Talk:Vector (spatial)

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1 Is Vector a kind of Tuple?
2 Euclidean?
3 Vector Symbols
4 Positive definite
5 Notation
6 The one thing missing
7 Zero Vector
8 Overhaul coming up
9 Graphic representation
10 Different articles dealing with vectors
11 Vector (physical)
12 contravariant
13 Cleanup
14 What is this article about?
- 14.1 A possible new intro section
- 14.2 Historical remarks
15 norms

[edit] Is Vector a kind of Tuple?

Well... is it?

I think it is, but I have no formal background in mathematics, so I will not put it in.

If it is, I wish someone would add this to the description, because it eases generalization of mathematical concepts, which is a pretty neat thing. —The preceding unsigned comment was added by 200.164.220.194 (talk) 02:53, 23 December 2006 (UTC).

Mathematically speaking, a vector can be represented by a tuple. See Coordinate_vector for more information.192.55.12.36 (talk) 21:33, 22 April 2008 (UTC)

Thanks for drawing my attention to this thread. See tuple. I am adding it to the disambiguation page vector. silly rabbit (talk) 21:23, 22 April 2008 (UTC)

[edit] Euclidean?

I feel very confused about this article. I agree the need of text about curvilinear coordinates, but that is not the most important. Why in this text are defined only vectors in Euclidean coordinate system (orthogonal basis) since it is a special case??

Eswen 23:05, 1 April 2007 (UTC)

[edit] Vector Symbols

Can anyone tell me what the difference between having the arrows above and below the vector means? i.e. $\overrightarrow{AB}$ and ${CD}_{\rightarrow}$ --pizza1512 ^{Talk Autograph} 18:10, 30 April 2007 (UTC)

The overarrow is the "American" style, the underarrow the "European" style. I use both, so I don't know what that makes me. Silly rabbit 06:27, 25 May 2007 (UTC)

Isn't the symbol $\overrightarrow{AB}$ a symbol for a ray? Professor Calculus (talk) 15:57, 6 March 2008 (UTC)

The ray from point A to point B is equivalent to a vector with the same slope and magnitude as the ray. In this context, we usually think of the vector as the parallel transposition of the ray, with A mapping to the origin of the coordinate system. I use $\overrightarrow{X}$ to distinguish the vector X from the variable X, and use $\overrightarrow{XY}$ for the ray from point X to point Y, myself. Pete St.John (talk) 18:07, 6 March 2008 (UTC)

Oh, thanks. I didn't realize that.Professor Calculus (talk) 00:51, 7 March 2008 (UTC)

For the representation of vectors perpendicular to a page, it is pretty clear that circle-dot (⊙) is coming up from the page, however for going down into the page, is it circle-x (⊗) or circle-cross (⊕)? The text says the latter but the big image shows the former. I think the text is right because the circle-dot is also a symbol for the Sun (hence up), while the circle-cross is also a symbol for the Earth (hence down). --George Hernandez (talk) 20:02, 17 April 2008 (UTC)

It's always a circle with an x in it, not a + or cross. The text should be corrected. --Steve (talk) 17:02, 18 April 2008 (UTC)

[edit] Positive definite

This article references the term positive definite, which is a disambiguation page. Please review this usage and determine which of the articles at the disambiguation is intended and adjust as appropriate. Chromaticity 02:17, 7 May 2007 (UTC)

[edit] Notation

As far as I know, the proper symbols for the magnitude of a vector are double vertical lines, i.e.:

$\left \| \mathbf{v} \right \|$

The use of single bars, i.e.:

$\left | \mathbf{v} \right |$

Is generally discouraged, because they are used for the absolute value of scalars and the determinants of matrices.

As for vectors themselves, the accepted notations are column matrix, row matrix, or ordered groups:

$\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ \vdots \end{bmatrix}$ , $\mathbf{v} = \begin{bmatrix} v_1 & v_2 & v_3 & \cdots \end{bmatrix}$ , $\mathbf{v} = (v_1, v_2, v_3, \ldots)$

The common notation using angle brackets, done in order to distinguish them from coordinates (an arguably unnecessary distinction), can result in confusion with inner products, especially in $\mathbb{R}^2$ :

$\mathbf{v} = \langle v_1, v_2 \rangle$	$\mathbf{v} \in \mathbb{R}^2$
$\langle \mathbf{u}, \mathbf{v} \rangle = \mathbf{u} \cdot \mathbf{v}$	$\qquad \mathbf{u}, \mathbf{v} \in \mathbb{R}^n$

At least some of this should probably be mentioned in the article.—Kbolino 07:42, 23 May 2007 (UTC)

[edit] The one thing missing

Examples of vectors being used to represent physical quantities! Overwhelmingly this page will be accessed by people taking introductory physics courses and yet this article fails to make one very simple and very needed connection: instantiation.

A good example

A man walks 4 meters east and then 3 meters north. How would we use a vector to represent his displacement? (insert picture) If we pick east to be the positive x direction and north to be the positive y direction, we can represent the man's displacement $\boldsymbol{x_1}$ by the vector (4,3). If we drew this vector as an arrow, it would have a length of $\sqrt{3^2+4^2}=5$ and point in a direction $\theta = \arctan{\frac{4}{3}}$ above the positive x axis.

If we then had the man walk 5 meters south and 5 meters east. Call this displacement $\boldsymbol{x_2}$ . Clearly

$\boldsymbol{x_2}$ = (5,-5)

which itself has a length of $\sqrt{5^2+5^2}=5\sqrt{2}$

Suppose we wanted to know the man's final displacement $\boldsymbol{x_f}$ after traveling through $\boldsymbol{x_1}$ and $\boldsymbol{x_2}$ . This would simply be the addition of the two displacements. Following the above rules for vector addition, we can see that

$\boldsymbol{x_f} = \boldsymbol{x_1} + \boldsymbol{x_2}=(4,3) + (5,-5) = (9,-2)$

Recalling our coordinate system, this means the total displacement $\boldsymbol{x_f}$ of the man is 9 meters east and -2 meters in the north direction, or 2 meters in the south direction.

