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Math Help - Vector Addition Problem

  1. #1
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    Vector Addition Problem

    1. The problem statement, all variables and given/known data

    Write the vector <1,7> as a sum of two vectors, one parallel to <2,-1> and one perpendicular to <2,-1>

    2. Relevant equations
    DOt Product


    3. The attempt at a solution

    I'm confused on where to begin this problem. Should I be using the dot product?

    Thanks
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  2. #2
    MHF Contributor Unknown008's Avatar
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    You can do it like this:

    \displaystyle a \binom{2}{-1} + b \vec{n} = \binom{1}{7}

    Where \displaystyle \vec{n} = \binom{x}{y}

    To find the perpendicular vector n, use the dot product:

    \displaystyle \binom{2}{-1}\cdot \binom{x}{y} = 0

    Find a value of x and y that fit in this dot product other than zero. Making a quick sketch will confirm your answer.

    Then, express your answer as the sum of the vectors.

    Post what you get!
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  3. #3
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    But you can do it using the dot product.
    The projection of vector \vec{u} on \vec{v} has length |\vec{u}|cos(\theta) where \theta is the angle between the two vectors. It is also true that \vec{u}\cdot\vec{v}= |\vec{u}||\vec{v}|cos(\theta) so that the projection of \vec{u} on \vec{v} has length |\vec{u}|cos(\theta)= \frac{\vec{u}\cdot\vec{v}}{|\vec{v}|}.

    In order to make that into a vector in the direction of \vec{v} multiply it by the unit vector in the direction of \vec{v}:
    \frac{\vec{v}}{|\vec{v}|} to get

    \frac{\vec{u}\cdot\vec{v}}{|\vec{v}|^2}\vec{v}

    That is the component of \vec{u} in the direction of \vec{v}. To get the conponent perpendicular to that direction, subtract that from \vec{u}.
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  4. #4
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    Here is some different notations.
    Suppose that U~\&~V are two non-parallel vectors.
    Then U = V_{||}  + V_ \bot where V_ \bot   = \frac{{U \cdot V}}{{V \cdot V}}V~\&~U_{||}  = U - V_ \bot  .
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