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Math Help - Vectors in 2D space

  1. #1
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    Vectors in 2D space

    question: ABCD is a quadrilateral. P,Q,R and S divide AB, AD, CD and CB respectively in the ration 2:1.
    (a) Find the position vectors p, q, r, s in therms of a, b, c, d.
    (b) Prove that PQRS is a parallelogram.


    (a) is easy:
    p = (a+2b)/3
    q = (a+2d)/3
    r = (c+2d)/3
    s = (c+2b)/3
    But how to do (b) ?
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  2. #2
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    Quote Originally Posted by afeasfaerw23231233 View Post
    question: ABCD is a quadrilateral. P,Q,R and S divide AB, AD, CD and CB respectively in the ration 2:1.
    (a) Find the position vectors p, q, r, s in therms of a, b, c, d.
    (b) Prove that PQRS is a parallelogram.


    (a) is easy:
    p = (a+2b)/3
    q = (a+2d)/3
    r = (c+2d)/3
    s = (c+2b)/3
    But how to do (b) ?
    Hello,

    you have to prove that the sides PQ \parallel RS~\wedge~|PQ| = |RS|
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  3. #3
    MHF Contributor red_dog's Avatar
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    PQRS paralelogram \Leftrightarrow p+r=q+s
    \displaystyle p+r=\frac{a+2b+c+2d}{3}
    \displaystyle q+s=\frac{a+2b+c+2d}{3}
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  4. #4
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    so far i have done:
    \overrightarrow{P Q} \ = (2a+2b+2d)/3
     \overrightarrow{R S} \ = (2c+2d+2b)/3



    it seems that i have to prove a = c in order to get

    \overrightarrow{P Q} \ = \overrightarrow{S R} \

    but i have no idea how to prove a=c
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  5. #5
    MHF Contributor red_dog's Avatar
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    You are wrong.
    \overrightarrow{PQ}=(q-p)=\left(\frac{2(d-b)}{3}\right)
    \overrightarrow{RS}=(s-r)=\left(\frac{2(b-d)}{3}\right).
    Then \overrightarrow{PQ}\parallel \overrightarrow{RS} and |\overrightarrow{PQ}|=|\overrightarrow{RS}|
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