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Thread: collinearity not equivalence relation??

  1. #1 collinearity not equivalence relation?? 
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    Hello!
    Recently, I found in my textbook that says collinearity is not equivalence relation on the set of all vectors. But I don't agree with this statement. Any two vectors are collinear if and only if they are parallel.

    So a || a is true.

    a||b => b||a is true.

    and finally:

    a||b ^ b||c =>a || c

    which is again true.

    What is the problem?


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  3. #2  
    Forum Ph.D. Leszek Luchowski's Avatar
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    I think the problem is the zero vector, which is collinear with any vector.


    Leszek. Pronounced [LEH-sheck]. The wondering Slav.
    History teaches us that we don't learn from history.
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  4. #3  
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    Quote Originally Posted by Leszek Luchowski
    I think the problem is the zero vector, which is collinear with any vector.
    I have taught about it, but actually you are right. The zero vector is parallel to any vector by definition, so it destroys the concept of transitivity. But this also means that parallelism is not equivalence relation.
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  5. #4 Re: collinearity not equivalence relation?? 
    . DrRocket's Avatar
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    Quote Originally Posted by scientist91
    Hello!
    Recently, I found in my textbook that says collinearity is not equivalence relation on the set of all vectors. But I don't agree with this statement. Any two vectors are collinear if and only if they are parallel.

    So a || a is true.

    a||b => b||a is true.

    and finally:

    a||b ^ b||c =>a || c

    which is again true.

    What is the problem?
    Leszek's comment is correct. The fundamental problem is the zero vector.

    There is also a more subtle issus. The concept of parallel vectors is a bit dicey,

    If you are talking about elements of a vector space, then the relationship among vectors is simply whether one is a scalar multiple of the other. If you exclude the zero vector you get an equivalence relationship among non-zero vectors and you can make the zero vector an equivalence class of its own. Or you can make the relationship dependent on scaling by a non-zero scalar up front which does the same thing.

    The concept oif parallel vectors, located at different points, is a bit more sophisticated, and really requires more machinery, In the most simple case of a plane, you simply look at the difference between the tip and tail, whcih gets you back to a simple vector space. In a more general setting you need the idea of parallel transport -- see a book on differential geometry.
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