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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)
(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)
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.
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.
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)
I suggest that for clarity we rename the article from Vector (spatial) to Vectors in three dimensions.-- Patrick 10:58, 13 November 2005 (UTC)
(further discussion at Wikipedia talk:WikiProject Mathematics/Archive13#Vector (spatial))
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)
I think it's perfectly accurate. How is a vector not a geometrical entity? -- Fropuff 06:41, 12 January 2006 (UTC)
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.
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)
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)
The two most common ways of describing a spatial vector are:
What is the mathematical representation of a curve as a vector?
Recent addition (bold):
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)
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)
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)
![]() | This page is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
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)
(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)
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.
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.
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)
I suggest that for clarity we rename the article from Vector (spatial) to Vectors in three dimensions.-- Patrick 10:58, 13 November 2005 (UTC)
(further discussion at Wikipedia talk:WikiProject Mathematics/Archive13#Vector (spatial))
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)
I think it's perfectly accurate. How is a vector not a geometrical entity? -- Fropuff 06:41, 12 January 2006 (UTC)
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.
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)
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)
The two most common ways of describing a spatial vector are:
What is the mathematical representation of a curve as a vector?
Recent addition (bold):
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)
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)
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)