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Thread: [SOLVED] Quick Communication Vector Question

  1. #1
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    [SOLVED] Quick Communication Vector Question

    a) If u and v are non-zero vectors, but Proj(u onto v) = 0, what conclusion can be drawn?
    b) If Proj(u onto v) = 0, does it follow that Proj(v onto u) = 0? Explain.

    Small, clear explanation would be fine thanks
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  2. #2
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    $\displaystyle v \ne 0\;\& \,proj_v u = \frac{{u \cdot v}}{{v \cdot v}}v\, \Rightarrow \,u \cdot v = 0$
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  3. #3
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    Vector and scalar product

    Hello narutoblaze
    Quote Originally Posted by narutoblaze View Post
    a) If u and v are non-zero vectors, but Proj(u onto v) = 0, what conclusion can be drawn?
    b) If Proj(u onto v) = 0, does it follow that Proj(v onto u) = 0? Explain.

    Small, clear explanation would be fine thanks
    (a) Use the scalar (dot) product formula to give the projection of $\displaystyle \vec{u}$ onto $\displaystyle \vec{v}$, which is:

    proj($\displaystyle \vec{u}$ onto $\displaystyle \vec{v}$)$\displaystyle =\frac{\vec{u}\cdot \vec{v}}{|\vec{v}|} = 0, |\vec{v}| \ne 0$

    $\displaystyle \Rightarrow \vec{u}\cdot \vec{v} = 0$

    $\displaystyle \Rightarrow |\vec{u}||\vec{v}|\cos\theta = 0$, where $\displaystyle \theta$ is the angle between the vectors $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$

    $\displaystyle \Rightarrow \theta = \tfrac{\pi}{2}$, since $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$ are non-zero.

    So the vectors are perpendicular.

    (b) Yes: proj($\displaystyle \vec{v}$ onto $\displaystyle \vec{u}$)=$\displaystyle \frac{\vec{u}\cdot \vec{v}}{|\vec{u}|} = 0, |\vec{u}| \ne 0$

    $\displaystyle \Rightarrow \frac{\vec{u}\cdot \vec{v}}{|\vec{v}|} = 0$

    $\displaystyle \Rightarrow$ proj($\displaystyle \vec{u}$ onto $\displaystyle \vec{v}$)= 0

    Grandad
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  4. #4
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    Much clearer, thanks Grandad
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