Add this and the article's usefulness increases enormously. The fact is there is not one example in this article of a vector actually being used for a physical quantity.--Loodog 05:50, 9 June 2007 (UTC)

[edit] Zero Vector

There is no mention of zero vectors <0,0,0> in this article. They're hardly important in the topics covered here, but they do pose a problem with the initial definition of vectors involving direction and magnitude, since zero vectors have no direction. The concept is covered decently in other articles so maybe a change is unnecessary, but it's just a thought.

OzymandiasOsbourne 18:59, 30 June 2007 (UTC)

No, you're right. A brief note would be prudent.--Loodog 20:32, 30 June 2007 (UTC)

[edit] Overhaul coming up

There is so much information here just thrown at the reader in a cluttered manner. I'm giving this a major overhaul to simplify it down to something more useful to a lay reader or freshman physics student, which is the primary audience.--Loodog 14:50, 2 August 2007 (UTC)

[edit] Graphic representation

My internal word bank may be screwed up today, so bear with.

My geometry teacher made it extremely clear that there is a difference between a ray (point, other point, then an arrowhead) and a vector (a point, and then HALF of an arrowhead at the second point). Is the latter conventional, used at all, or what? If it is used, even rarely, would it be worth noting? 97.86.248.2 22:00, 24 October 2007 (UTC)

[edit] Different articles dealing with vectors

(changed the name due to archiving an unrelated part Arcfrk (talk) 04:05, 29 February 2008 (UTC))

I just ran through "vector (spacial)", "vector (physical)", and "vector". So this is supposed to be the real math article defining vector, and "vector" is just a disambiguation page, and "vector (physical)" is Firefly's unique view, which has been suggested for merging but which is definitely not the topic of this article? Is that correct? Pete St.John (talk) 01:23, 28 February 2008 (UTC)

No, vector space should be the general article on vectors in mathematics. This article is on the common definition of vectors as things having a magnitude and "direction". As soon as you say that something has a "direction", you are implying a specific relationship to the spatial coordinates (rather than an arbitrary tuple of numbers) and thus you aren't talking about an arbitrary vector space; advanced texts make the concept of "direction" more precise by requiring the vectors to be contravariant, but this is just the precise form of the freshman calculus/physics definition. —Steven G. Johnson (talk) 05:54, 28 February 2008 (UTC)

I'm unclear on the distinction between "spatial coordinates" vs "arbitrary tuple" (do you mean, "vectors in Euclidean n-space for n up to 3"?) and I'm unclear on the distinction between "direction" and "unit vector" (my concept of "direction" is "vector divided by it's magnitude", so I don't grasp a vector space without directions). Are the differences just the phrasing for appropriate pedagogy for various backgrounds, e.g. undergraduate engineers vs math majors? Is the six dimensional vector describing the momentum and spin of a billiard ball bouncing off the table, a "spatial" vector? Thanks, Pete St.John (talk) 21:21, 28 February 2008 (UTC)

The differences are a precise version of what it means to have "direction", and are relevant for lots of things from the study of symmetry to differential geometry. No, momentum+spin is not a contravariant vector. See the "contravariance" section below; contravariance is a relationship between two vector spaces, one for spatial positions and another one for the contravariant vectors. 03:23, 29 February 2008 (UTC) —Preceding unsigned comment added by Stevenj (talk • contribs)

[edit] Vector (physical)

Vector (physical) has been nominated for deletion. As that article has an overlap with this article, I believe the discussion may benefit from commentary from editors interested in this area. The discussion may be found here. Thanks. --Sturm 15:30, 28 February 2008 (UTC)

[edit] contravariant

Is the defintion of contravariance complete? To me, it seemed first, that all vectors are contravariant (so I don't see how that can distinguish "spatial" vectors); and that the transformation M is interchangeable (in the paragraph) with the inverse of its transpose, suggesting that covectors are defined with respect to a set of vectors and to a particular transformation. I certainly defer to Johnson about mathematical physics, but the defintion is not clear to me. Pete St.John (talk) 22:39, 28 February 2008 (UTC)

It would seem not. I'm not sure how to address the problem, but I think that calling a "vector" a "contravariant vector" actually refers to the components of the vector. So in the definition, the vector space V must be given coordinates in order for the two notions to make sense. Actual elements of V would then be covariant, whereas the components in a basis are contravariant. Unfortunately, physicists don't usually work in general abstract vector spaces, so I think they have the variances switched up. I have no idea how to correct this problem in a way that would be vaguely intelligible. Is John Baez available for comment? Silly rabbit (talk) 23:03, 28 February 2008 (UTC)

I find that article a little unclear, I agree. Given a spatial coordinate system, a contravariant vector is one whose components transform under rotations in the same way as the spatial coordinates. That is, (from the current article, if you scroll way down to the end): if the coordinate system undergoes a rotation described by a rotation matrix R, so that any spatial coordinate vector x is transformed to x′ = Rx, then any other contravariant vector v must be similarly transformed via v′ = Rv. (In differential geometry, there are ways to define contravariance in a coordinate-free fashion IIRC, but most physics texts don't go into this.) This is just a formalization of the notion of "having a direction"—to have a "direction" implies a specific relationship to positions in space.

The important thing is that this is a relationship between two (or more) vector spaces: given one vector space defining spatial "positions" (or more generally some curvilinear coordinate system, but let's not get into curved manifolds), contravariant vectors are other vector spaces that are associated with the first vector space in the sense of transforming similarly under rotations. This is most certainly not true of all vector spaces—there are many vector spaces that have nothing to do with some other spatial coordinate system. (For example, you could make a vector space out of the charge density at three points in space, but a triplet of scalars does not change under coordinate rotations and hence does not define a "direction.")

