Talk:Euclidean vector/Archive 4

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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)

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)

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)

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)

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)

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)

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)

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