Talk:Vector (mathematics and physics)

Possible Error in Definition

The article currently states, "If n is an integer and K is either the field of the real numbers or the field of the complex number, then K^n is naturally endowed with a structure of vector space, where K^n is the set of the ordered sequences of n elements of K." The case where n = -1 (or any other negative number) doesn't seem to make sense, as you can't select a negative number of elements from K to order and construct K^n. — Preceding unsigned comment added by 76.16.195.106 (talk) 23:52, 26 November 2011 (UTC)

Fixed. D.Lazard (talk) 07:54, 27 November 2011 (UTC)

Ambiguous Disambiguation Page

This disambiguation page is confusing. The primary analogate of all these uses of "vector" is the mathematical "Euclidean vector." I'm a physicist who's never heard "vectors" called by a name that makes it sound like they don't exist in non-Euclidean geometries. Whatever the provenance of that name, it will be intuitively obvious to few people what that means, so people will putz around the page looking for the page on "vector" before clicking on a guess. I'm not sure what the solution is, but the current situation (with, e.g., so many links duplicated in the appropriate places in the "Euclidean vector" article) is just silly. JKeck (talk) 13:41, 24 June 2011 (UTC)

One of these people looking for the definition of vector as used in statistical programs like R. I was also only familiar with vector as used in physics (i.e. atribute with both value and direction) and could not figure out how to relate this to the vector data type in R. Not sure this page helped me at all. — Preceding unsigned comment added by 41.3.173.252 (talk) 11:02, 20 September 2011 (UTC)

I agree with both of you. This is why I have added the preamble recently. In my opinion, the remaining of the article has to be completely rewritten. About vectors as atributes with both value and direction, this is the intuitive definition of vector bundle. Your remark suggest to add to the preamble something like a vector may be a pair of a point in some space and a direction associated to the point, lying in a vector space which may or not depend on the point. This meaning has been formalized in the notions of vector bundle and vector field. D.Lazard (talk) 08:31, 27 November 2011 (UTC)

Definition of the gradient vector

For the definition of the gradient vector, I would prefer Gradient vector, the vector which has, as coordinates, the partial derivatives of a multivariate function. It is equivalent with the present definition (Gradient vector, the vector giving the magnitude and direction of maximum increase of a scalar field), but does not need to know what is a scalar field, and it precise what is the "magnitude". Maybe both definitions should be given as Gradient vector, the vector which has, as coordinates, the partial derivatives of a multivariate function, of, equivalently, the vector giving the magnitude and direction of maximum increase of a scalar field. D.Lazard (talk) 19:04, 15 January 2012 (UTC)

A CONCEPTDAB article is needed

The WT: WikiProject Mathematics ‎#“Vector” redirects discussion convinced me that there should be no disambiguation (set index) page here. There should be a WP:CONCEPTDAB article to which numerous redirects like component (vector) and vector sum could point. Instead of Vector (mathematics and physics), it also can be titled as outline of vectors. The current index content IMHO has to me moved to something like list of vector topics or list of vector quantities. Incnis Mrsi (talk) 11:35, 24 April 2013 (UTC)

Does Editorial Stupidity Never End?

Some anonymus editor has interjected "It has been suggested that some portions of this article be split into a new article titled list of vector quantities. Please discuss this on the article's talk page. (April 2013)"

"It has been suggested" certainly fits one's normal idea of weasel words, but I'm afraid I don't know how to interject one of those fun boxes on the sorry spot. Who suggested it? Where is there anyone silly enough to make that specific stupid suggestion?

The actual suggestion is inane: where in this universe would there be enough space for such a list? It might be thought useful to put an example in some of the classes of vectors, I suppose; I'm not one of the people who think so.

Could someone remove the stupid box, and then this, my comment on it, please?

David Lloyd-Jones (talk) —Preceding undated comment added 14:40, 12 January 2014 (UTC)

Please maintain a civil tone. Calling other editors names is not acceptable.

The wording in the box is boilerplate. If you want to complain about the specific wording, the appropriate forum would be the talk page for the template.

You can always find out who added any item to a page on Wikipedia. Just click the "View History" tab at the top of the page. A quick skim down the list (since the text tells you the box was added in April) shows that it was added by Incnis Mrsi on April 24.

