 degrees. I would agree that this
is the obvious interpretyation of addition in this this case (although
others are possible). All other axioms woukd be satisfied in this
interpretation.

As you point out u+v=v+u is not valid for general rotations in R^3 and
this is a reason why rotations in R^3 do not form a vector space.

I think the concept of a vector is a bit trickier than many people
realize. I've met math teachers who were unable to answer the following
questions, or incorrectly.
1. Let AB be the trip from Albany to Buffalo, let BC be the trip from
Buffalo to Chicago so AB+BC=AC in the sense that you have a trip from
Albany to Chicago. Is AB a vector?
Explain.

2. Are forces free vectors or a bound vectors?

3. Angular displacements in R^3 (i.e. rotations) are not vectors, so
why are angular veclocities in R^3 vectors?

There is a nice, inexpensive little book by physicist Banesh Hoffman
called "About Vectors" published by Dover (that I hope is still in
print) which you might find interesting.

jw
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mariano.suarezalvarez@gma
science forum addict


Joined: 28 Apr 2006
Posts: 58

PostPosted: Mon Jul 17, 2006 5:34 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

jw12jw12jw12@yahoo.com wrote:
Quote:
Given the previous posts I'm hesitant about adding anything, but here
goes:

Heh.

Quote:
1-D rotations ARE a vector space...this space is isomorphic to R^1.For
example, one of the axioms of a vector space is: u+v=v+u , and in
the case of 1-D rotations a rotation of (for example) 30 degrees
followed by a rotation of 40 degrees is equivalent to a rotation of 40
degrees follwed by a rotation of 30 degrees. I would agree that this
is the obvious interpretyation of addition in this this case (although
others are possible). All other axioms woukd be satisfied in this
interpretation.

As I said before, making sense of a statement such as
"1-D rotations ARE a vector space" requires that one at the
very least decide what are "1-D rotations". In particular, one
needs to be able to determine when two "1-D rotations" are
the same "1-D rotation".

Are the "1-D rotations" of 360 degrees and of 0 degrees the
same rotation? If yes, then your attempt at regarding the set
of 1-D rotations as a vector space is doomed to fail. If no,
well, then the notion of 1-D rotation would be more usefully
called "oriented angle", and yes, oriented angles can be seen
as a vector space in a natural way.

-- m
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ben1
science forum beginner


Joined: 16 May 2006
Posts: 9

PostPosted: Mon Jul 17, 2006 8:21 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

Rotations (in Euclidean space) do not form a vector space over the
reals because of the failure of the consistency axiom. If r,s are real
numbers and T is a rotation, then it is not necessarily the case that
r(sT) = (rs) T

For instance, let T denote rotation through one quarter-circle (90
degrees or pi/2 radians). Then
4T= Id, the identity rotation
1/4 ( 4T) = Id
which is not the same as
(1/4 * 4) T = T

Ben
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ben1
science forum beginner


Joined: 16 May 2006
Posts: 9

PostPosted: Mon Jul 17, 2006 8:29 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

The set of rotations of the plane form a group (although not a
vector-space), which we can denote O(1). Pairs of these form a group
O(1)xO(1). This group is not isomorphic to O(2); which is a fancy way
of stating a pair of 1D rotations is not the same as a 2D rotation. The
easiest way to see this is the case, however, is to note that one is a
commutative group and the other is not.

Quote:
I thought so; but hm ? So 2 x 1-D rotations is a vector space and 1-D
rotations are commutative. Where then from the axioms does the
non-commutaivity of 2-D rotations come from ? Puzzled I am.
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 10:54 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"ben" <benedict.williams@gmail.com> wrote in message
news:1153124461.756415.119060@s13g2000cwa.googlegroups.com...
Quote:
Rotations (in Euclidean space) do not form a vector space over the
reals because of the failure of the consistency axiom. If r,s are real
numbers and T is a rotation, then it is not necessarily the case that
r(sT) = (rs) T

For instance, let T denote rotation through one quarter-circle (90
degrees or pi/2 radians). Then
4T= Id, the identity rotation
1/4 ( 4T) = Id
which is not the same as
(1/4 * 4) T = T

Ben


Thank you. I'll try to make sure I understand what you say. Can you
provide an accessible reference text for the Consistency Axiom
as basic as possible. I don't recall it being mentioned in any of the basic
texts that I skim thru - I'll see if it is in Wikipedia.
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 10:55 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"Stephen Montgomery-Smith" <stephen@math.missouri.edu> wrote in message
news:CaCug.1084014$xm3.732932@attbi_s21...
Quote:
I am reminded of a story told to me by my driving instructor - blather

My reply to you was quite civil - but corrected your errors. It seems you
wish to change the subject to ad hominem junk. I understand the psychology
of your response but so what. You may not approve of my methods of
improving the average mathematical competence of this newsgroup but it
works. When this thread is complete you will all have a clearer
understanding of vectors & rotations - thanks to me. No need for gratitude,
I see it as a work of charity.

One person seems to be having no problem providing reasonable responses to a
reasonable question. I am truly grateful, but not a changed person.
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Terry Padden
science forum beginner


Joined: 17 Jun 2005
Posts: 28

PostPosted: Mon Jul 17, 2006 10:55 am    Post subject: Re: Rotations - why are they not vectors Reply with quote

"ben" <benedict.williams@gmail.com> wrote in message
news:1153124952.700118.157220@s13g2000cwa.googlegroups.com...
..
Quote:

I thought so; but hm ? So 2 x 1-D rotations is a vector space and 1-D
rotations are commutative. Where then from the axioms does the
non-commutaivity of 2-D rotations come from ? Puzzled I am.

I have tidied up your top-post


Quote:
The set of rotations of the plane form a group (although not a
vector-space), which we can denote O(1). Pairs of these form a group
O(1)xO(1). This group is not isomorphic to O(2); which is a fancy way
of stating a pair of 1D rotations is not the same as a 2D rotation. The
easiest way to see this is the case, however, is to note that one is a
commutative group and the other is not.

From my perspective this is cheating. We are not supposed to be discussing
groups - or any kind of Euclidean space.
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