Talk:Euclidean vector

Coordination among articles

The article linear algebra has been demoted to "start class". Several people are trying to fix it. But this article and the articles linear map, vector space, linear algebra, and matrix seem to have been written without reference to one another. A good goal would be to have all of these articles agree in terminology and style, and this article seems to be the place to start. There are probably other articles that should also be included in this project.

The first thing to consider is whether the title "Euclidean vector" is the best title for this article, leaving no article on the more general subject "Vector".

Rick Norwood (talk) 16:57, 9 February 2010 (UTC)

Considering the length of the disambiguation pages at Vector and Vector (mathematics and physics), I think the scope and title of this article -- for physical vectors in 2 and 3 real-world dimensions -- are not badly chosen.

There was extensive discussion on the title in /Archive 5. IMO the present title is better than any alternatives that were being canvassed at that time. Jheald (talk) 17:49, 9 February 2010 (UTC)

I don't have any problem with an article titled "Euclidean vector". My problem is with the lack of an article titled "vector (mathematics)". I haven't checked, but I suspect every mathematical encyclopedia has such an article. For example, this article at MathWorld http://mathworld.wolfram.com/Vector.html. At one point, Wikipedia explicitly wanted an article on every subject on MathWorld. Rick Norwood (talk) 14:20, 10 February 2010 (UTC)

Right now, vector (mathematics) redirects to vector space. I think that's a good solution: After all, a vector (strictly speaking) is just an element of a vector space, so you can't really discuss one without the other. Ozob (talk) 14:32, 10 February 2010 (UTC)

Consider the intelligent layperson who hears the word "vector" and wants to know what it means. To say that a vector is an element in a vector space is not helpful. I've been trying to find a good definition that will include vectors over an arbitrary field of scalars, and still be something a layperson can understand. Something like "a vector is a mathematical object that has both magnitude and direction, though in abstract mathematics the concepts of magnitude and direction may also be abstract." Rick Norwood (talk) 14:55, 10 February 2010 (UTC)

If you can think of a good article to put at vector (mathematics), then go for it. Myself, I don't know what could be put there; but I'll be interested to see what you come up with. Ozob (talk) 15:25, 10 February 2010 (UTC)

Sometimes its easier to define what something is by defining what it isn't.

It's hard to find an equationally defined class used in practice that wasn't originally motivated by its concrete instances. A Boolean algebra can be defined concretely as, any structure (with the appropriate operations) isomorphic to a subalgebra of a power set, or more abstractly as, any model of the equational theory of the two-element Boolean algebra. (That this theory has a finite basis, or finite axiomatization, is convenient but not the main point in the concrete-abstract distinction.)

The article Introduction to Boolean algebra begins with the finite axiomatization, but from the point of view of elementary algebra rather than abstract algebra. Section 5 on Boolean algebras (necessarily plural to avoid confusion) initially ignores the axioms and begins with concrete Boolean algebras (a) because they arise naturally and (b) to make the point that one can speak about at least the concrete kind without reference to any axiomatic definition of the concept.

The same should apply to vector spaces, with the parallels being strikingly clear when one considers that a vector space over GF(2), when equipped with a second constant 1 as the complement of the origin 0, is equivalent to a Boolean algebra via the evident translations in each direction between their respective languages. In particular, just as there is one finite concrete Boolean algebra 2ⁿ for each natural number n, so (given any field k) is there one finite-dimensional concrete vector space k ⁿ for each n. In both cases these are, up to isomorphism, the only finite/finite-dimensional such. (That the only non-free algebras here are some of the Boolean ones is an interesting but not central point.)

The benefit of the abstract definition, that it does not commit to a basis, can be had almost as well in the concrete case by defining an isomorphism of a concrete n-dimensional vector space to be a non-singular n×n matrix and pointing out that the resulting automorphism group elegantly links all bases in a way that makes the concrete concept basis-independent. One can then ask whether there might be an even neater approach to basis independence, which then leads naturally to the notion of an abstract vector space.