If you had clicked on the link that says "Discuss this", you would have been taken to the section above, where Incnis explains why he thought splitting this page would be a good idea.--Srleffler (talk) 16:35, 12 January 2014 (UTC)

This is already essentially a list, specifically a set index article. I'm removing the tag. 8ty3hree (talk) 00:26, 20 July 2014 (UTC)

Recent edits

I have recently restructured heavily the article: it was a WP:SIA, consisting of a weakly structured list of articles having "vector" in their title. On the other hand the term vector is often used without explicit reference to a vector space, and sometimes, it is used for quantities that can be added like vector. It was clear that WP needed a WP:broad-concept article, which should explain the relationship between all these concepts of vectors. This is why I have tried to with my edits. Essentially, I have kept almost all previous items, I have introduced a categorization, and for most categories of vectors, I have added an explanation of the categorization. I have also rewritten and/or expanded many items, specially when a short description cannot explain what about is the target (example: spinor). Also, I have tried to answer to the question "what is exactly a Euclidean vector?" This is clear in Euclidean vector, but needs reading a long part of the article, and is very confusing in many articles linking to Euclidean vector.

By the way, when editing the item p-vector, it appeared Multivector and Exterior algebra deserve to be merged. As I am not willing to do the merge myself, I have not created a merger proposal.

I have rated this new version of the article as "high importance" and "C class".

It remains certainly many things to do for improving the article: fixing my typos and improving my English; cleaning the very long "See also" section, adding sources, which are provided in the linked articles, etc.

Thank you for your future work on this article. D.Lazard (talk) 17:05, 21 September 2018 (UTC)

Tuples that aren’t really vectors?

Sorry, but the ‘tuples that aren’t really vectors’ part of this page seems a bit made up to me. The claims ‘the vector addition does not mean anything for these data, which may make the terminology confusing’ for points in R^n and ‘operations of vector spaces do not apply to them’ for direction/magnitude pairs are false/meaningless; these are both perfectly valid examples of vector spaces with specific definitions of the vector addition and scalar product. Indeed, all other examples listed in this section are vector spaces too, or at least subsets of vector spaces (often subspaces under some definition of the vector operations). Could someone clarify the intention of this section or consider removing it? 86.4.113.129 (talk) 17:02, 31 October 2021 (UTC)

Lol yes, I agree, this is totally made up. I will go ahead and fix this. Caleb Stanford (talk) 01:47, 12 November 2021 (UTC)

No, the previous beginning of the section was not made up. It is simply a (very) bad formulation of a true fact. That is: n-tuples form a vector space; data that do not belong to a vector space are often represented by tuples or, more generally by vectors; there are therefore often called vectors. I undid Caleb Stanford's edit for explaining this and changing the section heading to § Data represented by vectors. D.Lazard (talk) 10:15, 12 November 2021 (UTC)

@D.Lazard: I think the new text is much better and I made some improvements. However, the false part of what you said is "data that do not belong to a vector space are often represented by tuples". No, they do belong to a vector space, they just do not form a subspace. I hope you agree, it's perfectly valid to call an element of $\mathbb {R} ^{n}$ a vector just as it is perfectly valid to call a prime number an integer even though prime numbers are not closed under addition and multiplication. The previous text was even more egregious because for many of the listed examples (probability vectors and rotation vectors), the vector space operations do apply and these are actually explicitly studied in a linear algebra context. Caleb Stanford (talk) 14:05, 12 November 2021 (UTC)

Actually the article still needs some work to sort this out. Elsewhere in the article it lists the following as a legit example of a vector:

Unit vector, a vector in a normed vector space whose norm is 1, or a Euclidean vector of length one.

That is exactly the same concept as the rotation vector. Many of the examples under "Data represented by vectors" need to be moved up to the "Vector spaces" section.

@D.Lazard: Would you be interested in making a proposal for making the article coherent on this? Currently I am thinking we should remove the rotation vectors etc. from "Data represented by vectors" and instead places there examples where the entries are not real numbers. Logical vectors seems fine to list there. Caleb Stanford (talk) 14:09, 12 November 2021 (UTC)

IMO, you are wrong. A rotation vector is a vector that represents a rotation. The rotations do not form a vector space. In fact, they form a group. One can adding two rotation vectors, and the result represent some other rotation. But the the resulting rotation is not the sum, in any sense of the word, of the input rotations. So, one must definitively avoid formulations that that suggest that rotations form a vector space. Your suggestion to move rotation vectors to Vector space section is such a confusing formulation to be avoided.

