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Physics B‑class Top‑importance

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Mathematics B‑class Top‑priority

This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Misplaced Pages. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics B This article has been rated as B-class on Misplaced Pages's content assessment scale. Top This article has been rated as Top-priority on the project's priority scale.

Template:WP1.0 Talk:Vector (spatial)/Archive1

I am a little surprised that the first example of a vector (in the introduction) is force. I think the idea of velocity as a quantity with magnitude and direction is considerably more intuitive to the non-physics-educated person. Was there any reason for this choice? Dmharvey File:User dmharvey sig.png Talk 22:07, 3 Jun 2005 (UTC)

Maybe that it is difficult to meaningfully add two velocity vectors at a point to the velocity of something. Still that is no reason to exclude velocity vectors as an example --MarSch 00:17, 5 Jun 2005 (UTC)

Diagrams in Article

(NOTE: I am very sick at the moment and out for a bit of light reading, so while I make sense to me I may be typing gibberish.) Aren't the vectors defined in "Vector addition and subtraction" 3D, while the diagram, with the exact same vector names, is 2D? goofyheadedpunk 05:06, 7 August 2006 (UTC)

vector space

I can't believe vector space isn't mentioned in a prominent place or maybe at all. --MarSch 13:31, 12 Jun 2005 (UTC)

Okay I found it, but I can't believe that this article is really only about 3-dimensional real vectors; elements of TR. I can't believe that most of what is in this article is about the vector space structure of TR and about its Euclidean structure. I think we should chop this article up and merge with various other articles. I don't know if there is any info in here that is not anywhere else, but we'll see.

• For anyone to whom this is not clear:

Please explain to me the relation between the articles: vector (spatial), vector field, vector space, tangent bundle, tangent space, tangent vector(!) which are all about vectors. Then there are also the articles about vectors as tensors: scalar, scalar field, tensor, tensor field, Tensor_(intrinsic_definition), Intermediate_treatment_of_tensors, Classical_treatment_of_tensors.

• Does anyone have a grand vision? --MarSch 13:51, 12 Jun 2005 (UTC)

I've decided that it is probably better to start a cetralized discussion about this issue at Misplaced Pages talk:WikiProject Mathematics/related articles. Please contribute there. --MarSch 14:03, 12 Jun 2005 (UTC)

Rename from Vector (spatial) to Vectors in three dimensions

I suggest that for clarity we rename the article from Vector (spatial) to Vectors in three dimensions.--Patrick 10:58, 13 November 2005 (UTC)

I disagree. Yes, for a mathematician it is clear that a space is a more complicated beast than 3D, and Vectors in three dimensions would be more correct. However, I find the new name needlessly complicated, and Vector (spatial) gives just the right idea to most people, and mathematicians (should) have no problem figuring out what it means. Oleg Alexandrov (talk) 17:12, 13 November 2005 (UTC)

I also disagree. The proposed name feeds into the misconception that a 3-vector as used in physics and engineering is only special in that it has three components. It is not. The distinguishing feature of spatial vectors is not that they are three-dimensional, but that they transform as the spatial coordinates do under rotations. They would be equally distinct in this sense if space had two dimensions, or four. —Steven G. Johnson 19:18, 13 November 2005 (UTC)

(further discussion at Misplaced Pages talk:WikiProject Mathematics/Archive13#Vector (spatial))

Another rename suggestion

Would anyone be opposed if we renamed this page to Vector (geometry). After all, this page is discussing a vector as a geometric construct, an object with a magnitude and a direction. Somehow the word spatial in the title has always bothered me, though I can't quite put my finger on why. Maybe it's because I can't think of any other pages that would be disambiguated by a spatial context. -- Fropuff 06:22, 12 January 2006 (UTC)

"Geometry" implies a restriction to a particular field of mathematics, which isn't very accurate here. "Spatial" is a perfectly good adjective, I think. —Steven G. Johnson 06:26, 12 January 2006 (UTC)

I think it's perfectly accurate. How is a vector not a geometrical entity? -- Fropuff 06:41, 12 January 2006 (UTC)

A vector doesn't need to be geometical. Its simple an ordered list, and so can represent anything to that effect. Fresheneesz 08:41, 10 May 2006 (UTC)

Addition in curvilinear coordinates

I was wondering why no one bothers to tell people how to add vectors in non-cartesian coordinate systems. After all, vector addition is a fundamental operation, and yet when working in curvilinear coordinates addition of vectors is very non-intuitive. Someone should add this in. —The preceding unsigned comment was added by 128.135.36.148 (talk • contribs) 01:43, 2 February 2006.