The problem with starting with the abstract definition is that it comes with no intuition. The point that is often lost is that concrete vector spaces are still vector spaces, despite not being defined equationally. It is enough for an object merely to satisfy the equations for a vector space, however the object was defined. --Vaughan Pratt (talk) 20:20, 13 July 2010 (UTC)

I think there ought to be an article about vector spaces in general, and one about vectors as used in classical mechanics and engineering with a short mention about other kinds of vectors used in physics. The first is what is currently at vector space, and either the current title or vector (mathematics) would be fine for it. For the latter, the current Euclidean vector is a good start but vector (physics) would be a better title. (Note also that physical vectors are not mathematical vectors, rather they are described by mathematical vectors: in modern mathematics all elements of all sets -- hence all vectors -- must be sets themselves, but the gravitational force acting on me right now in my frame of reference is not a set... A. di M. (formerly Army1987) (talk) 12:45, 14 July 2010 (UTC))

History of vectors reduced to just one line?

The history, as we can see from the article, is only ONE LINE. Would someone help expand the histories? Thank you. KaliumPropane (talk) 09:05, 4 November 2010 (UTC)KaliumPropane

Now the vector is not just one line. I suggest you look at the article "Angular vectors in the theory of vectors" https://doi.org/10.5539/jmr.v9n5p71 . In this paper, it is shown that it is necessary to separate the vectors into rectilinear and angular vectors. We introduce the concept of an inverse vector, which allows vector division operations. I hope that after reading, do not remain indifferent and help spread this article. Ujin-X (talk), —Preceding undated comment added 05:27, 17 September 2017 (UTC)

Formal definition

I have removed the "formal definition" from the first paragraph of the article:

More formally, a Euclidean vector is any element of a Euclidean vector space, i.e. a vector space that has a Euclidean norm. A Euclidean vector space is automatically a type of normed linear space and a type of inner product space.

For one thing, this is rather at odds with the way the article introduces vectors as directed line segments in the usual Euclidean space (which is more properly speaking an affine space, not a vector space). This is typical of how most mathematical treatments deal with geometric vectors (see, for instance, the EOM entry). It's also important to observe WP:NPOV. When most people consider geometric vectors, e.g., in mechanics, they are usually not thinking of the "element of a Euclidean space" viewpoint, but rather are thinking of a vector in the sense described in this article: a directed line segment in a (naive) Euclidean space. It might be worth having more discussion somewhere to disambiguate the naive vectors described here and the elements of a Euclidean vector space, i.e., an inner product space. A perusal of the archive shows that there is substantial confusion over what the scope of this article is, with formalists often trying to impose the "rigorous" definition (which is not even mathematically the same notion that the rest of the article is talking about). Sławomir Biały (talk) 13:52, 15 January 2011 (UTC)

Indeed: once upon a time this page was titled (IIRC) vector (physics) and it was the counterpart to scalar (physics); look for example at the 500th-oldest revision. Then the mathematicians took over and completely messed up with both the scope and the title of the article. :-) Now it's about any three-dimensional vector space over the real numbers with a positive-definite inner product, regardless of its relationships to physical space.

I once even proposed to keep this article with its "new" scope and "new" title and to start another article which would then be the new counterpart to scalar (physics) (and wrote a draft of it), but there were too few physicists (or engineer) around :-) so no-one saw the need for such an article. --A. di M. (talk) (formerly Army1987) 14:45, 15 January 2011 (UTC)

I'm getting sick of it

I know Wikipedia is written for different readers, but this is just ridiculous. The sum of the null vector with any vector a is a (that is, 0+a=a) is way too obvious and useless to be here. I don't mind if we remind readers that 1 + 1 = 2, but 0 + 1 = 1 is a little extreme. I mean I can't understand 90% of the mathematics on Wikipedia, and even I think this is too basic. 173.183.79.81 (talk) 03:10, 30 March 2011 (UTC)

I disagree. The existence and behavior of the null vector is central to the notion of a vector space. Without it and it's admittedly trivial-seeming behavior that 0+a=a, you don't have a linear space, you have an affine space. Plenty of times in advanced mathematics, seemingly trivial things are very important and need to be mentioned. —Ben FrantzDale (talk) 12:18, 31 March 2011 (UTC)

