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Thread: Vocabulary in English: the direction and "sense" o

  1. #1 Vocabulary in English: the direction and "sense" o 
    Forum Ph.D. Leszek Luchowski's Avatar
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    Hello;

    Sorry about the truncated subject line; it was meant to be direction and "sense" of vectors.

    I am looking for the English names for two concepts concerning vectors. The problem is that when I take their Polish names (kierunek and zwrot) and look them up in a dictionary, I get "direction" as the English equivalent of both.

    For lack of a better pair of words, just for the purposes of this thread, I'll call one of them "direction" and the other "sense" (using a carbon copy of a French word).

    Two lines, or two vectors, have the same direction if they are parallel.

    Two vectors with the same direction have the same sense if they point the same way, or opposite senses if they point in opposite, eermph... um.... directions?

    (we never talk about "senses" in vectors of different directions)

    Lines as such don't have sense, just direction.

    I hope this picture will help clarify what I mean:



    Any help? I'm writing a text on projective geometry and keep running into this hurdle - how to say, in English, that a set of vectors have the same "direction" (just meaning they are parallel) but not necessarily the same "sense". Such as copies of the same vector multiplied by different scalars, some of them negative. I don't think they have the same direction in the usual, erm, sense of this English word.

    Thanks in advance - L.


    Leszek. Pronounced [LEH-sheck]. The wondering Slav.
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    I think the closest I could come would be to use "orientation" for what you are calling "direction" and "direction" for what you are calling "sense" .


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    Quote Originally Posted by Harold14370
    I think the closest I could come would be to use "orientation" for what you are calling "direction" and "direction" for what you are calling "sense" .
    The problem is that orientation has another technical meaning -- related to the right hand rule.

    In English treatments of the mathematics of vectors we say that vectors are parallel if one is positive multiple of the other. This is different from parallelism of lines and includes what, apparently in Polish, includes "sense". If the vectors point in opposite directions then we call them anti-parallel. (This terminology is not universal, and you should check on what definitions and ocnventions are being used by any particular author.)

    There is also the complication that vectors all have their origin at 0, and the notion that seems to be under discussion applies to what engineers and physicists sometimes call "free vectors" which are handled rigorously in mathematics by the concept of a vector field.

    In fact rather than try to solve this semantic puzzle, it is probably more productive to note that the vector concept embodies within it both the notion of "direction" and "sense" and it is the vector itself that is the more fundamental.
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    Forum Ph.D. Leszek Luchowski's Avatar
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    First, thank you both for your replies.

    DrRocket, I know that a vector has both an "orientation" and "sense" (kierunek and zwrot in Polish). I know it because that's the way I was taught, in my school days, to conceptualize vectors. Now that I am writing about them in English I am perplexed that, after receiving input from two highly educated, and helpful, proficient (probably native) English speakers I still don't know how to complete this simple sentence:

    Multiplication by a real nonzero scalar preserves the ........ (kierunek) but not necessarily the ...... (zwrot) of a vector.
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    Quote Originally Posted by Leszek Luchowski
    First, thank you both for your replies.

    DrRocket, I know that a vector has both an "orientation" and "sense" (kierunek and zwrot in Polish). I know it because that's the way I was taught, in my school days, to conceptualize vectors. Now that I am writing about them in English I am perplexed that, after receiving input from two highly educated, and helpful, proficient (probably native) English speakers I still don't know how to complete this simple sentence:

    Multiplication by a real nonzero scalar preserves the ........ (kierunek) but not necessarily the ...... (zwrot) of a vector.
    You can use whatever terminology you like, but that terminology is not standard.

    An orientation in mathematics is a selction of an ordering of basis vectors for a vector space. It is important for establishing, for instance, the meaning of the vector cross product in 3-space.

    http://en.wikipedia.org/wiki/Orientation_(mathematics)

    A single vector does not have an orientation.

    A vector does have a direction, and in fact "direction" is really defined in terms of vectors. Direction in this sense is preserved by positive multiples of a vector by not by negative multiples.

    So, I would think that in your sentence it is "direction" that would ot necessarily be preserved by nonzero scalar multiples.

    I have no idea what word applies to that characteristic that is preserved by all scalar multiples in your sentence and I have been a professional mathematician for over thirty years. This is simply not standard terminology and in fact it is not important.

    What is preserved by all scalar multiples is a line through the origin that would, geometrically, include both the vector and its opposite. If you want to call that "orientation" then you can make that definition, but you will not have much company.
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    Forum Ph.D. Leszek Luchowski's Avatar
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    Thank you again, DrRocket. Now my quest is over, my mind at peace that English has no word for the concept I was trying to use, and I shall henceforth turn the powers of my lonely grey cell to expressing differently what I am trying to say in my text.

    I have been thinking if this conceptualization of vectors that is so fundamentally incompatible with English is a uniquely Polish invention. It is not. The German Wikipedia (where the two features of a vector are called "Richtung" and "Orientierung") corroborates my intuition that we may have imported the idea from the country of Karl Gauss and Richard Dedekind. Which may in turn have taken it from France, as the language of Rene Descartes also distinguishes "direction" and "sens" of a vector (whence I took the placeholder words I used in my original post).
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    In fact, in French the word sens may mean sense, but it may also mean "way" or "direction".

