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needed to be able to point to direction of a vector.
Direction should make sense regardless wheather the vector space is normed or not!
In finite dimensional geometry (at least) one traditionally distinguishes direction and orientation; direction of a nonzero vector is the equivalence class of lines parallel to the vector and does not include the orientation (the latter division is a matter of taste). What you defined here is just a unit vector with the same direction as the original vector in the less fundamental setting of normed spaces.
When you edit the entry, please add citations to support your terminology. For what it’s worth Wikipedia agrees with my use of terminology here
Admittedly, that Wikipia entry is written in ambiguous language. The WolframMathWorld article here is unambiguous.
I made some changes and additions to the page to reflect both points of view. It seems that you locked the page for 5 minutes. Are you working on the page ?
I have no time to work on citations now, but last two years I spent teaching elementary geometry and went through most of the well known textbooks on the subject. Definitely it is too much to ask for product when one talks about a version of parallelness. Product enters when one needs orthogonality.
I am fixing it, yes. Please give me a citation for your use of “direction”. By your definition, saying “To get to the train station, either turn left or turn right” is giving directions. Who uses terminology this way? MathWorld disagrees. Also, if you Google “direction of a vector” you find plenty of high school tutorials that give the definition the way I use it.
Well in wikipedia you did not cite direction but the (unit) direction vector which is just a unit vector with the same direction (the latter being more fundamental). One could talk direction line as well. I do not challenge the notion of the unit direction vector as it is in wikipedia.
Urs, I do not talk about orientation or not. This is matter of taste, I seen more primary source one way, As I wrote, you can take equivalence class of oriented lines if you wish. But no norm needed.
Let’s call your concept “unoriented direction of a vector”.
I suppose what is biting you is that people also speak of “direction vectors” of a line, that’s what you seem to have in mind. But that’s different from “direction of a vector”.
Yes, exactly, it is different, so why do you cite that wikipedia page ?
Urs, you erased my text which for unoriented case worked over ANY field. For real orientation on complex lines one just takes a real subspace in it for every element in class, so the oriented case worked for real and complex spaces and some other rientations mutatis mutandis. The present definition is limited to real and complex numbers only!!!
Please go and create direction vector. It”s a different concept.
Unit direction vector can be attached to (the direction of a) vector or to (the direction of a) line, regardless. The notion is the same as the direction is pertinent to (oriented) lines and vectors equally (edit: and contain equivalent information), as it is treated in classical geometry books.
It is unnatural to have a separate page to a direction vector. These are just different facets of special cases of basic notion of parallelness among subspaces of different dimension in the vector space or in some absolute geometry, with equal treatment of oriented and unoriented cases. The very notion of a free vector in the affine space uses equivalences using parallel transport, which is the opening line in your wikipedia entry.
Here is one of the most standard references in classical geometry from a big name in the field:
page 18, Definition 2.2 The equivalence classes in p) associated with parallelism are called directions. The direction of a line is the equivalence class that it belongs to.
The same definition is in the linear algebra textbook by Horvatić which I studied as a physics student. Berger has parallelness of subspaces in his magnum opus Geometry in two volumes but I see no index to find “direction”.
It is strange how counter nPOV you are here. In another thread we discussed that nPOV should treat posets as 0-categories without a difference, but here it is also the equivalent notion in the same sense, weather you defined via oriented lines or via vectors (and higher dimensional analogues with polyvectors and oriented subspaces). So I see it is against the spirit to separate the notion if we express it in different (though isomorphic) terms…
I commented on the multidimensional case both in the idea section and in the comment on my favourite geometry book by J. Lelong-Ferrand who defines direction d’une variété linéaire affine. This definition is quite common but as Urs insists on the reference I attributed it to this in my opinion pedagogically superb book for high school teachers of geometry by a very talented female mathematician.
I improved the idea section to include all cases.
I didn’t follow all of this, but the current page looks good to me. I don’t see anything counter nPOV.
Yes, thanks. I just meant that splitting the way Urs was inclined too would be against. I was talking about the inclination expressed in our discussion, not the current page which however waits for Urs’s approval or changes.
I think the current page is a mess. The third paragraph directly contradicts the first now. The notation in the second paragraph is first undefined and then inconsistent with the first paragraph.
Why make a dead simple high school subject so opaque? If you seriously care about something as exotic as “unoriented direction of a vector”, make it a remark at the end of an otherwise straightforward entry. I vote for reverting to rev 3 and I hereby register a general complaint about Zoran’s behaviour. It’s not the first time.