History

The history of vectors focuses entirely on quaternions. A brief mention of quaternions is fine, but the section simply describes the history and properties of quaternions and leaves out the history of vectors entirely. The section obviously does not satisfy quality standards and if an experienced knowledgable editor doesn't revise it the section should be removed. — Preceding unsigned comment added by 174.109.94.64 (talk) 02:19, 11 September 2011 (UTC)

This is absolutely correct. I've removed the section. Someone who wants to add a relevant history section is welcome to do so. The removed text is copied below. --JBL (talk) 21:44, 17 April 2013 (UTC)

I disagree. We are an encyclopaedia not a text book so we need to cover a history of the subject. To properly discuss how the concept of vectors came about we need to discuss what came before, complex numbers and quaternions were important precursors. Also note this section shows where the word "vector" first appears, part of a quaternion. Yes it could use some editing Grassmann’s Calculus of Extension needs a mention. Crowe's A History of Vector Analysis [1] seems a good basis for extending the section.--Salix (talk): 07:00, 18 April 2013 (UTC)

(copied from my talk page)--Salix (talk): 22:07, 18 April 2013 (UTC)

Yes, of course the history of a topic is important in an encyclopedia, and it's always nice to see historical information in math articles. But we have here a history section that is the history of a different topic than the subject of the article: there were 4 out of 5 or 6 paragraphs written to emphasize quaternions, with vectors an afterthought at best. (I've now re-removed one of these paragraphs, a comparison of quaternion and complex multiplication.) Three such paragraphs remain. If you'd like to keep them, please rewrite them so that they are about vectors, not quaternions. (Or move content over to the quaternion article as appropriate.) Right now the section is extremely misfocused and misleading. --JBL (talk) 14:13, 18 April 2013 (UTC)

The history of vectors is quite convoluted. Many of the important features of vectors, like the dot and cross products and the del operator were arrived at through studying quaternions. There was also a parallel development following Grassmann which is closer to what we would recognise as vector, however at the time this was very marginal. It was not until the 1880's when Gibbs and Heaviside both publish works which we would recognise as vector analysis.--Salix (talk): 23:07, 18 April 2013 (UTC)

Removed section

The concept of vector, as we know it today, evolved gradually over a period of more than 200 years. About a dozen people made significant contributions. The immediate predecessor of vectors were quaternions, devised by William Rowan Hamilton in 1843 as a generalization of complex numbers. Initially, his search was for a formalism to enable the analysis of three-dimensional space in the same way that complex numbers had enabled analysis of two-dimensional space, but he arrived at a four-dimensional system. In 1846 Hamilton divided his quaternions into the sum of real and imaginary parts that he respectively called "scalar" and "vector":

The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion.

Whereas complex numbers have one number $(i)$ whose square is negative one, quaternions have three independent imaginary units $(i,j,k)$. Multiplication of these imaginary units by each other is anti-commutative, that is, $ij\ =\ -ji\ =\ k$. Multiplication of two quaternions yields a third quaternion whose scalar part is the negative of the dot product and whose vector part is the cross product.

Peter Guthrie Tait carried the quaternion standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator.

In 1878 Elements of Dynamic was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product. This approach made vector calculations available to engineers and others working in three dimensions and skeptical of the fourth.

Josiah Willard Gibbs, who was exposed to quaternions through James Clerk Maxwell's Treatise on Electricity and Magnetism, separated off their vector part for independent treatment. The first half of Gibbs's Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis. In 1901 Edwin Bidwell Wilson published Vector Analysis, adapted from Gibb's lectures, and banishing any mention of quaternions in the development of vector calculus.

Why is there an overview section?

Right now, this article has an excellent introduction, followed by an overview section that mostly repeats the same content, but with less clarity and a variety of issues (like the idea that "an arrow" is the definition). I suggest simply removing the "overview" part of the first section (I.e., before the subsection "examples in 1 dimension"). --JBL (talk) 15:55, 28 August 2012 (UTC)

Vectors, pseudovectors, and transformations

Does this section have contravariant and covariant backwards? Not only does the word contravariant seem to imply it should vary in the opposite way, but the description seems to say so to.

In other words, if the reference axes were rotated in one direction, the component representation of the vector would rotate in exactly the opposite way. Similarly, if the reference axes were stretched in one direction, the components of the vector, like the co-ordinates, would reduce in an exactly compensating way.

