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Math Help - Find point M (vectors)

  1. #1
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    Find point M (vectors)

    I have to solve this problem:

    Find point M in any quadrilateral ABCD such that is MA+MB+MC+MD = 0

    Since LaTex doesn't work, MA,MB,MC,MD and 0 are all vectors.


    Help anyone?
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  2. #2
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    Quote Originally Posted by OReilly View Post
    I have to solve this problem:

    Find point M in any quadrilateral ABCD such that is MA+MB+MC+MD = 0

    Since LaTex doesn't work, MA,MB,MC,MD and 0 are all vectors.


    Help anyone?
    I guess that is the center (where diagnols intersect).
    Since that is the only which makes sense.
    Check to see if that is true.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    I guess that is the center (where diagnols intersect).
    Since that is the only which makes sense.
    Check to see if that is true.
    Well, since diagonals are not always equal I can't say that is the right point.

    If it were equal then it could be MC=1/2AC, MA=-1/2AC, MD = 1/2BD, MB = -1/2BD and we would have two pairs of opposite vectors.
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  4. #4
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    Quote Originally Posted by OReilly View Post
    I have to solve this problem:

    Find point M in any quadrilateral ABCD such that is MA+MB+MC+MD = 0

    Since LaTex doesn't work, MA,MB,MC,MD and 0 are all vectors.
    Help anyone?
    Hello, OReilly,

    I've attached an image which show you what to do in 4 steps. The steps are numbered.

    1. Construct the midpoint of AD, call it P.
    2. Construct the midpoint of BC, call it Q.
    3. Construct the midpoint of PQ: That's M.

    If you add 2 vectors the sum can be represented by the diagonal of a parallelogramm. If you add MA + MD then AD must be the other diagonal of the parallelogramm. Do the same with MB + MC.

    In step #4 you see the complete result.

    Obviously (MA +MD) and (MB + MC) are pointing in opposite directions and have the same length. Thus the sum of (MA + MD) + (MB + MC) = 0

    EB
    Attached Thumbnails Attached Thumbnails Find point M (vectors)-vectorsum_null.gif  
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  5. #5
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    Quote Originally Posted by earboth View Post
    Hello, OReilly,

    I've attached an image which show you what to do in 4 steps. The steps are numbered.

    1. Construct the midpoint of AD, call it P.
    2. Construct the midpoint of BC, call it Q.
    3. Construct the midpoint of PQ: That's M.

    If you add 2 vectors the sum can be represented by the diagonal of a parallelogramm. If you add MA + MD then AD must be the other diagonal of the parallelogramm. Do the same with MB + MC.

    In step #4 you see the complete result.

    Obviously (MA +MD) and (MB + MC) are pointing in opposite directions and have the same length. Thus the sum of (MA + MD) + (MB + MC) = 0

    EB
    Thanks, very nice solution.
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  6. #6
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    one final remark

    Hello, OReilly,

    if you know the coordinates of the points A to D then you can easily calculate the coordinates of M. According to my construction you have to calculate the coordinates of a midpoint between 2 midpoints.

    If
    A(a1 , a2)
    B(b1 , b2)
    C(c1 , c2)
    D(d1 , d2)

    then

    M( (a1+b1+c1+d1), (a2+b2+c2+d2))

    EB
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  7. #7
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    Quote Originally Posted by earboth View Post
    Hello, OReilly,

    if you know the coordinates of the points A to D then you can easily calculate the coordinates of M. According to my construction you have to calculate the coordinates of a midpoint between 2 midpoints.

    If
    A(a1 , a2)
    B(b1 , b2)
    C(c1 , c2)
    D(d1 , d2)

    then

    M( (a1+b1+c1+d1), (a2+b2+c2+d2))

    EB
    Can this point be actually found in this way:

    It is the point of intersection of lines that are dividing two pairs of opposite sides of quadrilateral in half.
    Attached Thumbnails Attached Thumbnails Find point M (vectors)-untitled.gif  
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