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Math Help - Vectors

  1. #1
    Member GAdams's Avatar
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    Vectors

    I got AY = -3a + 1/2 b for the first one. The other two I can't work out.
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  2. #2
    Member GAdams's Avatar
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    Oops I forgot part (iii)!
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by GAdams View Post
    I got AY = -3a + 1/2 b for the first one. The other two I can't work out.
    why did you get 1/2 b? shouldn't it be 3/2 b?
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  4. #4
    Member GAdams's Avatar
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    Quote Originally Posted by Jhevon View Post
    why did you get 1/2 b? shouldn't it be 3/2 b?
    Sorry, I meant to type 1 1/2b , i.e. 1.5b
    Last edited by GAdams; July 6th 2007 at 11:49 AM.
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  5. #5
    MHF Contributor red_dog's Avatar
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    i) \overrightarrow{AY}=\overrightarrow{AO}+\overright  arrow{OY}=-\overrightarrow{OA}+\frac{1}{2}\overrightarrow{OB}  =-3\mathbf{a}+\frac{3}{2}\mathbf{b}.
    ii) \overrightarrow{OX}=\frac{1}{2}(\overrightarrow{OA  }+\overrightarrow{OB})=\frac{3}{2}\mathbf{a}+\frac  {3}{2}\mathbf{b}
    iii) \overrightarrow{AZ}=\overrightarrow{AO}+\overright  arrow{OZ}=-\overrightarrow{OA}+\frac{2}{3}\overrightarrow{OX}  =-2\mathbf{a}+\mathbf{b}.
    If A,Z,Y are colinear, then Z is baricentre, so \frac{AZ}{ZY}=2
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  6. #6
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    Quote Originally Posted by GAdams View Post
    I got AY = -3a + 1/2 b for the first one. The other two I can't work out.
    Hello,

    as Jhevon pointed out:

    \overrightarrow{AY} = -3 \vec{a} + \frac{3}{2} \vec{b}

    (ii):

    1. \overrightarrow{AB} = -3 \vec{a} + 3 \vec{b}

    2. \overrightarrow{OX} = 3 \vec{a} + \frac{1}{2} \overrightarrow{AB}

    3. \overrightarrow{OX} = 3 \vec{a} + \frac{1}{2} (-3 \vec{a} + 3 \vec{b}) = 3 \vec{a} - \frac{3}{2} \vec{a} + \frac{3}{2} \vec{b}) = \frac{3 \vec{a} + 3 \vec{b}}{2}

    As you can see \overrightarrow{OX} is the mean (or average) of A and B.

    (iii)

    \overrightarrow{AZ} = -3\vec{a} + \frac{2}{3} \cdot \overrightarrow{OX}

    \overrightarrow{AZ} = -3\vec{a} + \frac{2}{3} \cdot \frac{3 \vec{a} + 3 \vec{b}}{2} = -2\vec{a} + \vec{b}
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  7. #7
    Member GAdams's Avatar
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    Quote Originally Posted by earboth View Post
    Hello,

    as Jhevon pointed out:

    \overrightarrow{AY} = -3 \vec{a} + \frac{3}{2} \vec{b}

    (ii):

    1. \overrightarrow{AB} = -3 \vec{a} + 3 \vec{b}

    2. \overrightarrow{OX} = 3 \vec{a} + \frac{1}{2} \overrightarrow{AB}

    3. \overrightarrow{OX} = 3 \vec{a} + \frac{1}{2} (-3 \vec{a} + 3 \vec{b}) = 3 \vec{a} - \frac{3}{2} \vec{a} + \frac{3}{2} \vec{b}) = \frac{3 \vec{a} + 3 \vec{b}}{2}

    As you can see \overrightarrow{OX} is the mean (or average) of A and B.

    (iii)

    \overrightarrow{AZ} = -3\vec{a} + \frac{2}{3} \cdot \overrightarrow{OX}

    \overrightarrow{AZ} = -3\vec{a} + \frac{2}{3} \cdot \frac{3 \vec{a} + 3 \vec{b}}{2} = -2\vec{a} + \vec{b}
    Thanks, I got it except the last bit, why is it 2/3 * OX?
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  8. #8
    Member GAdams's Avatar
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    Quote Originally Posted by red_dog View Post
    i) \overrightarrow{AY}=\overrightarrow{AO}+\overright  arrow{OY}=-\overrightarrow{OA}+\frac{1}{2}\overrightarrow{OB}  =-3\mathbf{a}+\frac{3}{2}\mathbf{b}.
    ii) \overrightarrow{OX}=\frac{1}{2}(\overrightarrow{OA  }+\overrightarrow{OB})=\frac{3}{2}\mathbf{a}+\frac  {3}{2}\mathbf{b}
    iii) \overrightarrow{AZ}=\overrightarrow{AO}+\overright  arrow{OZ}=-\overrightarrow{OA}+\frac{2}{3}\overrightarrow{OX}  =-2\mathbf{a}+\mathbf{b}.
    If A,Z,Y are colinear, then Z is baricentre, so \frac{AZ}{ZY}=2


    I don't quite get the last part about colinear and baricentre
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  9. #9
    MHF Contributor red_dog's Avatar
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    If X is the midpoint of AB and Y is the midpoint of OB, then OX and AY are medians in the triangle OAB and Z=OX\cap AY. The intersection point of medians is the baricenter which splits each median in two segments in ratio 2:1.
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