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A summary of this article appears in Vector (mathematics and physics). |
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)
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)
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)
Sometimes its easier to define what something is by defining what it isn't.
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)
I have removed the "formal definition" from the first paragraph of the article:
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)
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)
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)
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. [1] 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":
Whereas complex numbers have one number whose square is negative one, quaternions have three independent imaginary units . Multiplication of these imaginary units by each other is anti-commutative, that is, . 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. [1] 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.
It has been suggested that portions of this section be
split out into another page titled
History of quaternions#After Hamilton. (
Discuss) |
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)
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 and the vector 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, , 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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
This
level-4 vital article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||||||||||||||||||||||
|
A summary of this article appears in Vector (mathematics and physics). |
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)
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)
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)
Sometimes its easier to define what something is by defining what it isn't.
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)
I have removed the "formal definition" from the first paragraph of the article:
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)
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)
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)
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. [1] 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":
Whereas complex numbers have one number whose square is negative one, quaternions have three independent imaginary units . Multiplication of these imaginary units by each other is anti-commutative, that is, . 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. [1] 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.
It has been suggested that portions of this section be
split out into another page titled
History of quaternions#After Hamilton. (
Discuss) |
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)
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 and the vector 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, , 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)
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)
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)
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)
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)
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)
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)
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)
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)
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)