Probably because the easiest way to add vectors in curvilinear coordinates is to convert to cartesian coordinates, add, and then convert back. Consider the relatively "simple" problem of trying to add two complex numbers in polar coordinates, and you might understand why. -- Fropuff 05:02, 2 February 2006 (UTC)

History of vector mathematics

I would be very curious to know when vectors were developed, by whom, and for what purpose. As a Physics teacher of mine used to say, 'Many great breakthroughs in science had to wait for the necessary mathematics to be developed.' Were vectors explicitly intended as a method of describing forces? Ingoolemo  05:40, 5 March 2006 (UTC)

Revert by MarSch

MarSch removed a little bit of bulleted text in the intro that said that a vector can be described by a magnitude, and one or more angles OR two or more magnitudes with prespecified directions. MarSch called it a falsity. What did you mean by that MarSch. I thought the bulleted part blended well, and was very helpful, in adition to it being true. Am I wrong? Fresheneesz 10:58, 23 April 2006 (UTC)

I think you mean this bulleted list:

The compenents that describe a vector can have one of two equivalent formats:
* a magnitude and one or more angles (which can be defined in a 3-dimensional space by the Euler angles), or
* two or more magnitudes that have predefined directions (these are called components).

The components of a vector depend on the coordinate chart used and there are infinitely many charts. On the other hand there really is no difference between a magnitude, an angle and a magnitude with predefined direction. Either way the bulleted list does not cut it. The only thing worth preserving is giving an example of a coordinate system, in this case spherical coordinates, and I did that and also included two other well-known charts: polar and Cartesian. --MarSch 12:07, 23 April 2006 (UTC)

In what way is there no difference between a magnitude and an angle? Obviously they are both numbers, but one indicates length, and one indicates a component of direction. ..? Fresheneesz 21:04, 23 April 2006 (UTC)

In spherical coordinates the magnitude is just the component in the r or ρ direction and the angles are the components in the θ and φ directions. The only difference is that in this chart the radial component is positive and the angle components range from 0 to π and from 0 to 2π. But using an arctan all those domains may be identified. They may differ in THIS coordinate chart, but there is nothing geometrically (intrinsically) different about them. When is a coordinate an angle coordinate? --MarSch 10:39, 24 April 2006 (UTC)

In spherical coordinates, there is a magnitude (p)and two angles (θ and φ) - spherical coordinates doesn't go against the claim I wrote in bullet points above. I have no idea what you mean by an "angle coordinate" btw. When you said "They may differ" I don't know who "they" is. When you say "coordinate chart" - I'm again at a loss for the meaning of that term. I don't understand your argument, sorry.

But if you could find a coordinate system that goes against my claim (up there in bullet points), I'll concede. But otherwise, I still don't understand the falsity of my claim. Fresheneesz 21:31, 25 April 2006 (UTC)

The "They" refers to the domains of the 3 coordinates in the coordinate chart that is usually referred to as "spherical coordinates". By "angle coordinate" I mean what I think you should say when you say angle, since you are trying to split coordinates in two groups: magnitudes and angles. If you have an inner product than you can define the angle between two vectors and this is not what we are talking about, so it is better to be specific and say angle coordinate when you do not mean angle. What I am saying is that there is no such split possible in a geometrically meaningful way. Just explain to me why p is a magnitude and θ and φ are angles. What is the difference? Also try to imagine a vector field which points in the p direction everywhere, one which points in the θ direction everywhere, and one which points in the φ direction everywhere.--MarSch 17:39, 28 April 2006 (UTC)

