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Math Help - Vector question

  1. #1
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    Vector question

    If A(-1,3,4), B(4,6,3), C(-1,2,1) and D are the vertices of a parallelogram, find all the possible coordinates for the point D

    I managed to find one point using the fact that the vector AB is parallel to the vector CD and hence finding the vector of D. However, I cannot find the remaining two points.
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  2. #2
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    D is adjacent to either A and B, B and C, or A and C.

    If D is adjacent to A and B, then it is opposite C. You can see that in this case D = C + (A-C) + (B-C) = A+B-C by drawing a picture.

    Permuting the roles of A, B and C gives all possibilities.
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  3. #3
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    Quote Originally Posted by maddas View Post
    D is adjacent to either A and B, B and C, or A and C.

    If D is adjacent to A and B, then it is opposite C. You can see that in this case D = C + (A-C) + (B-C) = A+B-C by drawing a picture.

    Permuting the roles of A, B and C gives all possibilities.
    Sorry, I still don't exactly understand.
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  4. #4
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    Quote Originally Posted by acevipa View Post
    Sorry, I still don't exactly understand.
    One of the possible points is given by A+B-C, as indicated above. However, it doesn't matter how you order A,B,C in that expression. So you can have A+C-B and B+C-A as other possible points, exhausting possibilities.
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  5. #5
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    Quote Originally Posted by hatsoff View Post
    One of the possible points is given by A+B-C, as indicated above. However, it doesn't matter how you order A,B,C in that expression. So you can have A+C-B and B+C-A as other possible points, exhausting possibilities.
    Yes but how do you get that? I mean don't we label points in order. So if we have a parallelogram ABCD:

    Isn't AB || CD and BC || AD

    So wouldn't we get:

    \vec{AB}=\lambda\vec{CD}

    \vec{b} - \vec{a}=\vec{d}-\vec{c}

    \Rightarrow\vec{b} +\vec{c} - \vec{a}=\vec{d}


    \vec{BC}=\lambda\vec{AD}

    \vec{c} - \vec{b}=\vec{d}-\vec{a}

    \Rightarrow\vec{c} +\vec{a} - \vec{b}=\vec{d}
    Last edited by acevipa; April 10th 2010 at 03:29 AM.
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  6. #6
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    Hello acevipa
    Quote Originally Posted by acevipa View Post
    Yes but how do you get that? I mean don't we label points in order. So if we have a parallelogram ABCD:

    Isn't AB || CD and BC || AD
    Yes, provided we are told that the parallelogram is ABCD. However, if we're just told (as we were in your original phrasing of the question) that A, B, C and D are the vertices of a parallelogram, then the parallelogram could also be ABDC or ACBD. So there are, as has already been said, three possible positions for D.

    Grandad
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