    Here in the UK, even colloquially we use "direction" to mean something like an arrow, say in the simple case, left to right, without any specific reference to "how far". So that right-to-left is a different direction; I suppose you might connect these two by saying they are different "senses" for the same non-directed line segment (although obviously one can change direction without changing sense)

    I agree that "orientation" would be an unfortunate term to use, but hey, notation is arbitrary.

    At a very elementary level one says in English that vectors have both magnitude and direction. You might point out that only in very specific cases may one equate magnitude with length, say by using the examples of velocity or force. Also that the concept of "direction" is hard to make rigourous without additional machinery....

    You might say something like this, for a beginner (although it is far too "wordy"):

    "We define a change in the sense of a vector to be simply changing its sign i.e. exactly reversing its direction. Multiplying a vector by a real non-zero scalar may have the effect of changing its magnitude but not its sense (), it may have the effect of changing its sense but not its magnitude (), or it may do both () but it may not have the effect of simply changing its direction. Compare this to the addition rule for vectors"

    Obviously to make this explicit you would need to give the usual abstract definition of a vector space, but if you are doing projective geometry, surely this will be necessary anyway?
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    Forum Ph.D. Leszek Luchowski's Avatar
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    Thank you Guitarist.

    My text is for students in an engineering curriculum; I try to avoid going too much into high mathematical abstractions and rely on the notion of vectors as they know from ordinary geometry and physics.

    If English had two separate words equivalent to the French "direction" and "sens", I would be glad to use them. As it has not, I'll find my way around without introducing artificial vocabulary.

    Cheers - L.
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    I will defer to the others for the strict mathematical definitions. For engineering purposes, it would not be unusual to use the words "orientation" or "alignment" to discuss the relationship of a line to some reference axis, when the direction is not important. For example, you might say the axis of the machine has a north-south orientation or alignment. If it mattered whether it was north instead of south you would probably use the word "direction."
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  11. #10  
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    Quote Originally Posted by Leszek Luchowski
    Thank you Guitarist.

    My text is for students in an engineering curriculum; I try to avoid going too much into high mathematical abstractions and rely on the notion of vectors as they know from ordinary geometry and physics.
    That is probably the source of the confusion.

    As I stated earlier this sounds as though you are dealing with what are called "free vectors" in engineering texts. That is vectors are thought of as arrrows and the "tail" of the vector is allowed to be anywhere. This gets back to the engineering idea of a vector as something with "magnitude and direction" (hence once might consider a charging elephant to be a vector).

    In this case you have the idea of a "direction" from any give point, and in that sense "direction" is from the "tail" of the vector to the "head". "Direction" in that sense is preserved by positive miultiples and is reversed by negative multiples. "Orientation" is not a word normally used in English-language texts, even for engineers.

    In a more rigorous mathematical setting one works with vector fields rather than "free vectors".

    One problem with this rather naive approach is that one's intuition fails a bit when one looks at vector spaces over the complex field, as multiplication by a complex number can be construed as a rotation in a geometrical sense.

    Basing one's concept of vectors on what one learns in geometry and elementary physics can become a serious problem when one needs to apply vector space methods to more sophisticated problems, as is done, for instance, in engineering studies of control systems or quantum mechanical systems in physics.
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    Forum Ph.D. Leszek Luchowski's Avatar
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    My vectors are another name for, or a graphic representation/interpretation of, -tuples of real numbers (with most of the time). They do not form vector fields.

    If I have to say someting about where their "tails" are, I'll say they are all at the origin of the coordinate system. But I usually just don't think about them as being somewhere, any more than I think about where the plane is when I do planar geometry.
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    Just to throw my 2 cents in, in computer science (or at least computer graphics), orientation means a facing and rotation about that facing together. That may or may not be worth concern.
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  14. #13  
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    Quote Originally Posted by Leszek Luchowski
    My vectors are another name for, or a graphic representation/interpretation of, -tuples of real numbers (with most of the time). They do not form vector fields.

    If I have to say someting about where their "tails" are, I'll say they are all at the origin of the coordinate system. But I usually just don't think about them as being somewhere, any more than I think about where the plane is when I do planar geometry.
    What you have is standard way to (correctly) envision vectors in a 4-dimensional real vector space.

    With that set up, once one has specified the vector you have more or less automatically specified all that there is to specify. You can talk about unit vectors in the direction of any vector by simply dividing by the norm (normalizing to unit length). That should adequately encapsulate any notion of either "direction" or "sense" that is required.

    There are, however, other and very useful, applications of linear algebra that require one to adopt a more abstract point of view. Vector spaces are prominent players when one deals with systems of ordinary differential equations, a situation that arises naturally in control systems and in classical dynamics. They also are prominent in even more abstract settings, such as quantum mechanics in which the vector spaces of interest are infinite-dimensional.

    Vector fields are critical to engineering and physics in many applications. In fact they are crucial. The study of electrodynamics is essentially a study of vector fields and operators on vector fields.

    So, while in your particular application, such concepts may not be required, they are quite useful and really essential to many areas of engineering and physics. It is worthwhile understanding these things and that requires that one be able to think about vectors in different ways, as may be appropriate to the issues at hand.
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