Perhaps I meant "angle coordinate", but i'm fuzzy on the difference. "Just explain to me why p is a magnitude and θ and φ are angles." - I'm very confused as to why *you* couldn't tell me that. P is a magnitude because it is simply the magnitude of the vector. If you convert that vector from spherical coordinates to any other coordinate system, the magnitude of the vector will still be that same magnitude p. As for φ and θ, those are angles because they *only* hold information about direction (not magnitude). In those ways, angles and magnitudes are very distinct. p defines a radius θ defines a plane, and φ defines a second plane. Together they define a point - a vector.

A vector field that points in the p direction everywhere? I can't quite see what you're getting at.

I'm sure you're aware that a vector is usually introduced as a construct that has magnitude and direction. Fresheneesz 10:56, 30 April 2006 (UTC)

I am disagreeing with you that coordinates can be split into magnitudes and angles, so I cannot explain the difference to you, because I think there is no such difference. Anyway perhaps the example you asked for is needed, so here goes (in 3 dimensions):

let (x, y, z) be Euclidean coordinates. Then this defines (r, θ, φ) as spherical coordinates.

${\displaystyle {x}=r\sin \phi \cos \theta }$

${\displaystyle {y}=r\sin \phi \sin \theta }$

${\displaystyle {z}=r\cos \phi }$

Now define new coordinates (a, b, c):

${\displaystyle a=\arctan r=\arctan({\sqrt {x^{2}+y^{2}+z^{2}}})}$

${\displaystyle b=\tan \theta =y/x}$

${\displaystyle c=\tan \phi ={\frac {\sqrt {x^{2}+y^{2}}}{z}}}$

Now, Fresheneesz, are the coordinates a, b and c magnitudes, angles or whatever? --MarSch 13:39, 30 April 2006 (UTC)

Ok, that was a good example. I think I finally see your point. However, if you precisely define your coordinate system, then my bullet points still hold merit. For example, one could define polar coordinates so that the θ axis is circular, or one could define it so that it is straight. In the first case, the vector components would be a magnitude and a direction, and in the second case, the components would be two magnitudes with predefined directions. Fresheneesz 22:19, 2 May 2006 (UTC)

The coordinate system I have defined above is precise and it does not fit in your list. --MarSch 10:04, 3 May 2006 (UTC)

What I meant by "precisely defined" is that your axes are defined. For example, if the axes are all parallel (say 3 axes) then the comonents fit the "magnitude with predefined direction" thing. If two axes are circular, or curved (as in spherical coordinates) then the coordinates corresponding to the curved axes are directions (if the axes aren't cicularly curved, then they are directions with additional magnitude depending on direction). This way, variables don't matter, only the orientation of axes matter. Fresheneesz 00:51, 9 May 2006 (UTC)

proposed information at top

I think this is less incorrect, and very useful for people new to vectors, as it describes two interpretations of vectors. Fresheneesz 09:11, 14 May 2006 (UTC)

The two most common ways of describing a spatial vector are:

• a magnitude and one or more angles (which can be defined in a 3-dimensional space by the Euler angles), or
• two or more magnitudes that have predefined directions.

A curve as a vector

What is the mathematical representation of a curve as a vector?

In Mathematics: More or Less?

Recent addition (bold):

In mathematics, a vector is considered more than a representation of a physical quantity. In general, a vector is any element of a vector space...

I don'k know about this - who's to say if it is more or less? It's an abstraction; as such it is less than the thing it is abstracted from (a physical quantity); it is generalized to higher dimensions; as such it is more. Should the addition simply be reverted, or rephrased?--Niels Ø 09:52, 24 November 2006 (UTC)

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).

