Search   Memberlist   Usergroups
 Page 3 of 4 [53 Posts] View previous topic :: View next topic Goto page:  Previous  1, 2, 3, 4 Next
Author Message
William Elliot
science forum Guru

Joined: 24 Mar 2005
Posts: 1906

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

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.
Stephen Montgomery-Smith1
science forum Guru

Joined: 01 May 2005
Posts: 487

 Posted: Mon Jul 17, 2006 2:00 am    Post subject: Re: Rotations - why are they not vectors 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
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Mon Jul 17, 2006 1:21 am    Post subject: Re: Rotations - why are they not vectors

"Sylvain Croussette" <sylvaincroussette2@yahoo.ca> wrote in message
 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.
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Mon Jul 17, 2006 12:49 am    Post subject: Re: Rotations - why are they not vectors

"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 !
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Mon Jul 17, 2006 12:49 am    Post subject: Re: Rotations - why are they not vectors

<abe.buckingham@gmail.com> wrote in message
 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.
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Mon Jul 17, 2006 12:49 am    Post subject: Re: Rotations - why are they not vectors

"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.
Lynn Kurtz
science forum Guru

Joined: 02 May 2005
Posts: 603

Posted: Sun Jul 16, 2006 8:34 pm    Post subject: Re: Rotations - why are they not vectors

On Sun, 16 Jul 2006 12:57:25 GMT, "Terry Padden"

 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
Abraham Buckingham

Joined: 10 Mar 2005
Posts: 98

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

they satisfy the the axioms of a vector space.
Sylvain Croussette
science forum beginner

Joined: 05 May 2005
Posts: 32

Posted: Sun Jul 16, 2006 3:04 pm    Post subject: Re: Rotations - why are they not vectors

G.E. Ivey wrote:
 Quote: Why should any one respond to any question by you in the future? Several people have responded, as best they could, to your somewhat vague questions. (What, exactly do you mean by a "1-D rotation". I can see "flipping", changing (x, 0, 0) to (-x, 0, 0) but how can you "rotate" through an angle in 1 dimension?) You immediately started calling them "fools". That's not a good way to convince people to answer your questions.

I think what he means by 1-D rotation is a rotation in 3 dimensional
space but around one axis only. This is the jargon used in some fields
like robotics where a 1-D rotation joint is a joint that rotates around
only one axis. This seems to be confirmed by his assertion that 2-D
rotations do not commute. To him this means a rotation in 3 dimensions
but around 2 different axes. It is true that they do not commute in
general. 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.
Stephen Montgomery-Smith1
science forum Guru

Joined: 01 May 2005
Posts: 487

Posted: Sun Jul 16, 2006 2:57 pm    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. 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.

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.

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.

Actual rotations are not described by vectors, but by matrices -
orthogonal matrices. The composition of two rotations is not any kind
of addition, but multiplication of the matrices. The orthogonal
matrices do not form a vector space. This is true even if you restrict
yourself to rotations about a fixed axis (what I think you mean by 1-D
rotations) because rotation by 360 degrees is the same as rotation by 0
degrees, and as such it is impossible to properly define multiplication
by a scalar. And if you don't restrict yourself to rotations about a
fixed axis, matrix multiplication doesn't even commute (as you point out
above).

However the word "rotation" could also mean the rate of rotation, for
example "18 degrees per second clockwise about a certain axis." In that
case rotations do add like a vector space. However you should be
careful, for example in 4 dimensional space, the rotations is a 6
dimensional space (in general n goes to n(n-1)/2).

By the way, the notion of rotation (in the first sense I described)
about "an axis" really only has meaning if the underlying space is two
or three dimensional. In 4 or 5 dimensional space, the generic rotation
will have two axes.

Stephen
Jürgen R.
science forum beginner

Joined: 06 Feb 2006
Posts: 12

Posted: Sun Jul 16, 2006 2:51 pm    Post subject: Re: Rotations - why are they not vectors

On Sun, 16 Jul 2006 04:42:30 GMT, "Terry Padden"

 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.

