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Math Help - Proving Vectors 2

  1. #1
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    Proving Vectors 2

    A) show geometrically that, for any scalar k and any vectors u and v,
    k(u-v)=ku-kv

    B) illustrate for k>0 that k(u+v)=ku+kv

    Note: bolded are vectors (arrows on top)

    ok so these two questions are quite similar.. I think, but I am still having trouble on "proving" them. I am not sure if my method is O.K. (no answers in back of book), so can someone please verify or demonstrate the correct method. Thanks in advance

    A) ku-kv=ku-kv
    is this ok? does it prove the q?

    B) well I thought about this as for any constant number k, it has to be greater than 0 because otherwise the resultant will be 0 and does not prove ku+kv
    what about this question?
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  2. #2
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    Quote Originally Posted by skeske1234 View Post
    A) show geometrically that, for any scalar k and any vectors u and v,
    k(u-v)=ku-kv

    B) illustrate for k>0 that k(u+v)=ku+kv

    Note: bolded are vectors (arrows on top)

    ok so these two questions are quite similar.. I think, but I am still having trouble on "proving" them. I am not sure if my method is O.K. (no answers in back of book), so can someone please verify or demonstrate the correct method. Thanks in advance

    A) ku-kv=ku-kv
    is this ok? does it prove the q?

    B) well I thought about this as for any constant number k, it has to be greater than 0 because otherwise the resultant will be 0 and does not prove ku+kv
    what about this question?
    If it's asking you to show something geometrically, it's asking you to draw a picture.
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  3. #3
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    ok, ill try to describe my drawings

    u and v are vectors |u|=|v|=1
    now k(u-v)
    not sure how to draw the k part but I drew the (u-v) first
    ill describe as a triangle: base is u, v is hypotenuse, (u-v) is height


    B) what does it mean by k>0 that k(u+v)=ku+kv
    how are u supposed to draw that? doesnt it mean that
    ku+kv=ku+kv? im not sure how i would draw these to prove this eqtn
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  4. #4
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    Quote Originally Posted by skeske1234 View Post
    A) show geometrically that, for any scalar k and any vectors u and v,
    k(u-v)=ku-kv

    B) illustrate for k>0 that k(u+v)=ku+kv

    Note: bolded are vectors (arrows on top)

    ok so these two questions are quite similar.. I think, but I am still having trouble on "proving" them. I am not sure if my method is O.K. (no answers in back of book), so can someone please verify or demonstrate the correct method. Thanks in advance

    A) ku-kv=ku-kv
    is this ok? does it prove the q?

    B) well I thought about this as for any constant number k, it has to be greater than 0 because otherwise the resultant will be 0 and does not prove ku+kv
    what about this question?

    To prove A)

    Draw two vectors, u and v.

    Draw the vector u - v.

    Draw a vector parallel to u - v, of a different length (k).

    This vector is k(u - v).


    Then draw the vectors ku and kv.

    Draw the vector ku - kv.

    It should be clear that the vectors k(u - v) and ku - kv are parallel and of the same length. Therefore they are equal.


    The same process is used to prove B).
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  5. #5
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    ok, i think I have part A) thanks,

    for part B), what does it mean by k>0..
    im not sure how to SHOW k>0 by drawing
    like.. does it mean that k must exist, for k>0, or, does it mean that -k (cant go in a diff direction) is not allowed?

    so I have:

    draw vectors ku and kv
    then draw ku+kv vector

    k(u+v)
    draw vectors u and v, then draw vector (u+v), then beside it (parallel) draw k(u+v) but of diff length.
    so now my question is, how do you know about the k>0 does it mean k must exist or does it mean that k cannot be going in a diff direction
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  6. #6
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    You are given that k> 0, you don't have to prove it. That means that kv goes in the same direction as v.
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