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Math Help - Calculus III Word Problem

  1. #1
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    Calculus III Word Problem

    I have absolutely no clue. Can someone at least get me started?

    Thanks a Million!!!

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  2. #2
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    Vectors

    Hello qbkr21
    Quote Originally Posted by qbkr21 View Post
    I have absolutely no clue. Can someone at least get me started?

    Thanks a Million!!!

    For the geometric argument, consider an arbitrary point in the plane, O, as origin. Then take two lines, l and m, through O parallel to \vec{a} and \vec{b} respectively (effectively setting up a system of non-perpendicular* coordinate axes). Then consider point C whose displacement from O is represented by \vec{c}. From C, draw a line parallel to l, meeting m at P. Then \vec{PC} = s\vec{a} for some scalar s, and \vec{OP} = t\vec{b} for some scalar t, and then:

    \vec{OC} = \vec{OP} + \vec{PC}

    \Rightarrow \vec{c} = s\vec{a} + t\vec{b}

    For the components argument, let \vec{a} = a_1\vec{i} + a_2\vec{j}, and \vec{b} and \vec{c} similarly. Then consider separate components and show that equations

    c_1 = sa_1 + tb_1

     c_2 = sa_2 + tb_2

    have a solution for some s and t.

    Grandad

    * PS. Not necessarily perpendicular is what I mean. Instead of being covered by a grid of unit squares, as in Cartesian geometry, the plane is spanned by parallelograms with sides \vec{a} and \vec{b}.
    Last edited by Grandad; January 21st 2009 at 11:59 PM. Reason: PS added
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