Note that this relationship between vector spaces arises in physical laws, but also arises in pure mathematics; e.g. the gradient of a real-valued function of (x,y,z) is covariant with the (x,y,z) coordinates. (The difference between covariant and contravariant vector spaces disappears as long as one is talking about rotations in Cartesian coordinates, which are given by orthogonal matrices.)

(One subtlety arises when one includes improper rotations, in which case one obtains both contravariant vectors and pseudovectors in three dimensions, from curls and cross products.)

The tricky thing, in this article, is to be precise and yet remain accessible at the level of first-year physics and calculus students, where vectors "having a direction" are usually introduced. The article needs to make it clear that there is a more precise definition of what it means to "have a direction", as well as other kinds of general vector spaces that aren't associated with "directions" per se, without scaring off the readers by forcing terms like "contravariant" on them.

—Steven G. Johnson (talk) 02:13, 29 February 2008 (UTC)

Thanks for the input. At the risk of going offtopic, I would like to cleanup the covariance and contravariance article somewhat, and I would like a physicist's input on one thing. In mathematics at least, something is contravariant if it varies inversely with respect to a change in some reference elements (see for instance contravariant functor). The term and usage go back at least to the mid 19th century writings of Sylvester. However, physicists seem to use the word with the opposite meaning: something is contravariant if it varies in the same way as the changes in coordinates. I would like to give some indication for the reason for this peculiar usage of the term. I have privately speculated about it, and indeed have a reasonable hypothesis. However, perhaps you know of some standard answer to the question. Silly rabbit (talk) 15:08, 29 February 2008 (UTC)

I don't know; the usage of the term "contravariant" in this context always seemed a bit strange to me, but all of my books just define the terms without giving any explanation of their origin. —Steven G. Johnson (talk) 18:57, 29 February 2008 (UTC)

[edit] Cleanup

Silly rabbit asks that I list criticisms of this article to assist with a rewrite that he is planning. I had been thinking of waiting until the AFD of Vector (physical) was over but it's good to jot these points down while they are fresh in my mind.

Starting at the top, we have the title, for which there is some previous discussion. The term spatial vector does not seem right. Its usage seems to occur in the specialist fields of ECG analysis and soliton waves. Feynman talks of a space-vector which is plainer English but begs the question of what sort of space we are talking about - the general reader might suppose this is Outer space. In a case like this, where the qualifying word in parentheses indicates a domain of knowledge or context, the title Vector (mathematics) might be better.

If we look at the content, we see that the article does not just describe vectors and their representation, but also includes simple operations like scalar multiplication and vector addition. In some cases, these topics are redirected to this article, e.g. Vector sum. My preference would be to break these operations out into separate articles as they seem to overload this one. But if they are retained, they indicate that the title perhaps ought to be broader - what Feynman calls vector algebra.

The opening sentence uses the word object which is a poor choice when discussing an abstraction. A fundamental concept of this sort should be explained with more care and the lede should be polished word-by-word.

The article lacks good examples and there is some previous discussion of this too. There is one example of a velocity of 5 metres per second upwards and the notation (0,5) is used for this. This seems poor in that it does not explain the notation, the origin and the use of just two dimensions. And there is no discussion of the fact that your upwards is not the same as mine, because we are standing on a sphere in a gravitional field. Oversimplification of this sort, which is not well-linked to the physical world, will tend to confuse the general reader. I would like to see at least one genuine well-developed example - perhaps something from aeronautics where they talk of thrust vectors and intercept vectors.

The article is poorly sourced. A good general guideline is that there should be a citation to a reliable source per paragraph.

Wikipedia is not a textbook and so we should be careful to avoid the exact style of works which are, such as Feynman's lectures which I allude to above. Also, we are not trying to impress but inform and should consider that our readership is the world. When I was young, I used to enjoy reading works such as the Children's Encyclopedia and so I suppose that our readership includes bright children of age 10 or younger. The article currently seems too dry for this readership. For example, the original text used the word arrow which seems natural. The article now uses the term line segment. This perhaps improves the mathematical rigour but that's not good if causes people to stop reading.

Note that I criticise no particular editor. The previous editors are to be congratulated upon having got the article to this point.

Colonel Warden (talk) 08:22, 1 March 2008 (UTC)

I agree that the title is suboptimal. However, before anything else we first need to agree on what the article is about. The original point of this article, as I understand it, was to be about the concept that is informally introduced (at a freshman level) as something with "magnitude and direction", and is formalized at a more advanced level (defining what it means to "have a direction") as being a vector space contravariant with spatial coordinates under rotations.

(Such spaces are always finite-dimensional in practice, and always have a dot product...they need one in order to define the rotation group, and also need one to define the "magnitude" of the vector...and being concepts in differential geometry they also have two-forms, or cross products in 3d. And all vector spaces have addition and multiplication by scalars—if you don't have these, it's not a "vector" by any definition in mathematics or physics, so your suggestion that these don't belong in the article does not seem right to me.)

Vector (mathematics) is not right unless we are talking about general vector spaces. There is no need for such an article, since we already have vector space; also, this concept is considerably more abstract than the "magnitude and direction" notion of vectors (e.g. it includes infinite-dimensional vector spaces and vector spaces not over the real or complex numbers), and need not include a dot product (a dot product, or at least a norm, is necessary to have a "magnitude").

Vector (physics) is not right either, as there are many types of vector spaces used in the physical sciences, both finite and infinite-dimensional and both contravariant and more abstract.