The definition of a unit vector is perfectly correct, and is it correctly stated that such a vector is defined only in vector spaces on which a norm is defined. There is nothing to be fixed there. D.Lazard (talk) 14:48, 12 November 2021 (UTC)

If you think I am wrong, then prove me wrong: please provide the definition you have in mind of "Data represented by vectors" that applies to rotation vector and probability vector, but does not apply to unit vector?

What you wrote in the article is literally false: "scalar multiplication of vectors are not valid operations on these data" -- these are valid operations on a probability vector. For the record, you disagree that it's valid to add probability vectors and to view a probability vector as an element of $\mathbb {R} ^{n}$?

Would also appreciate a response to this point: "it's perfectly valid to call an element of $\mathbb {R} ^{n}$ a vector just as it is perfectly valid to call a prime number an integer even though prime numbers are not closed under addition and multiplication." Do you agree or disagree?

Also, please be patient with me here, I'm waiting for a consensus and will happily concede when you explain your point of view and prove me wrong. In the mean time, do not edit war because it's two people who have pointed out the problem and you are 1 person attempting to maintain the, allegedly, incorrect and incoherent status quo. Again I am happy with explaining this if it is explained correctly. Probability and rotation vector do not belong here. Caleb Stanford (talk) 14:58, 12 November 2021 (UTC)

When one says that "a rotation vector is a vector that represents a rotation" is is clear that the rotation is the data represented by the vector. I do not understand what is unclear to you.

"Scalar multiplication of vectors are not valid operations on these data": The scalar multiplication of a probability vector by a number larger that one does provide a probability vector. So, it is true that scalar multiplication is not a valid operation on probability vectors.

It is certainly not valid to call a prime number an integer, as a prime number is an integer, but, as $\mathbb {R} ^{n}$ is a vector space, it is valid to call its elements vectors. D.Lazard (talk) 15:30, 12 November 2021 (UTC)

Repeating my question: what definition you have in mind of "Data represented by vectors" that applies to rotation vector and probability vector, but does not apply to unit vector?

As you state, scalar multiplication does not preserve probability vectors. It does not preserve unit vectors either.

Similarly, adding two unit vectors does not preserve the fact that it's a unit vector. Adding two probability vectors does not preserve the fact it's a probability vector. What's the difference? And if there is none (as I contend), why does the article make a big deal about it? Caleb Stanford (talk) 16:22, 12 November 2021 (UTC)

I have no specific definition in mind for a common English phrase. A rotation vector represents a rotation exactly as the sequence of digits 12 represents the number 12. If a rotation is given, this is a data that requires a representation for being specified. A rotation vector is such representation. See Data representation for a (not very clear) definition.

Being a unit is a property of a vector. Addition, scalar multiplication, and linear combinations of vectors that are unit vectors are widely used in the definition of Cartesian coordinates. On the other hand, being a probability vector is not a property of a vector. Addition and scalar multiplication of probability vectors may be considered, but nobody do this in probability theory as well as in linear algebra, because this lacks of any useful meaning. This is a very big difference.

Thanks for your comments. I understand what you are saying better now. I agree with your comments on the data vs. the representation (the vector refers to the vector representation of the data), i.e. I agree with:

• "If a rotation is given, this is a data that requires a representation for being specified. A rotation vector is such representation."

I also agree with:

• "Being a unit is a property of a vector. Addition, scalar multiplication, and linear combinations of vectors that are unit vectors are widely used in the definition of Cartesian coordinates."

I disagree with, and contend that the following is demonstrably false:

• "being a probability vector is not a property of a vector. Addition and scalar multiplication of probability vectors may be considered, but nobody do this in probability theory as well as in linear algebra, because this lacks of any useful meaning."

Probability vectors can be -- and are -- added, scalar multiplied, and manipulated just as other vectors. That is the whole point of representing the data as a vector. Same with rotation vectors. Caleb Stanford (talk) 19:03, 12 November 2021 (UTC)

Introduction to Vectors for Non math everyday people

I propose the creation of a new article that explains what a vector is for non math people. ScientistBuilder (talk) 01:07, 19 February 2022 (UTC)

We should not have two articles on the same subject. People looking for simpler articles can visit the simple english wiki. - MrOllie (talk) 01:42, 19 February 2022 (UTC)

Ray or directed line segment?

There seems to be an error, or at least a severe discrepancy between the use of the word Ray on this page and the page it refers to (a section on the page Line which defines a ray as a half-line.) Further down on this page, a ray and a directed line segment seem to be considered the same. This can't be right.