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

 Quote: 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.

 Quote:
Narcoleptic Insomniac
science forum Guru

Joined: 02 May 2005
Posts: 323

Posted: Sun Jul 16, 2006 1:37 pm    Post subject: Re: Rotations - why are they not vectors

On Jul 16, 2006 7:57 AM CT, Terry Padden wrote:

 Quote: "G.E. Ivey" wrote in message news:627837.1153051489453.JavaMail.jakarta@nitrogen.mathforum.org... Why should any one respond to any question by you in the future? Several people have responded, as best they could, to your somewhat vague questions. (What, exactly do you mean by a "1-D rotation". I can see "flipping", changing (x, 0, 0) to (-x, 0, 0) but how can you "rotate" through an angle in 1 dimension?) You immediately started calling them "fools". That's not a good way to convince people to answer your questions. 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 that explains the hostility and senility.

 Quote: But what rreally makes me see red is the results of all that expenditure on mathematical education - the results as displayed by prior respondents and you.

Clearly you're mistaken; from the little time I've spent
on these forums I've come to know both William and G. E.
indirectly and they both happen to be very knowledgeble.

 Quote: What is unclear about "1-D rotations about a fixed axis" ? It is specified by a 1-tuple x an anglular displacement; just like a 1-D translation is specified by a 1-tuple x a linear displacement. Rotations and Vectors have nothing intrinsically to do with Rn or any similar pre-supposition. Rotations are just angular displacements - nothing necessarily to do with translations at all ! Vectors are just things which satisfy the axioms.

I definately agree with you on this last paragraph.
Although, when I mentioned earlier that spherical
rotations could be viewed as mappings on vector spaces
you called it "mumbo-jumbo" and called me a fool. I guess
I must be a fool for thinking that rotations are just
transformations that *act* on vector spaces and are *not*
vectors themselves.

 Quote: My standards are high but not unreasonably so considering my forced financial investment. If they seem unreasonable to you and others remember that as GBS told us all progress depends on unreasonable people.

I would like to forget anything that any Bush has told us.

 Quote: My question is clear and straightforward. If you don't understand it blame your maths professors; not me.

The topic of this thread is almost as meaningful as

"Colors - why are they not feelings".

 Quote: By the way the usual physical measurement of 1-D rotations is Time in hours, minutes, seconds, etc. Perhaps ypu have heard of Babylonian Maths; perhaps you know that miniutes and seconds also measure angles etc. So far all I have got back is the maths of Babel. PLEASE can the next reply be from someone who understands the question.
science forum beginner

Joined: 17 Jun 2005
Posts: 28

Posted: Sun Jul 16, 2006 12:55 pm    Post subject: Re: Rotations - why are they not vectors

"G.E. Ivey" <george.ivey@gallaudet.edu> wrote in message
news:627837.1153051489453.JavaMail.jakarta@nitrogen.mathforum.org...
 Quote: Why should any one respond to any question by you in the future? Several people have responded, as best they could, to your somewhat vague questions. (What, exactly do you mean by a "1-D rotation". I can see "flipping", changing (x, 0, 0) to (-x, 0, 0) but how can you "rotate" through an angle in 1 dimension?) You immediately started calling them "fools". That's not a good way to convince people to answer your questions.

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.

But what rreally makes me see red is the results of all that expenditure on
mathematical education - the results as displayed by prior respondents and
you.

What is unclear about "1-D rotations about a fixed axis" ? It is specified
by a 1-tuple x an anglular displacement; just like a 1-D translation is
specified by a 1-tuple x a linear displacement.

Rotations and Vectors have nothing intrinsically to do with Rn or any
similar pre-supposition. Rotations are just angular displacements - nothing
necessarily to do with translations at all ! Vectors are just things which
satisfy the axioms.

My standards are high but not unreasonably so considering my forced
financial investment. If they seem unreasonable to you and others remember
that as GBS told us all progress depends on unreasonable people.