Vector (contravariant) would be correct, but would be off-putting and unfamiliar to undergraduate students who should otherwise be the target readers for an article on this subject; it also might be a bit too general—we should probably stick to contravariance in Euclidean/Cartesian space under rotations, and leave curvilinear coordinates, curved manifolds, and different groups (e.g. the Lorentz group in relativity) to other articles. Vector (polar) is another term for this concept, but it is even more obscure and will easily be confused with polar coordinates. Since we are primarily talking about contravariance with respect to the rotation group in three dimensions, in relativity they would occasionally be called "three vectors" in contrast to "four vectors" which are contravariant with respect to the Lorentz group in spacetime, but this terminology is also unfamiliar to most users of such vectors.

Firefly suggested Vector (Gibbs-Heaviside) in reference to the historical originators of this concept of vectors (although the precise notion of contravariance didn't come later, as I said this is just a formalization of the earlier notion of "having a direction"). However, this terminology does not seem widespread.

Vector (magnitude and direction) is a bit wordy, but would be reasonably clear and uses the most common (albeit informal) terminology used to describe the concept. Vector (spatial) was an attempt at an abbreviated reference to the fact that contravariant vectors have a specific relationship to spatial positions (hence "direction"), but is not a widespread terminology. My vote, as the best compromise title I can think of, would be for Vector (magnitude and direction), unless someone can come up with a better suggestion (or unless we are going to change the subject of this article entirely).

(References are not a problem to add. There are dozens (at least!) of undergraduate calculus and physics textbooks covering this stuff at a basic level. And there are plenty of books covering the definitions at a more sophisticated level; e.g. Arfken & Weber, Mathematical Methods in Physics has a particularly clear discussion without going too far into differential geometry.)

—Steven G. Johnson (talk) 17:14, 1 March 2008 (UTC)

I'm surprised that you see this as an undergraduate/freshman topic. The basics of vectors are introductory material for science/maths at a secondary level in the UK and so pupils will meet this at age 13 or so. I suppose that it is much the same in the USA, i.e. high school. No? Colonel Warden (talk) 19:59, 1 March 2008 (UTC)

It's a while since I did my O-levels, but I'm pretty sure we didn't cover vectors as directed lines segments (A-level), vector addition (also A-level) and scalar products (that would be A-level again). What we did cover was that vectors have direction and magnitude, and that velocity, acceleration etc were vectors. I find it difficult to believe that today's 13-year-olds are doing stuff I did at A-level. However, I do believe it is the case that some aspects of mathematics (calculus, for example) which US students only meet at the age of 18 are on current A-level syllabuses; it's possible these include the basics of vectors. --Sturm 20:46, 1 March 2008 (UTC)

In the UK, the current National Curriculum specifies the following at Key Stage 4 for Mathematics which is ages 14-16:

Vectors

understand and use vector notation; calculate, and represent graphically the sum of two vectors, the difference of two vectors and a scalar multiple of a vector; calculate the resultant of two vectors; understand and use the commutative and associative properties of vector addition; solve simple geometrical problems in 2-D using vector methods.

The equivalent stage for Science is less clear on this subject but pupils are expected to understand "the difference between speed and velocity".

Now these are core subjects and so this is universal education that everyone is supposed to get (though many fall by the wayside, of course). Wikipedia should be operating primarily at this level since our readership is universal. When Steven says that this is a college-level topic, we have a significant difference of expectation. The UK has developed a National Curriculum to make such expectations clearer but most academic institutions are familiar with the concept of a syllabus. That's what Wikipedia needs perhaps - some method of breaking topics into introductory, intermediate and advanced levels. Colonel Warden (talk) 22:05, 1 March 2008 (UTC)

What you've just outlined appears to be from the key stage 4 higher programme of study "Ma3 shape, space and measures". Higher programmes are recommended to be taught to students who've achieved "a secure level 5" or level 6 at key stage 3. They aren't recommended to be taught to those achieving lower than this, so I'm not seeing this as a "core subject". (And that's without going to the issue of those who are "taught" but still arrive at university without understanding.) --Sturm 23:00, 1 March 2008 (UTC)

You're right, this is covered in high school too in the US, to a limited extent. Certainly, the article should begin by being as accessible as possible, although farther down in the article it can have subsections on more advanced concepts. In any case, both the high school and the freshman level informal definitions are the same (things with "magnitude and direction"), and the question becomes where to go from there. You have to talk about addition of vectors and multiplication by scalars, both of which are fundamental to the concept of "vector", and you have to talk about dot products (to get a magnitude). And you have to give at least some indication of how these informal concepts connect to more precise and/or general notions of vectors. In particular, at some point the article needs to say precisely what it is about, what "magnitude and direction" really means, and what types of vector concepts are included (so that the article isn't constantly being tugged into different directions from different generalizations of the vector concept). —Steven G. Johnson (talk) 22:26, 1 March 2008 (UTC)

[edit] What is this article about?

I think the question which Steven Johnson raises above is a good one, and probably deserves to be addressed in its own section. As a model for the original article, I used the introduction and first chapter of Pedoe's Geometry, in which the "magnitude and direction" aspect of spatial vectors is mentioned, although not necessarily emphasized to the exclusion of all other points of view. Pedoe treats vectors as synthetic geometric objects on their own: directed line segments (or equivalence classes thereof). The Encyclopedic Dictionary of Mathematics and the Springer Encyclopedia of Mathematics also take this approach. (Curiously, the Springer article is called vector (geometric), which may be another possible title.) I found the approach quite appealing because it readily permits the idea of spatial vector to be discussed outside of a purely Euclidean context. Thus a vector in special relativity, as a directed line segment, is on equal footing with a vector in classical mechanics.

The idea of "magnitude and direction" is a good one to guide intuition, and for the purposes of explanation, but as Steven points out, it is rather poor as a definition. Looking again at Tom Apostol's Calculus, he seems to be aware of issues such as these, and identifies three distinct approaches:

The analytic approach: Essentially do everything in coordinates.
The geometric approach: Directed line segments
The axiomatic approach: Vector spaces.