Zoran, I don’t know what “splitting” you mean. The direction vector is a different concept from the direction of a vector, if that’s what you’re referring to; saying that they’re “isomorphic” doesn’t even typecheck. One is a representative of the other.
Urs, thanks for pointing out that contradiction, I had missed that. I tried to fix up the page a little.
Thanks, Mike. Thanks a lot.
added the example of wave fronts, here, and cross-linked with relevant entries.
Also, I gave the Definitions numbered environments, accordingly rearranged the text slightly, and added more hyperlinks.
Mike 23
Zoran, I don’t know what “splitting” you mean.
Edit: Splitting into two separate web pages as Urs suggested.
The direction vector is a different concept from the direction of a vector, if that’s what you’re referring to; saying that they’re “isomorphic” doesn’t even typecheck. One is a representative of the other.
I agree with that. Parallelness and parallel are different concepts but belong to the same entry. For isomorphic I meant the direction of oriented lines vs. direction of vectors. Direction of an affine subspace (in particular line, oriented or not, this is not important) should not be separated from the direction of a vector.
Urs 22
is a mess
It might have been. You requested me (I am not in a project now on this subject unlike you) to provide references and do other work immediately, I spent two hours in a busy day fullfilling your request and trying to include all points of view as we usually do in nLab and was in the end not able to do careful enough proofreading. For this one needs a time distance, say to reread after one sleeps. Sorry for that.
Why make a dead simple high school subject so opaque?
Please delve a bit into this before saying this. I think you find it such because you find not worthy to look at the concept after you concluded that it is easy and you know it already. Bit if given a little thought you’d see that what I added is more or less trivial, and very useful in elementary geometry. I provided major references as you asked. Lelong-Ferrand is one of the most recognized textk books precisely for high school teachers. Anywhere else in lab we try to include many facets of the notion including advanced ones, why do you find this now a problem ? I think all the facets are related in the entry.
And finally if I tell you that I spend one semester in last year and one in this on elementary geometry and that the definitions I expose are common there, why don’t you simply believe me. Don’t you think that if I spend time on some $n$Lab details that I have some reason why I care about them ? It was in this case related to my one year of work (prompted by teaching the subject) (I mentioned the fact above), which you dismiss at one minute of a look ?
Mike 23
One is a representative of the other.
Mike, I think this is orthogonal to what Urs complained to my definition. Urs’s point of view at the beginning was that the direction is certain unit vector, not the equivalence class. I started by emphasizing the equivalence class. Here the “type” is different. In 13, if I understood him properly, Urs did not complain weather we have an equivalence class or its representative but weather the direction is defined from input which is (oriented) line as opposed to input which is vector. I mentioned that some vector is in the direction of a line if it has a representative of the form $\vec{A B}$ where $A$ and $B$ are in the line (and $A$ is before $B$ in the ordering on the line, if the line is oriented, that is the prefer ordering is chosen).
Urs 22
If you seriously care about something as exotic as “unoriented direction of a vector”
Urs, I don’t. In the very first comment #2 I said “does not include the orientation (the latter division is a matter of taste)”. Including it or not is a matter of taste, not essence. The thing which I emphasized and cared are
the scalar product or even norm does NOT matter for the notion of direction (parallelness does not need it, oriented or not, orthogonality needs it) but OK the representative can be and often is represented by the unit vector representative
the best initial definition used an equivalence class (using multiples or parallelness)
one has to emphasize that the vector is nonzero
direction can be attached not only to vectors but also to oriented lines (and even subspaces)
All the four were missing in the original treatment before my intervention. I complained loudly only about 1, corrected 3, and tried to add as a new material POV 2 and commented 4. Do you really want to revert on the 4 points ?
The versions for line in the literature usually dwell more on unoriented version that is why in those line and subspace versions I copied the unoriented definitions from Choquet, Lelong-Ferrand and others. But it is clear that it does not matter. Only it is less often covered (for vectors, one naturally more often includes orientation in the direction, for lines, naturally one more often looks at unoriented versions, but not always).
There is also point 5 and that is that to subspaces with structure (say you are in symplectic vector space or in complex and you look at the real orientation) one can attach more complicated versions of orientation (and still the equivalence class of fancy oriented parallel subspaces make sense). This is important when one has a bundle of something and that has a direction which moves from point to point. But without studying it more careful I am not able now to add point 5, but I had given some hint toward it. This mention may be opaque point as I am not fully understanding it.
