Talk:Pseudovector

message: Shouldn't there at least be a mention of the microprocessor application of pseudo-vectors? (John2kx 21:05, 27 April 2007 (UTC)
thread_title:  Microprocessors 

message: I have no idea what this article is talking about; can it be explained in a manner comprehensible to somebody who majored in a subject other than mathematics?  As it stands now, the article is of use only to those who already understand the topic.
69.140.159.215 (talk) 03:22, 10 January 2008 (UTC)
replies: text: I think you are right. Furthermore, I want a more technical definition of what this is all about. The "transforms like" is very vague. I sounds like it has a life of its own and "transforms" out of free will! —Bromskloss (talk) 16:11, 18 January 2008 (UTC)
replies: text: I tried to clarify the lead. However, we can expect people to read up on what a vector is, and how vectors transform. I've opted to put links to the relevant articles in the lead in so far as they were not already present. Before you learn to run, first learn to walk. It's useless to memorise the differences between vectors and pseudovectors if you don't have a general idea of how vectors behave in the first place. Shinobu (talk) 00:47, 28 November 2008 (UTC)
thread_title:  too technical 

message: It would be a good addition if the more general use of "pseudovector" were addressed. Brews ohare (talk) 22:30, 7 January 2010 (UTC)
replies: text: Yes, in geometric algebra or Clifford algebra the pseudovector is the dual of the vector, so its dimension is n - 1, where n is the dimension of the space and algebra. So in 3D bivectors are pseudovectors. In 4D (including spacetime that stuff like the Dirac algebra is over) trivectors are pseusovectors, etc. The pseudoscalar is similarly defined to be the dual of the scalar, so is a trivector in 3D and 4-vector in 4D. Those are the only two commonly used.--John Blackburne (words ‡ deeds) 21:02, 7 January 2010 (UTC)
thread_title: Higher dimensional spaces

message: I added a footnote on Baylis and another on the connection to Dirac algebra. These notes are relevant to the geometric algebra, and so are not for everyone. However, a heads up on how geometric algebra uses these terms and some WP links are appropriate.
The geometric algebra idea is to remain coordinate free, so it isn't appropriate to characterize its terminology in terms of right and left handed coordinate systems. It should stand within its own axioms. Brews ohare (talk) 17:26, 8 January 2010 (UTC)
replies: text: \boldsymbol {ab} = \boldsymbol {a \cdot b} +\boldsymbol {a \wedge b} \ , 
where the leading term is the customary vector dot product and the second term is called the wedge product. Using the postulates of the algebra, all combinations of dot and wedge products can be evaluated. A terminology to describe the various combinations is provided. For example, a multivector is a summation of k-fold wedge products of various k-values. A k-fold wedge product also is referred to as a k-blade.
In the present context the pseudovector is one of these combinations. This term is attached to a different mulitvector depending upon the dimensions of the space (that is, the number of linearly independent vectors in the space). In three dimensions, the most general 2-blade or bivector can be expressed as a single wedge product and is a pseudovector.
 In four dimensions, however, the pseudovectors are trivectors.
In four dimensions, such as a Dirac algebra, the pseudovectors are trivectors. 
 In general, it is a (n - 1)-blade, where n is the dimension of the space and algebra.
The transformation properties of the pseudovector in three dimensions has been compared to that of the vector cross product by Baylis.
  To paraphrase Baylis: Although the terms axial vector and pseudovector are often treated as synonyms, a distinction often is useful, as follows. Given two polar vectors (that is, true vectors) a and b in three dimensions, the axial vector composed from a and b is the vector normal to their plane given by . On the other hand, the plane of the two vectors is represented by the exterior product or wedge product, denoted by  = i , where i=  is called the unit pseudoscalar, and the basis vectors are the eℓ. In this context of geometric algebra, this bivector is called a pseudovector, and is the dual of the cross product.
 For further detail see the article Comparison of aʌb and a×b. If the vectors a and b are inverted by changing the signs of their components while leaving the basis vectors fixed, both the pseudovector and the axial vector are invariant. On the other hand, if the components are fixed and the basis vectors eℓ  are inverted, then the pseudovector is invariant, but the axial vector changes sign. This behavior of axial vectors is consistent with their definition as vector-like elements that change sign under transformation from a right-handed to a left-handed coordinate system, unlike polar vectors.
;Notes
Any comments? Brews ohare (talk) 18:40, 8 January 2010 (UTC)
replies: text: I added this section and removed a couple of footnotes that now appear in this section. Hope this action doesn't raise furor; I'm leaving town & thought maybe I'd save some trouble, assuming the section to be OK. Brews ohare (talk) 19:51, 8 January 2010 (UTC)

text: I'd like to add some examples like this & this of the physics use of geometric-algebra pseudovectors, but I am forbidden to do so. Maybe someone can help? Brews ohare (talk) 22:15, 8 January 2010 (UTC)
thread_title:  Geometric algebra 

message: *I think this subsection would be helpful in guiding the interested reader through some of the WP geometric algebra morass to find what a pseudovector is in this context. Roughly, the basic extension geometric algebra provides is the simultaneous consideration of multiple subspaces within the same notation. Those that aren't interested can skip it, of course. Brews ohare (talk) 18:40, 8 January 2010 (UTC)
replies: text: I added this section and removed a couple of footnotes that now appear in this section. Hope this action doesn't raise furor; I'm leaving town & thought maybe I'd save some trouble, assuming the section to be OK. Brews ohare (talk) 19:51, 8 January 2010 (UTC)
replies: text: I'd like to add some examples like this & this of the physics use of geometric-algebra pseudovectors, but I am forbidden to do so. Maybe someone can help? Brews ohare (talk) 22:15, 8 January 2010 (UTC)
thread_title: Comments on proposed section