Why then does it say, just above that, that they "transform like the coordinates" and the give math transformation both the coordinates $x'=Mx$ and the vector $v'=Mv$ the same. Also why, if they are both transformed by the forward transformation, is the need for an inverse to exist mentioned. Combined with the fact that covariance and contravariance of vectors page give the contravariant transformation in terms of the inverse, $v[fA]=A^{-1}v[f]$, I think this section got the transformations crossed over for part of it somehow. 207.112.55.16 (talk) 04:50, 5 February 2013 (UTC)

I'm surprised this rather obvious mistake wasn't fixed sooner. I've gone ahead and changed it.Paulmiko (talk) 17:08, 9 March 2015 (UTC)

I've reverted this edit. Covariant means to vary like the basis does. Contravariant means the opposite. The set of basis vectors is covariant (by definition), and the vector of coordinate scalars that is multiplied by the basis to get an invariant vector is contravariant. The wording in the article is confusing and should be fixed, but the meaning must not be switched around. —Quondum 17:37, 9 March 2015 (UTC)

vector subtraction

The article states: "to subtract b from a, place the end points of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. That arrow represents the vector a − b, as illustrated below:". However the section on representations tells us that the tip and the endpoint are synonymous. I presume that it should read "place the tails of a and b at the same point" or similar (this is what is illustrated). This subject is pretty fresh to me so I will leave it to somebody else to make the change. Kelly F Thomas (talk) 16:59, 1 April 2013 (UTC)

Quite right. I've made everything "head"s and "tail"s in that section. (In general this article is something of a mishmash and needs someone to go through and sort it out. Not volunteering, though.) --JBL (talk) 21:59, 1 April 2013 (UTC)

WT: WikiProject Mathematics #“Vector” redirects and next subsection

Please, contribute to these discussions. See also talk: Vector (mathematics and physics) #A CONCEPTDAB article is needed. Incnis Mrsi (talk) 07:58, 25 April 2013 (UTC)

History

Where should the history of a mathematical subject appear in a Wikipedia article? Checking three articles at random, in calculus, algebra, and trigonometry the history section follows immediately after the Table of Contents. I've moved the history section in this article to follow that example. Rick Norwood (talk) 23:06, 19 May 2013 (UTC)

Acceptable, although I'd rather place it at the end of the article, because I assume people is more interested in everything else... Paolo.dL (talk) 10:30, 20 May 2013 (UTC)

A further sampling of articles shows that the placement of the history section (when it is included) does not have a standard place. For example, in Speed of light (a featured article), it is at the end, before the footer sections (See also, Notes, References, etc.). In Algebra, it is in fact the second section. In Ring (mathematics), it is after the initial Definition and illustration section. In Logarithm (another featured article), it is the fourth section. In Matrix (mathematics), it is at the end, just before the footer sections. As such, I don't see that the placement in other articles should be used as a guide, but rather, a fresh motivations should be sought.

Being placed at the start may simply reflect a liking for the sequence to match the chronology – after all, the history is what led up to what the current state of the discipline is. Yet, the start-at-the-top format of WP suggests that the content be in the order most useful for access: first the lead summary, then a compact treatment of the topic, then a the more expansive related details. This argument would suggest that the history section for technical reference articles (and this would include essentially all the science and maths articles), a history section should almost never be near the top, since it is undoubtedly secondary (i.e. supportive) and not the primary content of the article. My own feeling is that it would generally best placed at the end of the article, at the start of the footer sections. In some sense it is natural to group it with the references and notes. To place it first is not a good idea to me: it makes accessing of the real meat of the article more clumsy, as one would have to skip over it every time the article is opened. — Quondum 11:41, 20 May 2013 (UTC)

It depends on whether most people come to technical articles to learn the subject, or to learn about the subject. A layperson who wants to learn about Euclidean vectors will appreciate a little historical context. A student who wants to actually learn the rules of manipulating Euclidean vectors can easily skip to those rules (which are more technical than the non-student will care to read). Rick Norwood (talk) 11:54, 20 May 2013 (UTC)