We could link to a page pencil to discuss more fully the concept of a collection of affine subspaces…
For isomorphic I meant the direction of oriented lines vs. direction of vectors.
That also doesn’t typecheck. And I also disagree in principle that those are the same thing even intuitively: a vector comes naturally with a direction, whereas a subspace has to be “equipped” with one as additional structure to become oriented.
28 Thank you, David. I hesitatingly created few lines of pencil with what I guessed you had in mind: considering parallelness class of affine lines in the plane as (a pencil through) a point at infinity of the projective completion. But maybe you had in mind something more profound so please complement/overwrite it with your version.
Mike 29 Thank you very much.
That also doesn’t typecheck.
It clearly does not as a function of vector or oriented line. But as the type of the result, I think that both directions are expressed by the isomorphic data, aren’t they ? It is even used as the same datum when we say that vector is parallel to a line (and, one may say, in equivalent direction class with an oriented line). The same data define equivalence class of oriented lines – or equivalence class of half-lines. In some literature both are then called just “directions” (slightly different “directions” would be attached to pseudovectors I guess).
As I am not sure that we assume the same additional data for the oriented line let me explain the triviality which I have in mind (I apologize if the following is not interesting to you) – in synthetic (=axiomatic) affine geometry one often starts like this: lines and points are primitive objects of say plane geometry and then an important axiom says that every line in space is equipped with two preferred orderings. Oriented line is a choice of one of the two preferred orderings. Half-line consists of a point on the line called vertex and of all points which are after the vertex in one of the two preferred orderings. Thus half-line is determined by the oriented line plus a choice of the vertex. Appropriate equivalence classes of half-lines thus define (oriented) directions as well.
a vector comes naturally with a direction, whereas a subspace has to be “equipped” with one as additional structure to become oriented
I surely agree, precisely this I meant in 27 where I said “for vectors, one naturally more often includes orientation in the direction, for lines, naturally one more often looks at unoriented versions, but not always”. As my apology for starting the thread with unoriented version (apart from the fact that I learned it that way from the axiomatic literature), this explains that from the axiomatic study of lines and points in affine space (where my recent background and initiative to contribute to the entry comes from) one naturally comes out viewing the unoriented direction of lines, while from the analytic point based on vector spaces and formulas one naturally comes at oriented direction of vectors. But in both cases both variants can be defined, though one of them in each case is less intuitive as you rightly point out.
Yes, the direction of a vector and the direction of an oriented line are both the same kind of thing, namely an oriented line. But “the direction of a vector” refers to the function, not the type of the result.
Surely, this is clear. The question was on splitting the page or not. i do not doubt what the phrase is about but what the scope of the page is. My assumption is, and I might be wrong, that the page has been named by Urs direction of a vector just to be distinguished from the direction in order theory and that one includes e.g. theorem that the vector is determined by the norm plus direction. Or you want to split by type reasons?
I no longer know what we’re talking about. Is there a change that anyone wants to make to the current version of the page?
Not one of our finer starts to a page!
In flat geometry the parallel vectors…
We don’t say what ’flat’ means, ’parallel’ hasn’t a page to link to, and vectors is disappointing to say the least.
Well, the nonexistence of parallel and the disappointingness of vector aren’t the fault of direction of a vector. I’m not sure what “flat” means; does it just mean “in a vector space” (or “in an affine space”)? If so we could just say that.
As a former contributor to the page, I am guilty for adding the rudimentary idea section including saying “flat”. The rationale of ’flat’ is to allow open perspective at the idea level to include several classical geometries which are considered non-curved and allow parallelness classes: the affine and vector space cases, as well as the flat tori (of any dimension), and possibly (though I did not honestly think it through) a plethora of finite geometries where parallelness classes make sense. For example, Lelong-Ferrand mentions planes of affine type which if I recall right satisfy less axioms than affine planes in the standard sense. This may not be sound to you, hence of course you are free to limit the ambition to the cases so far realized in the main part of the entry, say the vector space and affine case only and list them by an extensive definition of considered scope of geometries (which however definitely goes beyond the vector space case).
33: I don’t, sorry I just thought your typecheck complaint was just asking us to consider the splitting along the suggestion in 13.
Well, given that the current definitions don’t apply beyond the vector/affine case, nor do we even have anything more general on a page about parallelism, I’m inclined to remove “flat” for now.
Surely.
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