Talk:Euclidean vector/Archive 2

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 TR³. I can't believe that most of what is in this article is about the vector space structure of TR³ 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 Wikipedia 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 Wikipedia talk:WikiProject Mathematics/Archive13#Vector (spatial))

This topic may be just the source of my own confusion. So, a "spatial" vector is one which "transforms as spatial coordinates do under rotations". I doubt you mean (x, y, z, ict). Could you give me an example of a nonspacial vector? In mathematics the terminology would be very unfortunate because vectors are defined as points in vector spaces :-) Pete St.John (talk) 22:04, 26 February 2008 (UTC)

Incidentally, I couldn't find the section "Vector (Spatial)" at the link above; it just directs to Archive 13, and the only use of the word "vector" on that page is something else. Pete St.John (talk) 22:23, 26 February 2008 (UTC)

Here's an example, which I put in the last section of the article a while ago. A triplet consisting of the length, width, and height of a rectangular box could be regarded as the three components of a (mathematical) vector. Then there's a three-dimensional vector space characterizing the set of all possible boxes, and each box is element of that space. But these vectors are not spatial vectors, since rotating the box does not appropriately transform the vector. Does that help? --Steve (talk) 03:44, 27 February 2008 (UTC)

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 ^talk 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.

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

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

${z}=r\cos \phi$

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

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

$b=\tan \theta =y/x$

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

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)

"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?

The notion of having a "magnitude" and "direction" is formalized by saying that the vector has components that transform like the coordinates under rotations. That is, if the coordinate system undergoes a rotation described by a rotation matrix R, so that a coordinate vector x is transformed to x′ = Rx, then any other vector v is similarly transformed via v′ = Rv. This ensures the invariance of the operations dot product, Euclidean norm, cross product, gradient, divergence, curl, and scalar triple product, and trivially for vector addition and subtraction, and scalar multiplication.

More generally, a vector is a tensor of contravariant rank one. In differential geometry, the term vector usually refers to quantities that are closely related to tangent spaces of a differentiable manifold (assumed to be three-dimensional and equipped with a positive definite Riemannian metric). (A four-vector is a related concept when dealing with a 4 dimensional spacetime manifold in relativity.)

Examples of vectors include displacement, velocity, electric field, momentum, force, and acceleration.

Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations). The magnitude of any vector is a scalar.

A related concept is that of a pseudovector (or axial vector). This is a quantity that transforms like a vector under proper rotations, but gains an additional sign flip under improper rotations. Examples of pseudovectors include magnetic field, torque, and angular momentum. (This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying symmetry properties.) To distinguish from pseudo/axial vectors, an ordinary vector is sometimes called a polar vector. See also parity (physics).

For example, because the cross product depends on the choice of handedness it changes sign under mirror reflection (see parity), its result is referred to as a pseudovector. In physics, cross products tend to come in pairs, so that the "handedness" of the cross product is undone by a second cross product. Likewise, from the point of view of improper rotations, the scalar triple product is not a scalar, it is a pseudoscalar since handedness comes into its definition. It changes sign under inversion (that is when x goes to −x).

--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 Wikipedia 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)