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Math Help - vector question

  1. #1
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    vector question

    OAB is a triangle with OA = a and OB = b; C is the mid-point of OB, D is the mid-point of AB and E is the mid-point of OA; OD and AC intersect at F.
    If AF = hAC and OF = kOD show that h = k = 2/3
    well ive got that
    AC = 1/2(b - a)
    and
    OD = (1/2)b - (1/2)a

    AF = h/2(b - a)
    and
    OF = k((1/2)b - (1/2)a)

    and this what i think i need to do get the intersection
    h/2(b - a) = k((1/2)b - (1/2)a)

    but i dont know where to go from there ?
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  2. #2
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    Quote Originally Posted by renlok View Post
    well ive got that
    AC = 1/2(b - a)
    and
    OD = (1/2)b - (1/2)a

    AF = h/2(b - a)
    and
    OF = k((1/2)b - (1/2)a)

    and this what i think i need to do get the intersection
    h/2(b - a) = k((1/2)b - (1/2)a)

    but i dont know where to go from there ?
    1. Draw a sketch!

    2. \overrightarrow{AC} = -\vec a + \frac12 \vec b

    \overrightarrow{OD} = \frac12 (\vec a + \vec b)

    3. \overrightarrow{OF} = \vec a + h \cdot \overrightarrow{AC}

    \overrightarrow{OF} = k \cdot \overrightarrow{OD}

    Consequently you get:

    k \cdot \frac12 (\vec a + \vec b) = \vec a + h \cdot \left(-\vec a +\frac12 \vec b \right)

    Expand and rearrange:

    \frac12 k \vec a + \frac12 k \vec b = (1-h) \vec a + \frac12 h \vec b

    4. That means you know now:

    \left|\begin{array}{rcl}\frac12 k&=& 1-h \\ \frac12 k &=& \frac12 h \end{array} \right.

    5. Solve this system of equations for h, k.
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  3. #3
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    oh so you have to get if for the same vector i get it now
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