I believe that the consensus is that other articles satisfactorily address the third of these possibilities, and that this article should do no more than perhaps point the way to them.

I think that this article should handle both the geometric and the analytic approaches to spatial vectors. There are some arguments against this, I realize. The chief among them is perhaps that there already are articles on the analytic concept of vectors (e.g., coordinate vector), and that, furthermore, this article will never be able to break its current duality unless it selects one definite approach to take. My replies to these objections are as follows. Firstly, none of the existing articles (coordinate vector, vector calculus, etc.) appears quite ready to take up the torch of presenting analytic vectors in a coherent and well-motivated way. Furthermore, modern readers are likely to see both concepts developed in tandem rather than separately, and I think the vector (spatial) article should keep with this tradition. Secondly, when one gets into the physics (or deeper into the geometry), the coordinate approach and the geometric approach become inextricably linked. In physics, one deals with the ideas of covariance under changes in the coordinates, and in geometry with similar notions via the Erlanger program. In each case, the coordinates are not what one uses to model the vector per se (they are not an "intrinsic feature" of the vector's description), but rather are a tool for modeling the space around the vector. Thus, in physics and geometry, one thinks of the vector as existing independently of the coordinate system. Hence I believe that the geometric approach is the one which must be given precedence in the article. Silly rabbit (talk) 18:37, 1 March 2008 (UTC)

Even restricting considerations to the "analytic" approach, there is a tension between the definition of a "vector" as essentially anything in $\mathbb{R}^n$ , and the physics/differential-geometry notion that "having a direction" implies a specific relationship to some notion of position (i.e. contravariance). The latter approach is taken e.g. in Arken & Weber, Mathematical Methods in Physics, although they also describe abstract vector spaces as the alternative.

Put another way, starting from the informal concept of "magnitude and direction", there are essentially two routes one can follow towards more advanced and precise concepts. One is the concept of a Banach space (not just a vector space if you want to have a "magnitude"), and the other is the concept of contravariant vector spaces, tensors, and differential geometry (which formalize the notion of "direction"). The article, at least, needs to make it clear that both of these routes are possible, and distinct choices, and needs to make it clear how far along either route we intend to go in this article.

The typical physics point of view (as e.g. in Arken & Weber) is that contravariance is the natural formalization of the specific concept of "magnitude and direction", whereas abstract vector/Banach spaces are a more general concept (that includes contravariant vectors as a special case).

—Steven G. Johnson (talk) 19:16, 1 March 2008 (UTC)

Yes, that is another issue that I had attempted to address during an earlier incarnation of this article: how and when to introduce "magnitude" and "direction". I didn't get very far beyond consolidating all discussion of the dot product and norm into a section of its own, and miscellaneous other movement of text around. I pretty much gave up trying to make a decent "overview," but perhaps more editors can now come up with something better. (It's sort of a shambles now, I know, but it is significantly better than how it started out about a year and a half ago.)

Magnitude and direction can both be built into a transformation-group (Erlangen or covariance-based) approach to vectors, and so have appropriate generalizations to basically any geometry. Two vectors have the same "magnitude" if they are congruent under the action of the Euclidean group; they have the same direction if they are congruent by a scaling-translation.

I was reluctant to ascribe "magnitude" a particular mathematical meaning in the beginning of the article, but it seemed impossible to abandon the term completely. The Encyclopedic Dictionary cleverly dodges the problem by saying something like: "The concept of vector originated with the physical notions of velocity and force, which are quantities having both a magnitude and a direction." The article then goes on to treat vectors geometrically until enough background has been developed to introduce magnitude safely.

However, you now raise yet another interesting point. As long as one is willing to grant that magnitude means length, then various sorts of normed spaces and inner product spaces are natural generalizations. Even in finite dimensions, it is trivial to write down norms which are not the usual Euclidean length. However, these more general spaces are probably never associated with the idea of a spatial vector, and this article should do little more than indicate the existence of such generalizations, and provide appropriate links. Silly rabbit (talk) 19:49, 1 March 2008 (UTC)

The persistent problem with this article is that it gets tugged in too many directions, with too many possible generalizations and notions of "vector".

I really think that this article should as much as possible stick to the elementary concept of a vector as things with magnitude and direction, with the specific meaning of "magnitude" in terms of the ordinary norm, with the usual operations on such vectors, and some typical applications, with a short section saying that "direction" can be formalized in terms of contravariance, and with another short section giving brief pointers to other articles on different generalizations.

The audience for this article should be people who are trying to understand "vectors" as they are used in elementary calculus and physics courses, and it should therefore only give brief pointers to other articles for generalizations that are not used in such contexts. (Contravariance/covariance, on the other hand, is not a generalization so much as a precise definition of "having a direction".)

—Steven G. Johnson (talk) 20:07, 1 March 2008 (UTC)

Yes, I basically agree with you. I wasn't advocating including a bunch of generalizations in the early parts of the article, but was rather trying to rationalize the definition of a vector as a "directed line segment" as being the geometrically correct one. We can still have magnitude and direction, of course, but I don't think that should be advanced as the definition since it is inadequate for a variety of different reasons. Silly rabbit (talk) 20:31, 1 March 2008 (UTC)

This seems unnatural to me. How is a velocity, or an electric field, or a gradient, a "directed line segment"? (Who defines it this way, except for displacement vectors?) My suggestion would be to give the "magnitude and direction" as the informal definition (this is totally standard, almost universal in introductory courses), and say that this can be formalized and made precise in a variety of ways—the standard way (in physics and differential geometry, when talking about things with "magnitude and direction" as opposed to other abstract types of vector spaces) being as contravariant vectors—whereas there are also abstract generalizations of "vector" that aren't naturally described as having either a "direction" or a "magnitude" (i.e. a norm) and which are only mentioned in the article. —Steven G. Johnson (talk) 21:57, 1 March 2008 (UTC)

