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Math Help - vector

  1. #1
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    vector

    ABCDEFGH is a regular octagon. State in terms of a single vector, the sum of the following vector
    +AC+AF+AH+HG (sorry that I don't know how to type the arrows out..)
    If | | =2, evaluate |AF| (vector too)
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  2. #2
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    Re: vector

    Hello, Trefoil2727!

    ABCDEFGH\text{ is a regular octagon. }\,\text{ State in terms of a single vector,}
    \text{the sum of the following vector: }\:\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AF} + \overrightarrow{AH} + \overrightarrow{HG}
    Code:
                  A       B
                  o-------o
                 /|  *     \
                / |     *   \
               /  |        * \
            H o   |           o C
              |   |           |
              |   |           |
              |   |           |
            G o   |           o D
               \ *|          /
                \ | *       /
                 \|    *   /
                  o-------o
                  F       E
    Note that: . \overrightarrow{AC} \,=\,\overrightarrow{GE},\;\overrightarrow{AB} \,=\,\overrightarrow{FE}

    We have: . \overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AF} + \overrightarrow{AH} + \overrightarrow{HG}

    . . . . . . =\;\left(\overrightarrow{AH} + \overrightarrow{HG} + \overrightarrow{AC}\right) + \left(\overrightarrow{AF} + \overrightarrow{AB}\right)

    . . . . . . =\;\left(\overrightarrow{AH} + \overrightarrow{HG} + \overrightarrow{GE}\right) + \left(\overrightarrow{AF} + \overrightarrow{FE}\right)

    . . . . . . =\;\overrightarrow{AE} + \overrightarrow{AE}

    . . . . . . =\; 2\overrightarrow{AE}




    \text{If }|\overrightarrow{AB}| = 2,\,\text{ evaluate }|\overrightarrow{AF}|.

    \text{If the side of the octagon is }|\overrightarrow{AB}| = 2
    . . \text{then }|\overrightarrow{AF}|\text{ is the side of the circumscribing square.}

    Code:
                  A   2   B
              * - o-------o - *
            _ :  /|        \  :
           √2 : /2|        2\ :
              :/  |          \:
              o   |           o
              |   |           |
             2|   |           |2
              |   |           |
              o   |           o
            _ :\  |          /:
           √2 : \2|        2/ :
              :  \|        /  :
              * - o-------o - *
                  F   2   E
    We see that: . |\overrightarrow{AF}| \:=\:2+2\sqrt{2}
    Thanks from Trefoil2727
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