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)

Vector Symbols

Can anyone tell me what the difference between having the arrows above and below the vector means? i.e. ${\displaystyle {\overrightarrow {AB}}}$ and ${\displaystyle {CD}_{\rightarrow }}$ --pizza1512 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)

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)

Notation

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

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

The use of single bars, i.e.:

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

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:

${\displaystyle \mathbf {v} ={\begin{bmatrix}v_{1}\\v_{2}\\v_{3}\\\vdots \end{bmatrix}}}$, ${\displaystyle \mathbf {v} ={\begin{bmatrix}v_{1}&v_{2}&v_{3}&\cdots \end{bmatrix}}}$, ${\displaystyle \mathbf {v} =(v_{1},v_{2},v_{3},\ldots )}$

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 ${\displaystyle \mathbb {R} ^{2}}$:

${\displaystyle \mathbf {v} =\langle v_{1},v_{2}\rangle }$	${\displaystyle \mathbf {v} \in \mathbb {R} ^{2}}$
${\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\mathbf {u} \cdot \mathbf {v} }$	${\displaystyle \qquad \mathbf {u} ,\mathbf {v} \in \mathbb {R} ^{n}}$

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

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 ${\displaystyle {\boldsymbol {x_{1}}}}$ by the vector (4,3). If we drew this vector as an arrow, it would have a length of ${\displaystyle {\sqrt {3^{2}+4^{2}}}=5}$ and point in a direction ${\displaystyle \theta =\arctan {\frac {4}{3}}}$ above the positive x axis.

If we then had the man walk 5 meters south and 5 meters east. Call this displacement ${\displaystyle {\boldsymbol {x_{2}}}}$. Clearly

${\displaystyle {\boldsymbol {x_{2}}}}$ = (5,-5)

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

Suppose we wanted to know the man's final displacement ${\displaystyle {\boldsymbol {x_{f}}}}$ after traveling through ${\displaystyle {\boldsymbol {x_{1}}}}$ and ${\displaystyle {\boldsymbol {x_{2}}}}$. This would simply be the addition of the two displacements. Following the above rules for vector addition, we can see that

${\displaystyle {\boldsymbol {x_{f}}}={\boldsymbol {x_{1}}}+{\boldsymbol {x_{2}}}=(4,3)+(5,-5)=(9,-2)}$

Recalling our coordinate system, this means the total displacement ${\displaystyle {\boldsymbol {x_{f}}}}$ 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)

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)

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)

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)

"Vectors and transformations" section

The following section was deleted a few months ago (August 27) by Silly rabbit, with the comment, "No one touched this useless section in a few months. Deleting." I don't see why, and I propose putting it back in, perhaps with a bit of rewriting for clarity. Certainly this gives useful information about vectors as they're used by physicists. I noticed the omission, for example, and it motivated me to recently add a section on pseudovectors (which could be merged with this). Does anyone know anything that I don't, or have some opinion?

--Steve 19:36, 30 November 2007 (UTC)

Done. Thoughts? --Steve 05:09, 3 December 2007 (UTC)

Explanation: At some point, I rewrote most of the article, and moved all of the pseudovector cruft and vectors qua transformation laws stuff out to a section of its own since it was junking up the rest of the article. I made very little attempt to make this new section suitable for public consumption. After a few months, I deleted it since I was shocked that anyone would leave such a terrible section in the article. I see you restored it, but still left the legwork to someone else (me) to clean it up. Now I've done my best to tie it all together and make a decent, presentable piece out of it. I still think it can be improved; for instance, I never liked how the old section focused on rotational rather than general covariance. But I can accept this small compromise for the sake of readability over maximum generality. Silly rabbit (talk) 07:52, 29 January 2008 (UTC)

What's the story, silly rabbit?

You just reverted two perfectly good minor edits. The comma is NOT part of the symbol system. It is an English-language comma but in that location must be mistaken for part of the symbol by those who do not know it is not. In the second edit, "quantity" IS in fact the right word. Vector analysis is quantitative and deals with quantities and is not interested in concepts except insofar as they are of quantities. Epistemology deals with concepts, not vector analysis. What I am doing here is an English-language edit. Now, I am not acquainted with the history of this article or your involvement in it and at the moment I am not going to be. I would like to suggest that you reconsider the English edit and put them back in place. Perhaps you acted in haste. I took you at first for a vandal but on looking over this quickly I see you have had some involvement with the article and it has been contentious. Well, we can't always avoid contention but on the other hand if we overreact we hold the article back. Please reconsider.Dave (talk) 00:26, 16 February 2008 (UTC)