Even to the layperson, the history is still only context, not the core content, and such person can just as "easily skip to" the history section. It is the student who will be repeatedly accessing the article, not the layperson, so that "skipping" adds up. In this particular article, I see the typical reader as the high school student or layperson who quickly wants to check facts about vectors, to learn or to refresh their memories. I do not see the bulk of those who go beyond the lead as wanting primarily the history section. Nevertheless, I am glad to see you are arguing on the basis of usage, not chronological order, and not still on typical section order. I have no strong opinion on the matter for this article, and was merely trying to suggest a "history first" general pattern as the norm would not be ideal. — Quondum 13:26, 20 May 2013 (UTC)

I agree with Quondum. Most people (both laypersons and others) are interested in the "real meat" and will be forced to skip the history section if it is placed at the beginning. Paolo.dL (talk) 16:28, 20 May 2013 (UTC)

It is not that big a deal either way, because the history section, like any other section, is just a click away. But I doubt many people come here looking for "real meat". A layperson wants a general idea about a subject, without any technical details. And a mathematician, scientist, or engineer learned the "real meat" in a college course, and isn't going to look for it in Wikipedia, except maybe as a reminder of something they have forgotten. Rick Norwood (talk) 18:19, 20 May 2013 (UTC)

Tangent Vectors

In general, a tangent vector is not the same thing as a vector. The section on viewing vectors as directional derivatives should clarify what is meant, in the context of Euclidean space. — Preceding unsigned comment added by 99.19.84.64 (talk) 23:41, 7 November 2013 (UTC)

I don't get your point. A more general tangent is not necessarily a vector. The tangent to a curved surface for example. But the two examples given, of a scalar valued function of position (a scalar field) and a curve parameterised by a scalar, both have well defined vector valued derivatives, even though they are not themselves vectors.--JohnBlackburne^words_deeds 00:15, 8 November 2013 (UTC)

If you take the identity map from Rⁿ to Rⁿ then the set of tangent vectors is identical to the set of vector in Rⁿ. See Tangent space#Tangent vectors as directional derivatives. For a non-singular curved surface the tangent plane will be spanned the tangent vectors.--User:Salix alba (talk): 07:37, 8 November 2013 (UTC)

Geometric vectors are not always Euclidean

The hatnote refers to vectors used in Physics, but in Relativity the vectors used are Lorentzian rather than Euclidean. Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:33, 26 June 2017 (UTC)

I agree, geometric vector would be a better title, and this article shouldn’t imply that vectors can only describe Euclidean situations. Geometric vectors need not even be metrical, e.g. displacement vectors in a generic affine space don’t have any notion of distance. –jacobolus (t) 17:49, 9 November 2022 (UTC)

Subtraction

In order to enable a larger group of readers to understand what is happening in the 'explanation' of the subtraction of vectors, I added extra explanations. Obviously I disagree with the removal of my additions. As it stands now, I think that too many readers won't understand it. Bob.v.R (talk) 22:41, 14 January 2019 (UTC)

The description given now is extremely simple. Your additions complicated it by tying the definition of subtraction to the definition of opposite and of vector addition. But that is not necessary, and doing so diluted the point of the paragraph (which is to define vector subtraction). --JBL (talk) 22:50, 14 January 2019 (UTC)

I agree with JBL. Your text would make sense in a context where operations on vectors would be defined geometrically (that is coordinate free). As the choice here is to start from coordinates, one has not to prove again the properties; one has only to show that the geometrical definition is equivalent with the algebraic definition. So, your addition is confusing. D.Lazard (talk) 23:19, 14 January 2019 (UTC)

That the definition of subtraction should be based on the definitions of opposite and of vector addition is indeed the case, in my opinion. And in that way a reader who is new to the topic will understand how subtraction of vectors is defined, though it would still be better if an illustration would be added that shows the construction of (-b) + a. In the current version there is a drawing where a vector is constructed based on two other vectors, and then it is stated that this third vector is in fact a − b. So, clearly lacking is any explanation as to why this construction should be a − b. In fact, in the current text there is not even the slightest attempt to explain the construction to the reader. The reader reads a statement and that's that. Bob.v.R (talk) 00:44, 15 January 2019 (UTC)

Orientation and sense

It is worth noting that "direction" of a vector is sometimes "split" into "orientation" and "sense". 195.187.99.60 (talk) 09:18, 8 September 2020 (UTC)