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Math Help - Sum of a vector parallel to b and a vector orthogonal to b

  1. #1
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    Sum of a vector parallel to b and a vector orthogonal to b

    Vectors are
    <br />
\begin{array}{l}<br />
 \underline v  =  < 2, - 4 >  \\ <br />
 \underline b  =  < 1,1 >  \\ <br />
 \end{array}<br />


    As far as a parallel vector to vector b went it would be just in the same direction and thus for expressing vector v as the sum of a vector parallel to vector b. I thought it would be
    <br />
\underline v  + \underline b  = \left\langle {(2 + 1)} \right.,\left. {( - 4 + 1)} \right\rangle  = \left\langle 3 \right.,\left. { - 3} \right\rangle <br />

    and for expressing the vector v as the sum of a vector orthogonal to vector b would be just
    <br />
 \bot \underline {b = \left\langle { - 1,\left. 1 \right\rangle } \right.} <br />
    and so
    <br />
\underline v  +  \bot \underline b  = \left\langle {(2 + ( - 1)),\left. {( - 4 + 1)} \right\rangle  = \left\langle {1,\left. { - 3} \right\rangle } \right.} \right.<br />

    However the answer given for the orthogonal part of the question is <br />
\left\langle { - 1,\left. { - 1} \right\rangle } \right.<br />
    I'm not sure what I have done wrong in the later part of the question.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Craka View Post
    Vectors are
    <br />
\begin{array}{l}<br />
 \underline v  =  < 2, - 4 >  \\ <br />
 \underline b  =  < 1,1 >  \\ <br />
 \end{array}<br />


    As far as a parallel vector to vector b went it would be just in the same direction and thus for expressing vector v as the sum of a vector parallel to vector b. I thought it would be
    <br />
\underline v  + \underline b  = \left\langle {(2 + 1)} \right.,\left. {( - 4 + 1)} \right\rangle  = \left\langle 3 \right.,\left. { - 3} \right\rangle <br />

    and for expressing the vector v as the sum of a vector orthogonal to vector b would be just
    <br />
 \bot \underline {b = \left\langle { - 1,\left. 1 \right\rangle } \right.} <br />
    and so
    <br />
\underline v  +  \bot \underline b  = \left\langle {(2 + ( - 1)),\left. {( - 4 + 1)} \right\rangle  = \left\langle {1,\left. { - 3} \right\rangle } \right.} \right.<br />

    However the answer given for the orthogonal part of the question is <br />
\left\langle { - 1,\left. { - 1} \right\rangle } \right.<br />
    I'm not sure what I have done wrong in the later part of the question.
    please state the question as it appears in your book

    am i to understand that you are given vectors v and b and you want to express v as a sum of vectors parallel to b and then as a sum of vectors orthogonal to b?

    if that is the case, why are you adding v to b and v to something parallel to b? it seems that you are not even attempting to do what the problem i think is asking. you want v = (sum of vectors parallel to b) and v = (sum of vectors orthogonal to b) right? how did you end up with v + b on the left side?
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  3. #3
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    Sorry the question states "Let v=<2,4> and b=<1,1>. Express v as the sum of a vector parallel to b and a vector orthogonal to b."
    v and b above are vectors
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Craka View Post
    Sorry the question states "Let v=<2,4> and b=<1,1>. Express v as the sum of a vector parallel to b and a vector orthogonal to b."
    v and b above are vectors
    Hint: if \bold{a} is a vector perpendicular to \bold{b}, then \bold{a} \cdot \bold{b} = 0

    let \bold{a} = \left< x,y \right>. then we want \left< x,y \right> \cdot \left< 1,1 \right> = 0

    also, any vectors parallel to \bold{b}, are of the form n \bold{b}. where n is a scalar.

    thus you want to write \bold{v} in the following way

    \bold{v} = n \bold{b} + m \bold{a} for some scalars m,n


    i suppose you can continue
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