I reverted one edit, putting the comma in the correct place in the sentence. See WP:MOSMATH. As for the word "quantity," I am not so sure the word quantity applies: According to the Misplaced Pages article, "a quantity exists as a multitude or magnitude." Is a triangle a quantity, for instance? Is a line a quantity? If these are all quantities, then I stand corrected. If not, then I stand by my opinion that "geometric object" is a much better characterization of what a vector is, rather than a quantity. (By the way, I didn't revert your edit.) Silly rabbit (talk) 00:42, 16 February 2008 (UTC)

A quote from Vectors by Moon and Spencer to sway Silly Rabbit that Vectors are indeed Quantities

Moon and Spencer analyze around 20 definitions for vector--from Gibbs to M.R. Spiegel to Wilson. They go on to state (p. 321)

"Evidently the 'vector' of tensor analysis has little in common with the 'vector' of vector analysis. The latter is definitely limited to Euclidean 3-space and may even require rectangular coordinates. We have seen that there are at least five definitions of 'vector' advocated in vector analysis. The word is defined as a quantity

 (a) Having magnitude and direction, 
 (b) Specified by three numbers, 
 (c) Given by a directed line segment,
 (d) Allowing parallelogram addition, 
 (e) Having certain transformation properties. 

Since these various defintions all act as an introduction to the same subject of vector analysis, one expects them to refer to the same entity. But this can hardly be the case. A quantity (b) specified by three numbers is merely a univalent holor . Every time the coordinate system is changed, the three numbers change, and there is nothing in the definition to tell how they must change to make vector analysis a meaningful discipline.

The geometric definitions (a) and (c) perhaps imply vaguely the idea of a geometric object--the arrow--which is more basic than the coordinate system and which is invariant with respect to coordinate transformation. Such an idea may be implied but it certainly is not stated. And no indication is given as to the GROUP of transformations under which this arrow is to remain invariant. In fact, one finds the whole subject of vector analysis built implicity on one coordinate system. Gradient, divergence, and curl are usually defined only in rectangular Caresian coordinates, as if no other coordinates were ever necessary. Yet practical applications to FIELDS nearly always require curvilinear coordinates. So the ordinary treatment, which has based its proofs on rectangular coordiantes, is forced into highly questionable evasions when it attempts to patch up its systems for curvilinear coordiantes. An example occurs with the -operator, as shown in appendix C.

Apparently sensing a lack of rigor, some mathematicians have introduced (d), which requires that the directed line segments shall add by parallelogram addition. Obviously, this modification does not touch the fundamental defect involving transformations.

Recently the word 'vector' has also become prominent in the term 'vector space'. This term has only the remotest connection with familiar meansings of either 'vector' or 'space'.

'A vector space is any set of geometric objects X, Y, Z, ... called VECTORS that can be 'added' to each other and 'multiplied' by real numbers a, b, c ...-the resulting sums and products being again vectors in the vector space--provided these abstract operations of addition and multiplication obey certain of the laws of ordinary arithmetic'. -- R.H.Crowell and R.E.Williamson (p.53 Calculus of vector functions, 1962, Prentice Hall)

The 'space' is non-metric and the 'vector', being closely related to the general univalent holor , is completely different from the heavily restricted 'vector' of vector analysis.

As a final example of the word 'vector', we might mention the electrical engineer's 'vector diagram'. It involves voltage and current or power, represented by directed line segments in a 2-space. It has no bearing on vector analysis; and fortunately the word 'vector' is now being replaced in this connection by 'phasor'." --Firefly322 (talk) 10:37, 21 February 2008 (UTC)

If anything, this lengthy discussion only supports my contention that vectors should not be regarded as quantities. A quantity, I contend, is a number or perhaps a list of numbers; i.e., what most people would call a scalar. There is, after all, a rather unfortunate trend in this article to coordinatize the article. I think we agree that this is a bad idea, and much of the coordinate-free treatment here is my own work.

