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Math Help - Vectors - Finding the midpoint

  1. #1
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    Vectors - Finding the midpoint

    Find the midpoint of AD.

    The position vectors of points A, B, D are respectively a, b, -a+2b.

    I know -a+2b is equivalent to points (-1,2), but not sure what to do from here. I know how to do it when I know the points of A, but there's no numbers, so that's why I'm confused! Help is much appreciated.
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  2. #2
    Senior Member BAdhi's Avatar
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    \overrightarrow{AD}=(-\underline{a}+2\underline{b})-\underline{a}=2(-\underline{a}+\underline{b})

    if mid point of \overrightarrow{AD} is E

    then \overrightarrow{OE}=\overrightarrow{OA}+\overright  arrow{AE}=\overrightarrow{OA}+\frac{\overrightarro  w{AD}}{2}
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  3. #3
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    Quote Originally Posted by BAdhi View Post
    \overrightarrow{AD}=(-\underline{a}+2\underline{b})-\underline{a}=2(-\underline{a}+\underline{b})

    if mid point of \overrightarrow{AD} is E

    then \overrightarrow{OE}=\overrightarrow{OA}+\overright  arrow{AE}=\overrightarrow{OA}+\frac{\overrightarro  w{AD}}{2}
    So is the answer [2(-a+b)/2] = -a+b or am I not reading it right?
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  4. #4
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    Quote Originally Posted by brumby_3 View Post
    So is the answer [2(-a+b)/2] = -a+b or am I not reading it right?
    Unfortunately that's wrong.

    1. As BAdhi wrote you have to calculate:

    \overrightarrow{OE}= \overrightarrow{OA}+\frac12 \cdot \overrightarrow{AD}

    which simplifies to:

    \overrightarrow{OE}=\vec a + \frac12 (2(-\vec a + \vec b))=\vec b

    2. In short: The staionary vector of the midpoint of a distance is the mean of the startpoint and the endpoint vectors of the distance:

    \overrightarrow{OE}=\dfrac{\vec a + (-\vec a + 2\vec b)}2 = \vec b
    Last edited by earboth; March 11th 2011 at 03:04 AM.
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