Several sources define it this way. The sources I have been looking at include the Springer Encyclopedia of mathematics, the Encyclopedic Dictionary of Mathematics, Dan Pedoe's Geometry, and Tom Apostol's Calculus. These are all fairly standard mainstream sources. I think the first two are especially pertinent because they are from encyclopedias, which must strive to give an encyclopedic definition. However, I also understand your objection to the arrow definition, that it perhaps is not encompassing enough to all quantities which would be relevant to physicists (which "have a magnitude and direction"). If one considers space as the physical position space, then indeed velocity, force, etc., are not arrows in that space but instead in some other auxiliary Euclidean space (a tangent space or other such). That's obviously treading into untenable territory, and should be avoided as well. This seems to be the source of the duality in the article, then: neither approach is correct. One can either give a suitable (and reasonably rigorous) definition of a geometrical vector, or a hand-waving "magnitude and direction" characterization which covers the uses in physics but is wholly inadequate as a definition. Silly rabbit (talk) 22:16, 1 March 2008 (UTC)

(restarting indenting)

The problem is, any definition which appears to exclude velocity as a "vector" is going to alienate a large fraction of the audience for this article (people who learned, or are learning, these concepts in the context of high-school/undergraduate math and science courses). What's wrong with handwaving about "magnitude and direction" (calling it an "informal" definition) while stating that there is a more precise definition as well as other generalizations, and then giving the more precise definition of "direction" via contravariance (in the simple case of Cartesian coordinates under rotations) for advanced readers in a subsection? (You don't need to go into tangent spaces as long as you're willing to represent vectors by components.)

This is not really any different from, say, real number. You neither want nor need to start with any rigorous definition in terms of Dedekind cuts etc.; you start with an informal notion, and then eventually give a precise meaning for those (relatively few) readers who will be sophisticated enough to appreciate it.

—Steven G. Johnson (talk) 22:34, 1 March 2008 (UTC)

I can think of a few problems with putting "magnitude and direction" ahead of other definitions. The first of which is that the article really should have a definition, if possible, as early on as possible. Readers shouldn't have to wait until the last few sections. Secondly, the magnitude and direction characterization, while immensely useful for saying what things in physics can be described by vectors, does not say what can be done with vectors. For instance, to add two vectors, you either need to commit to a coordinate system (which I think should be avoided if possible), or you need to represent them as arrows in a Euclidean space so that you can apply the parallelogram rule. Perhaps we can come to some suitable language which will encompass both points of view. Silly rabbit (talk) 22:44, 1 March 2008 (UTC)

[edit] A possible new intro section

Just to be more concrete, let me propose a possible new introduction:

In elementary mathematics and physics, a vector is informally defined as something described by a magnitude (a non-negative number) and a direction. Geometrically, vectors are often represented by directed line segments ("arrows") with length proportional to the vector magnitude and pointing in the direction of the vector. A typical example is velocity, which has both a magnitude (the speed, e.g. 100 km/h) and a direction (e.g. "north"), and might be represented geometrically by an arrow pointing in the direction of motion with a length proportional to the speed. Since its earliest mathematical formulation by Gibbs and Heaviside in the 1880s, this vector concept has become widely used in science and engineering to describe numerous physical variables (such as velocity, force, and electric fields) and also forms the foundation of vector calculus.

This article is about this basic notion of vectors with magnitudes and directions: their applications, the standard operations on such vectors, and their typical representations either geometrically or in terms of vector components. However, the subject of vectors leads towards many more advanced and abstract concepts, which are described in more detail by other articles.

First, the informal concept of "having a direction" is made precise, in physics and differential geometry, by defining a specific relationship, called contravariance, between the vectors (more precisely called contravariant vectors) and a separate notion of spatial "positions." The simplest definition of contravariance, given below, is for Cartesian coordinates in ordinary Euclidean space under rotations, but in differential geometry this is extended to curved manifolds via tangent spaces. The concept of contravariant vectors can be further generalized to tensors, as well as to vectors contravariant with more than spatial positions (e.g. four vectors in special relativity, which are contravariant with both space and time under Lorentz transformations).

Second, there are also much more abstract generalizations of the "vector" concept that are not directly associated with "directions" or "magnitudes," but which share the operations of vector addition and rescaling: vector spaces. In an abstract vector space, the vectors may not have a magnitude (length); the most common abstract definition of vector spaces that do include vector "magnitudes" is called a Banach space. In this viewpoint, contravariant vectors form only one specific class of Banach spaces, in which the vectors have a precise relationship to spatial "directions."

Comments? Besides nitpicks over individual phrasings, is the overall structure of the intro and the definition of the article topic something we can mostly agree on? (Note the important distinction between saying that a vector is a directed line segment versus saying that a vector can be represented by a directed line segment.)

(The brief discussion here of more advanced topics is necessary, both to give readers interested in such topics pointers of where to go, and to clearly demarcate the subject of this article from other articles on Wikipedia...the history of this article shows that, without such a clear up-front demarcation and an explanation that there is a formal meaning of "having a direction," this article will descend into confusion every time an editor who is a math major comes across it. At the same time, by clearly stating that the last two paragraphs are about advanced topics, I think we can avoid intimidating elementary readers.)

Once we agree on the subject of the article, then it will be easier to discuss the title.