message: The lead sentence of this article is a verbatim quote from the AbsoluteAstronomy.com web page. Brews ohare (talk) 01:53, 19 January 2010 (UTC)
replies: text: The reason is because it came from Wikipedia in the first place!  See the bottom of that page where it says "The source of this article is wikipedia, the free encyclopedia."  I think it's funny ... until it happens to me! :)  CosineKitty (talk) 03:15, 19 January 2010 (UTC)
replies: text: It's funny how wikipedia can create its own truth...like in the Argusto Emfazie hoax article, where an article about a made-up person was on wikipedia for four years...at the end of the four years, this "person" had 4000 google hits! :-) --Steve (talk) 04:42, 19 January 2010 (UTC)
thread_title:  Lead sentence 

message: See this edit. I feel bad undoing it, because someone did a lot of work, but I think it has to be undone...
I checked the first ten google books results. Nine of them used the notation a × b for cross product, and the tenth used the notation a × b. None of the ten used the notation ab, which is the notation introduced by this edit.
Obviously we want the article to use the same notation that is almost-universally used in the world, in order to help readers easily understand the article (among many other reasons). So I am undoing the edit to restore the common notation  a × b. --Steve (talk) 21:34, 31 May 2012 (UTC)
replies: text: I agree with your revert: most of the formatting changes were unnecessary while the ones you identify were plain wrong. I noticed that the geometric algebra section had similar formatting issues, which may be why the editor tried to 'fix' the rest of the article, so I've tidied up that section to match the rest of the article and the manual of style.--JohnBlackburnewordsdeeds 21:58, 31 May 2012 (UTC)
thread_title:  Undoing a notation-changing edit 

message: The definition at the start of the lead
::a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection
applies only in an odd number of dimensions, yet the concept makes sense in any number of dimensions in the sense of geometric algebra. This statement must either be qualified (and I'd propose restricting it to three dimensions), and using a general definition (that does not make reference to sign flips) for any more general number of dimensions.  — Quondum 13:53, 5 October 2013 (UTC)
thread_title:  Constrain the first definition to 3D 

message: Substituted at 03:30, 30 April 2016 (UTC)
thread_title: Assessment comment

message: Before we get started, let me just say that vectors and scalars and tensors are the same thing. Technical details available upon request. None of the arguments made in Pseudovector, Pseudoscalar, and Pseudotensor make any sense at all because they assume the vector, scalar, and tensor counterpart is somehow oriented wrong. Therefore, I'm going to boldly propose that these 3 articles be deleted and redirected to vector, scalar, and tensor. Let me know what you think. Brian Everlasting (talk) 07:37, 22 April 2018 (UTC)
replies: text: I’m sorry, but you clearly do not understand the contents of the articles Pseudovector, Pseudoscalar, and Pseudotensor. They are distinct and well understood topics, with rigorous mathematical definitions and concrete physical applications. Looking at just this article, see its lengthy and authoritative list of references. I suggest you consult some of them to gain a better understanding of the topic. Until you do please do not make the edits you proposed. Such would be effectively deleting the articles, and could be seen as disruptive without a proper deletion discussion.--JohnBlackburnewordsdeeds 08:16, 22 April 2018 (UTC)
replies: text: , I have consulted the references and I believe I am knowledgable about this topic. You resort to personal attacks "you clearly do not understand" instead of discussing the issue. Please familiarize yourself with WP:NPA before we continue our discussion. Thanks. Brian Everlasting (talk) 17:26, 22 April 2018 (UTC)

text: OK, I hereby request the technical details that support your claim "that vectors and scalars and tensors are the same thing". I'm also interested in whether you figured this out all by yourself, or if you learned it from a book or other reference, and if the latter, please tell me where to find it. Thanks, --Steve (talk) 17:55, 22 April 2018 (UTC)
replies: text: , thank you for your message and your great contributions to wikipedia. Because this is a very complicated issue, I will require a little more time to give you a good response. Sorry I am so slow. Brian Everlasting (talk) 17:39, 24 April 2018 (UTC)
thread_title:  Pseudovector and Pseudoscalar and Pseudotensor factual errors 

message: In the traditional physics that I learned during the mid-20th century under 
the United States federally-funded "Nation Defense Education Act" (of 1958), 
all vectors were like arrows, with a direction and a length. Moreover they 
could be, "connected head-to-toe" in single-link chains as we may please. 
As this is how vectors were, and this is how they would ever be. 
Algebraically, this required nothing more abstract than a monoid.  
My next class was, "Introduction to Linear Algebra". Newton was right, but 
he thought only in the large scale of the planets that orbit Sol, our sun. 
Linear Algebra is The Rock of practical mathematics in this era. Computers 
can do a thing a human will never do: multiply matrices, the basic blocks 
of linear algebra, at very high rates. This allows practical matrix math. 
The truth is in mathematical vector spaces; physicists call them, "pseudo". 
Vector Spaces will never do the head-to-toe thing like they should, *sigh*. 
And they don't look much like arrows do mostly, neither. These realities are plainly seen with simply opened eyes. 
After the Einstein-Bohr debates (1925), and I see Bohr as Newtonian-minded, Einstein worked with Elie Cartan on the fundamentals of a radial field that originated at a point, such as starlight for a fundamental example. This lead 
to the center of the star, and what would it look like? Suppose you collapse 
the radius of a sphere toward zero: what will the center look like? "A point!" 
is what the Newtonian would exclaim, "That is plain to see, most obviously." 
The kernel found by Einstein and Cartan was not a traditional point at all. 
Cartan's original 1913 paper on Spinors was difficult to read, but correct. CFjohnny1955 (talk) 07:15, 21 April 2024 (UTC)
thread_title:  Vectors as seen in a Moniod Algebraic context.