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Thread: Decompose a Vector

  1. #1
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    Decompose a Vector

    Let n = (-2, 1). Decompose the vector g = (0, -9.8) into the sum of two orthogonal vectors, one parallel to n and the other orthogonal to n.

    I'm not sure how to go about doing this.
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  2. #2
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    Quote Originally Posted by RedKMan View Post
    Let n = (-2, 1). Decompose the vector g = (0, -9.8) into the sum of two orthogonal vectors, one parallel to n and the other orthogonal to n.

    I'm not sure how to go about doing this.
    From $\displaystyle \vec n = (-2,1)$ you'll get the orthogonal vector as $\displaystyle \vec o = (1,2)$ since $\displaystyle \vec n \cdot \vec o = 0$.

    Now you are asked to find values of $\displaystyle s, t \in \mathbb{R}$ such that

    $\displaystyle s \cdot (-2,1) + t \cdot (1,2) = (0, -9.8)$

    Can you handle it from here?
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  3. #3
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    Hello, RedKMan!

    Let $\displaystyle \vec n \,=\,\langle-2, 1\rangle$
    Decompose the vector $\displaystyle \vec g \,=\, \langle0, -9.8\rangle$ into the sum of two orthogonal vectors,
    one parallel to $\displaystyle \vec n$, the other orthogonal to $\displaystyle \vec n$
    Are you familiar with "projections"?


    Given vectors $\displaystyle \vec u, \vec v$

    The projection of u onto v is given by: .$\displaystyle \overrightarrow{w_1} \;=\;\frac{\vec u \cdot\vec v}{|\vec v|^2}\,\vec v$

    . . This is the component of $\displaystyle \vec u$ that is parallel to $\displaystyle \vec v.$


    And: .$\displaystyle \overrightarrow{w_2} \;=\;\vec u -\overrightarrow w_1$

    . . This is the component of $\displaystyle \vec u$ that is orthogonal to $\displaystyle \vec v.$



    We have: .$\displaystyle \vec u \:=\:\langle 0,-9.8\rangle,\;\vec v \:=\:\langle-2,1\rangle$

    . . $\displaystyle \overrightarrow{w_1} \;=\;\frac{\langle0,-9.8\rangle\cdot\langle-2,1\rangle}{(\sqrt{(-2)^2 + 1^2})^2}\,\langle-2,1\rangle \;=\;\frac{-9.8}{5}\langle-2,1\rangle \;=\;\langle3.92,-1.96\rangle $

    . . $\displaystyle \overrightarrow{w_2} \;=\;\langle 0.-9.8\rangle - \langle3.92,-1.96\rangle \;=\;\langle-3.92,-7.84\rangle $



    Therefore: .$\displaystyle \boxed{\;\vec g \;\;=\;\;\langle 3.92,-1.96\rangle + \langle -3.92,-7.84\rangle\;} $

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