—Steven G. Johnson (talk) 05:56, 2 March 2008 (UTC)

I do not like the proposed version, and prefer the current one, despite its perceived lack of applicability to physics. Here are a few comments:

This one may be a bit nitpicking, but why is it legitimate to say that velocity (etc.) are vectors, whereas directed line segments only represent vectors. If anything, this conflates the noumenon (a vector) and phenomenon (the object of human measurement: velocity). I think the line segments are vectors, and velocity (etc.) are represented as vectors. For an object to have a magnitude and direction (in the sense agreed to above) means that the object lives in some Euclidean space, and is a directed line segment there. This is also the meaning of the complicated covariant description of vectors, whether you believe it or not (see Klein geometry). I think, contrary to your suggestion, we can be upfront with a reasonably rigorous definition without descending into confusion as long as it is done properly. At the moment, however, it seems that we are unable to break the stalemate on the issue of the definition of a vector. Some outside input might be helpful.
I think the history is wrong. Gibbs and Heaviside did not originate the idea of a spatial vector. That honor should probably be given to Hamilton who certainly thought of his quaternions in much the same way as our spatial vectors. Use of the word "vector" in its modern form can certainly be found in the writings of Clifford in the early 1870s, though I am not certain if he was the first to do so. Vectors as lists of numbers are conventionally credited to JJ Sylvester, but these are not "spatial" vectors. Certainly Hermann Grassmann's 1844 work bears mentioning (although no-one seems to have read it until vectors were already commonplace). (Note: Someone really should write a history section.)
It is too long, and spends too much time on things which are barely mentioned in the article. Three paragraphs (!) are spent detailing the generalizations of the subject of the article, rather than discussing the subject itself. My original intent (about a year ago) was to move the information about generalizations, covariance, and so forth out of the lead and into the "Overview" section. Certainly, a brief mention of covariance can be given, but it should be proportional to its relative treatment in the article.
The remaining parts of the proposed lead do very little to summarize the content of the article. The lead should at least mention coordinates (in a way that will be understandable to someone who doesn't know what a "rotation" is). It should also mention the fundamental operations defined on vectors and, I feel, say what those operations mean. The bulk of the article is dedicated to such things, and they need to be specifically enumerated in the lead, per WP:LEAD.
One thing the new version does quite unambiguously is to specify the scope of the article. It effectively says that the article is not going to do anything rigorously, and that other, more rigorous definitions can be found in other articles. I suspect that this is overcompensation for my philosophical discussions above regarding the current structure of the article, and definition of a vector as a directed segment in Euclidean space. However, I do not believe that rigor needs to be sacrificed for clarity in this case.

Just my two cents. Silly rabbit (talk) 15:04, 2 March 2008 (UTC)

Regarding your points, in order.

First, the important thing is not your opinion, or my opinion, it is what is in standard references. And there are many, many texts and articles etc. that say that velocity is a vector. This is standard usage of the term, philosophical quibbles aside. (This reminds me of the endless discussion of whether a vector can be described as a "quantity"....the point is that people do use the word this way in published references, whether or not you agree with it.)

Again, with regard to history, most sources seem to say that the modern formulation of vectors is due to Gibbs and Heaviside. Certainly, the terminology and the enumeration of the operations is due to them; quaternions were expressed fairly differently (with a "vector part" and a "scalar part", both of them were considered parts of the same thing). Certainly, we can have a history section that goes into the history in more detail, and discusses the antecedents of the modern formulation.

Actually, only two paragraphs are spend discussing generalizations. Do we count differently? And it seems kind of important to me for the introduction to explain what the topic of the article is (rather than to fit in all of the content of the article), and how it relates to other concepts of "vector" in math and physics.

It does mention coordinates. Defining what they mean will take too long for the introduction and has to be left to the text within the article, as will defining all the standard operations (addition, scaling, subtraction, dot products, cross products). (And please don't lecture me that cross-products are two-forms...at this level, vectors are at most three-dimensional and cross-products are treated as one of the standard vector operations.)

Actually, the introduction does say that a rigorous definition of contravariant vectors, at least for Euclidean space, will be given in the article. (This is fairly easy to do at an undergraduate level if you are willing to work in terms of components.) Honestly, though, most readers will have no use for that level of rigor, which is why it should go in only one subsection. Nor is that level of rigor typically found in elementary treatments of this subject. Nor should this article be about abstract vector spaces. Nor, honestly, is it all that "un-rigorous" to define the operations both geometrically and in terms of components...are you confusing "rigorous" with "abstract and general"?

—Steven G. Johnson (talk) 16:15, 2 March 2008 (UTC)

I think you've missed the point, at least the first one. There may be some vector we call "velocity" and it may represent some particular magnitude and direction of motion (in some inertial frame of reference), but it is called velocity because it represents what is actually velocity. If I had a function P(X) which represents my profit, what is my profit? I might say "P(X)" but it isn't really profit - it's a function. Profit is money in the bank. P(X) just happens to share the name "profit" because it represents profit - by the same token, the money in my bank is certainly not a function - P(X) is the function. Likewise, "velocity" may be represented by a vector, but it is not a vector. A vector is an abstraction. Vectors do not exist in real life. --Cheeser1 (talk) 16:24, 2 March 2008 (UTC)

Regarding the history, is it really true that most sources agree that the modern formulation of vectors is due to Gibbs and Heaviside? Granted, they are the ones traditionally credited with the assembly of the subject currently known as vector analysis. But the conception of vectors as objects with a magnitude and direction clearly predates their work. See for instance Clifford, Preliminary sketch of biquaternions, Proceedings of the London Mathematical Society 1871 s1-4(1):381-395. A direct quote:

The vectors of Hamilton are quantities having magnitude and direction, but no particular position; the vector AB being regarded as identical with the vector CD when AB is equal and parallel to CD and in the same sense.