My question is clear and straightforward. If you don't understand it blame

By the way the usual physical measurement of 1-D rotations is Time in hours,
minutes, seconds, etc. Perhaps ypu have heard of Babylonian Maths; perhaps
you know that miniutes and seconds also measure angles etc. So far all I
have got back is the maths of Babel.

PLEASE can the next reply be from someone who understands the question.
G.E. Ivey
science forum Guru

Joined: 29 Apr 2005
Posts: 308

 Posted: Sun Jul 16, 2006 12:04 pm    Post subject: Re: Rotations - why are they not vectors Why should any one respond to any question by you in the future? Several people have responded, as best they could, to your somewhat vague questions. (What, exactly do you mean by a "1-D rotation". I can see "flipping", changing (x, 0, 0) to (-x, 0, 0) but how can you "rotate" through an angle in 1 dimension?) You immediately started calling them "fools". That's not a good way to convince people to answer your questions.
Narcoleptic Insomniac
science forum Guru

Joined: 02 May 2005
Posts: 323

Posted: Sun Jul 16, 2006 11:58 am    Post subject: Re: Rotations - why are they not vectors

On Jul 16, 2006 5:56 AM CT, Terry Padden wrote:

 Quote: "Narcoleptic Insomniac" i_have_narcoleptic_insomnia@yahoo.com> wrote in message news:7171135.1153043965570.JavaMail.jakarta@nitrogen.mathforum.org... 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 -

It's not "mumbo-jumbo" but fairly basic linear algebra...

...you know, the stuff you learn *after* you see those
axioms.

Hahahaha, seriously, get some new material. For your sake
I hope that you're just trolling and doing this for fun.

 Quote: Rotations have nothing to do with R3.

Actually, you cut out the part in this thread where *you*
began considering simple rotations of a sphereical object.
If you would look past your beloved axioms for a moment
you'd see that any rotation of a sphereical object can be
described as a transformation of R^3 -> R^3.

Going back to your original topic (simple rotations (in
1-D) of a sphereical object and why they're NOT vectors),
it suffices to just consider a simple rotation about the
x-axis since we can transform any arbitrary axis to the
x-axis.

We can the described this rotation by the matrix A(t) =

[1 0 0]
[0 cos(t) sin(t)]
[0 -sin(t) cos(t)].

The set of all rotation matrices of this type, along with
the 3x3 identity matrix, will form a *group* structure,
but I can't see why you would think this could generate
a vector space.

Regards,
Kyle Czarnecki

 Display posts from previous: All Posts1 Day7 Days2 Weeks1 Month3 Months6 Months1 Year Oldest FirstNewest First
 Page 3 of 4 [53 Posts] Goto page:  Previous  1, 2, 3, 4 Next View previous topic :: View next topic
 The time now is Sat Mar 23, 2019 11:11 am | All times are GMT
 Jump to: Select a forum-------------------Forum index|___Science and Technology    |___Math    |   |___Research    |   |___num-analysis    |   |___Symbolic    |   |___Combinatorics    |   |___Probability    |   |   |___Prediction    |   |       |   |___Undergraduate    |   |___Recreational    |       |___Physics    |   |___Research    |   |___New Theories    |   |___Acoustics    |   |___Electromagnetics    |   |___Strings    |   |___Particle    |   |___Fusion    |   |___Relativity    |       |___Chem    |   |___Analytical    |   |___Electrochem    |   |   |___Battery    |   |       |   |___Coatings    |       |___Engineering        |___Control        |___Mechanics        |___Chemical

 Topic Author Forum Replies Last Post Similar Topics Best fit orthogonal basis for list of vectors chengiz@my-deja.com num-analysis 4 Wed Jul 19, 2006 6:16 pm Null rotations Greg Egan Research 1 Sun Jul 02, 2006 12:08 pm vectors from l_2 and their difference isn't in l_1+uncoun... eugene Math 7 Fri Jun 23, 2006 10:52 pm notation for vectors and points pluton Math 3 Sat Jun 03, 2006 4:15 pm ? e-vectors of sum of rank one matrices Cheng Cosine Math 1 Sat Jun 03, 2006 6:36 am