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Math Help - Vector projection and perpendicular

  1. #1
    M.R
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    Vector projection and perpendicular

    Hi,

    I am trying to solve the following problem:

    1. Find the projection of u=2i+j+3k on v=-i+3j+2k. Hence resolve u into two vectors, one parallel to v and the other perpendicular to v.

    I can solve the the first part, just by using the projection formula = (u . v / |v|) . v / |v|

    But how do I get it perpendicular to v? I know that if the dot product of two vectors = 0, then they are perpendicular. So:

    projection dot ??? = 0. How do I calculate what to dot it with?
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  2. #2
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    Re: Vector projection and perpendicular

    By trial and error i+j-k is perpendicular to v
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    Re: Vector projection and perpendicular

    What about uXv (croos product)?
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    Re: Vector projection and perpendicular

    Quote Originally Posted by M.R View Post
    1. Find the projection of u=2i+j+3k on v=-i+3j+2k.
    Hence resolve u into two vectors, one parallel to v and the other perpendicular to v.
    I can solve the the first part, just by using the projection formula = (u . v / |v|) . v / |v|
    But how do I get it perpendicular to v?
    You need to know these two:

    {u_{||}} = \frac{{u \cdot v}}{{v \cdot v}}v\;\& \;{u_ \bot } = u - {u_{||}}

    Those two are perpendicular and their sum is u.
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  5. #5
    M.R
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    Re: Vector projection and perpendicular

    Quote Originally Posted by Plato View Post
     \;{u_ \bot } = u - {u_{||}}
    That's what I needed. Thanks
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  6. #6
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    Re: Vector projection and perpendicular

    Quote Originally Posted by M.R View Post
    Hi,

    I am trying to solve the following problem:

    1. Find the projection of u=2i+j+3k on v=-i+3j+2k. Hence resolve u into two vectors, one parallel to v and the other perpendicular to v.

    I can solve the the first part, just by using the projection formula = (u . v / |v|) . v / |v|

    But how do I get it perpendicular to v? I know that if the dot product of two vectors = 0, then they are perpendicular. So:

    projection dot ??? = 0. How do I calculate what to dot it with?
    This is a more geometrical explanation (if I understand your problem correctly!):

    The vectors \vec u und \vec v span a plane. You are looking for a vector \vec p which lies in this plane and is perpendicular to \vec v.

    \displaystyle{\vec p = (\vec u \times \vec v) \times \vec v}

    will do.
    Thanks from M.R
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  7. #7
    M.R
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    Re: Vector projection and perpendicular

    Quote Originally Posted by earboth View Post
    This is a more geometrical explanation (if I understand your problem correctly!):

    The vectors \vec u und \vec v span a plane. You are looking for a vector \vec p which lies in this plane and is perpendicular to \vec v.

    \displaystyle{\vec p = (\vec u \times \vec v) \times \vec v}

    will do.
    Would that also mean that it is perpendicular to both vectors \vec u and \vec v?
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    Re: Vector projection and perpendicular

    Quote Originally Posted by M.R View Post
    Would that also mean that it is perpendicular to both vectors \vec u and \vec v?
    Yes that is correct. The answer I gave you is a standard in as much as it is the decomposition of \vec u into the sum of two vectors one parallel to \vec v the other perpendicular to \vec v.
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  9. #9
    M.R
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    Re: Vector projection and perpendicular

    Thanks
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