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Math Help - Position vector

  1. #1
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    Position vector

    OABCDE is a regular hexagon. The points A and B have position vectors a and b respectively, referred to the origin O.
    Find, in terms of a and b, the position vectors of C, D and E.

    I've no idea how to do this. If I sketch a hexagon where will be C, D, E? And what are their position vectors?
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  2. #2
    MHF Contributor red_dog's Avatar
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    \overrightarrow{OC}=2\overrightarrow{AB}=2\left(\o  verrightarrow{OB}-\overrightarrow{OA}\right)=2\left(\overrightarrow{  b}-\overrightarrow{a}\right)

    \overrightarrow{OD}=\overrightarrow{OC}+\overright  arrow{CD}=2\left(\overrightarrow{b}-\overrightarrow{a}\right)-\overrightarrow{OA}=2\overrightarrow{b}-3\overrightarrow{a}

    \overrightarrow{OE}=\overrightarrow{OD}+\overright  arrow{DE}=\overrightarrow{OD}-\overrightarrow{AB}=
    =\overrightarrow{OD}-\left(\overrightarrow{OB}-\overrightarrow{OA}\right)=-2\overrightarrow{a}+\overrightarrow{b}
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  3. #3
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    Hello, geton!

    OABCDE is a regular hexagon.
    The points A and B have position vectors \vec{a} and \vec{b} resp., relative to the origin O.
    Find, in terms of \vec{a} and \vec{b}, the position vectors of C, D\text{ and }E.
    Code:
               A   b-a   B
               *- - - - -*
              /       *   \
           a /     *       \
            /   * b         \
           / *               \
        O * - - - - * - - - - * C
           \ *      c        /
            \   *           /
           e \   d *       /
              \       *   /
               *- - - - -*
               E         D

    We have: . \vec{a} = \overrightarrow{OA},\;\;\vec{b} = \overrightarrow{OB}

    Let: . \vec{c} = \overrightarrow{OC},\;\;\vec{d} = \overrightarrow{OD},\;\;\vec{e} = \overrightarrow{OE}

    Note that: . \overrightarrow{AB} \:=\:\vec{b}-\vec{a}


    Since OABCDE is a regular hexgon, \overrightarrow{OC} \:=\: 2\!\cdot\overrightarrow{AB}

    . . Therefore: . \boxed{\vec{c} \;=\;2(\vec{b} - \vec{a})}



    \vec{d} \:=\:\overrightarrow{OD} \:=\:\vec{c} + \overrightarrow{CD}

    Since \vec{c} \:=\:2(\vec{b}-\vec{a})\text{ and }\overrightarrow{CD} \:=\:-\vec{a}

    . . \vec{d} \:=\:2(\vec{b} - \vec{a}) - \vec{a} \quad\Rightarrow\quad\boxed{ \vec{d} \:=\:2\vec{b} - 3\vec{a}}



    \vec{e} \:=\:\overrightarrow{OE} \:=\:\vec{d} + \overrightarrow{DE}

    Since \overrightarrow{DE} \:=\: -\overrightarrow{AB} \:=\:\vec{a}-\vec{b}

    . . \vec{e} \;=\;(2\vec{b}-3\vec{a}) + (\vec{a} - \vec{b}) \quad\Rightarrow\quad\boxed{ \vec{e} \:=\:\vec{b} - 2\vec{a}}

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  4. #4
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    Quote Originally Posted by Soroban View Post
    Hello, geton!


    Code:
               A   b-a   B
               *- - - - -*
              /       *   \
           a /     *       \
            /   * b         \
           / *               \
        O * - - - - * - - - - * C
           \ *      c        /
            \   *           /
           e \   d *       /
              \       *   /
               *- - - - -*
               E         D

    We have: . \vec{a} = \overrightarrow{OA},\;\;\vec{b} = \overrightarrow{OB}

    Let: . \vec{c} = \overrightarrow{OC},\;\;\vec{d} = \overrightarrow{OD},\;\;\vec{e} = \overrightarrow{OE}

    Note that: . \overrightarrow{AB} \:=\:\vec{b}-\vec{a}


    Since OABCDE is a regular hexgon, \overrightarrow{OC} \:=\: 2\!\cdot\overrightarrow{AB}

    . . Therefore: . \boxed{\vec{c} \;=\;2(\vec{b} - \vec{a})}



    \vec{d} \:=\:\overrightarrow{OD} \:=\:\vec{c} + \overrightarrow{CD}

    Since \vec{c} \:=\:2(\vec{b}-\vec{a})\text{ and }\overrightarrow{CD} \:=\:-\vec{a}

    . . \vec{d} \:=\:2(\vec{b} - \vec{a}) - \vec{a} \quad\Rightarrow\quad\boxed{ \vec{d} \:=\:2\vec{b} - 3\vec{a}}



    \vec{e} \:=\:\overrightarrow{OE} \:=\:\vec{d} + \overrightarrow{DE}

    Since \overrightarrow{DE} \:=\: -\overrightarrow{AB} \:=\:\vec{a}-\vec{b}

    . . \vec{e} \;=\;(2\vec{b}-3\vec{a}) + (\vec{a} - \vec{b}) \quad\Rightarrow\quad\boxed{ \vec{e} \:=\:\vec{b} - 2\vec{a}}


    Thank you so much
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