That said, I agree that the present article may not give a completely general or satisfactory definition of spatial vector. I'm glad that you seem to be knowledgeable enough to understand some of the more subtle issues; e.g., why we do not define a vector as an element of a vector space. Let me also caution, however, that while the article should strive to be as complete as possible, it must also be readable by first-year undergraduates (or advanced highschool students) studying physics. Bearing this general warning in mind, I am interested to know what recommendations you have for the article. Silly rabbit (talk) 11:57, 21 February 2008 (UTC)

3 Points to Discuss with Silly Rabbit

In that case, I think there are a few things we should discuss. First, the origin of this criteria of "readable by first-year undergraduates studying physics". Second, the notion of a Gibbs-Heaviside vector (i.e., the mathematical-physical object from the Vector Analysis school of thought) which also happens to be the concept found in all first year University level Calculus and Physics textbooks. Third, the meaning of your comment "unfortunate trend in this article to coordinatize the article"

1) This is a wikipedia article and as such unless this article is to be changed to vector for those in freshman physics courses the criteria of "readable by first-year undergraduates studying physics" can have no legitimate basis. So as another contributer just asked you: what's the story, Silly Rabbit?

2) The length, direction, and orientation of a Gibbs-Heaviside vector from Vector Analysis can always be measured. Perhaps not as easily as a scalar quantity, but it can definitely be measured and compared with other vectors with the same units. So Silly Rabbit, are you arguing that vectors can be measured when you write "vectors should not be regarded as quantities" ?

3) The readers of 1st year Physics and Calculus textbooks are implicitly asked to accept a dual system: there is a Finite-Dimentional Cartesian System with an origin matching the zero vector from the Gibbs-Heaviside vector system. Both systems can in fact independently be used to describe objects in Finite-Dimentional motion. Any vector function can be completely represented by a set of scalar cartesian functions. The problem is that this idea is inconsistent with your other comment about the criteria of first-year undergraduate physics majors. For the realization that vectors exist independently of a cartesian space is far beyond the target audience of your criteria. So Silly Rabbit, your ideas contain a paradox, a contradiction. --Firefly322 (talk) 18:44, 21 February 2008 (UTC)

Re: "What's the story Silly rabbit?" I don't appreciate this kind of snarkiness from you or some of the other drive-by editors here. Somehow, someone changes one or two words in the article, and then they feel entitled to behave condescendingly towards me. Whereas I have personally invested many hours of work into this article, and would appreciate a little bit more respect. Please keep the discourse civil if you want to participate in a discussion with me.

On to your points. First of all, my original version of the article aimed for a high-brow approach to the subject. However, another editor User:Loodog and I subsequently reduced the article and made it more suitable for readers at an elementary level. There is nothing wrong with going into further details later in the article, but the lead and introductory material should indeed be understandable by a general audience. This is the general editorial advice of Misplaced Pages:Wikiproject Mathematics for all of the mathematics articles here; unfortunately, it is seldom followed.

Secondly, yes vectors do have a length associated with them, and their length can be measured. That does not, of course, make the vector itself a quantity. (Buildings have a height. Is a building a quantity?) Also, I would prefer to keep properties such as length separate from the definition of the vector. This makes discussion of the generalizations more natural in my opinion. (Affine vectors, pseudovectors, 4-vectors in special relativity, and so forth.) Silly rabbit (talk) 18:34, 21 February 2008 (UTC)

Note: The above text by Firefly was edited after my response to it.

Thirdly, I don't fully understand your point. As your original post suggests, there are several different (but equivalent) ways of regarding a vector: as Euclidean line-segments, as triples (or n-tuples) of coordinates, as coordinate-equivariant lists of quantities (i.e., "tensors"). My "position" contains no contradiction at all. I just want to avoid, as much as possible, commitment to any particular definition of a vector. If you can think of another way to do this, I would be happy to listen. If you just want to continue attacking me, then discussion seems rather unproductive. Silly rabbit (talk) 19:00, 21 February 2008 (UTC)

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