This is precisely the modern conception of a spatial vector. Silly rabbit (talk) 16:32, 2 March 2008 (UTC)

For anyone whose English, particularly historical-mathematical-English, might be lacking, "equal" means equal magnitude, "parallel" means what it normally means, and "same sense" means pointing the same direction (since "parallel" could mean same direction or directly opposite). --Cheeser1 (talk) 16:39, 2 March 2008 (UTC)

Another point, which I may not have explained clearly enough, is that I regard it as perfectly rigorous to define a vector as a directed line segment in Euclidean space. One can then proceed to also define the addition, subtraction, and scalar multiplication in this context. That is what I mean when I say "rigorous": Rather than using a non-definition ("magnitude and direction") let's at least advance a definition ("arrow in Euclidean space"). In fact, although Steven may not feel it is suitably rigorous or encompassing of everything he would like to call a vector, it turns out to mean precisely the same thing as the definition using covariance properties under the Euclidean group. (One could take this as the definition of Euclidean space.) However, using an arrow strikes me as a much more accessible definition than the one using transformation groups. And yes, there are many many reliable sources that define a vector this way. (In fact, I just peaked at Misner, Thorne, and Wheeler, and even they adopt this definition of a spatial vector.) Silly rabbit (talk) 16:43, 2 March 2008 (UTC)

(my comment here is probably a bit of a non-sequitor) But FWIW, only history and a lot of experimental confirmations have cleared up many of the original mathematical questions and unknowns that initially arose in the wake of the two very successful, very agreeable, yet somewhat historically separated scientific areas: electromagnetics and relativity. For example, mathematical invariance--as has already been pointed out--has been around since at least the 1850's, yet its true usefulness and popularity didn't come to light until the phenomenal success of relativity. On the other hand, as has already been mentioned, even without an upfront and clear application of the condition of invariance, the science of electromagnetics somehow also managed--in the 19th century decades before relativity came into being--to not only accurately predict the physical constant of light, but also to also be one of the few 19th century sciences whose original equations hold true even today under relativistic conditions. Invariance is no doubt a sufficient condition to ensure accurate physical application of entities with magnitude and direction (and this condition can and does do so in, I will be the first to admit, the more graceful manner in terms of expression and mathematics), but a sort of existential quantifier on the history of science nearly proves the existence of an equivalent set of ideas that shouldn't be ruled out altogether (even if not quite as graceful and as well-discussed as invariance). --Firefly322 (talk) 02:59, 3 March 2008 (UTC)

[edit] Historical remarks

I think that we should tread very lightly around the history of vectors: e.g. Kreyszig, Differential geometry, University of Toronto Press, 1959, p.9 (reprinted by Dover, 1991), claims that

The concept of a vector was first used by W. Snellius (1581 – 1626) and L. Euler (1707 – 83).

My understanding is that the contribution of Gibbs (and later, of Heaviside) was to advance vector calculus based on what we'll today call "vector space approach", as opposed to the quaternion approach prevalent at the time. But it's very tangential to the main goal of the article, which is to explain modern point of view on vectors, not to give a genetic introduction following their long and convoluted history. Arcfrk (talk) 03:24, 3 March 2008 (UTC)

I agree about treading very lightly around the history of vectors. My understanding, and hope this isn't too much to share, is that several ancient Greek authors wrote about the parallelogram of velocities in two dimensions (including the unknown Greecian author of Mechanica, Archimedes, and Hero of Alexandria). And yes more than a few details would probably exceed the still incipiently fleshed-out goals of the article. --Firefly322 (talk) 05:18, 3 March 2008 (UTC)

Vectors as now taught follows Vector Analysis (Gibbs/Wilson) as a dumbing down of the quaternion studies in hypercomplex numbers. Evidently the mathematical abstraction needed to be reigned in for clear application to physical science. Shedding the multiplicative structure of algebras leads to linear algebra, a field we need to improve in WP. The history of linear algebra is caught up in its applications, the driving force. It is surprising how Euclidean-inspired intuition continues to dominate linear algebra education. Fortunately the WP editors have helped move understanding along by the department-free nature of our colaboration. For the history of Vectors then, we have the early period noted by Firefly322 and others, and then the explosion of the 1890's which produced a Quaternion Society (1899 - 1913) and a conservative reaction curtailing studies to three-dimensions.Rgdboer (talk) 20:41, 6 March 2008 (UTC)

I wouldn't define a group as a ring without multiplicative structure; nor a set as a list without order. Generally we want to build upwards from the simple, specific, and concrete, towards the complex, general, and abstract; so to me defining vectors from quaternions would be backwards (if not unhistorical). Pete St.John (talk) 21:14, 6 March 2008 (UTC)

Putting aside the historical perspective on vectors, doesn't axiomatic math start from the rigorous, the simple, the general, and the abstract and move towards the specific and the concrete? Perhaps the approach of physical science is what is being mentioned here? --Firefly322 (talk) 11:47, 7 March 2008 (UTC)

Indeed it's not so simple as I made it out. "Simple, specific, concrete" may clash; what's simple may be general, etc. However, in axiomatic systems, defintions get more complex with development, as they have more referents (lemmas); so we define Ring (with two operations) after we define Group (with one). So there is a tendency to increasing complexity. But we also define matrix (as an array of numbers) before we define linear transformation, the former being simpler, but also less general; that is, simple and specific and concrete (a technique for solving systems of simultaneous equations) and get more complex, general, and abstract later (inner product spaces). But there are whorls in the current. Pete St.John (talk) 17:51, 7 March 2008 (UTC)

[edit] norms

Is the main distinction between "spatial vectors" and "vectors", that between normed vector spaces and vector spaces? I think part of my own confusion was the sense from the lead that we were talking about vector spaces in general, when I surmised from the discussion that we were not. So perhaps:

Spatial Vectors are used in the sciences to represent magnitudes together with directions, such as momentum or velocity, and comprise normed vector spaces in theoretical mathematics.

But perhaps their are more requirements, e.g. a euclidean inner-product space? I can accept the popular motivation of "magnitude and direction" in the definition, but if there is reference to the actual mathematical object that defines the subject, it would be unambiguous as well as pedagical. There is no harm in mentioning links to the general from an article on the particular, and we can keep at the undergraduate level. Pete St.John (talk) 19:41, 3 March 2008 (UTC)