Results 1 to 5 of 5

Math Help - Vectors question.

  1. #1
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782

    Vectors question.

    Suppose that v_1, v_2,......,v_n are orthogonal non-zero vectors in Euclidean n-space, and that a vector v is expressed as

    v=\lambda_1 v_1 + \lambda_2 v_2+.......+\lambda_n v_n,

    Show that the scalars \lambda_1, \lambda_2,......., \lambda_n are given by

    \lambda_i=\frac{v.v_i}{||v_i||^2},    i=1,2,.....,n

    What are \lambda_i if the vectors v_1,v_2,.........,v_n are orthonormal?
    Well this is what i'm stuck on. The first step of my method was to multiply both sides by v_i. However, the question seems to hinge on what v has to be equal to. I can't see what this is. My initial thought was v=0, but this does not make sense since the vectors can have different magnitudes.

    Anyone have any ideas?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1
    I may not understand your difficulty. Put I hope this helps.
    i \ne j \Rightarrow v_i  \cdot v_j  = 0\quad \& \quad v_i  \cdot v_i  = \left\| {v_i } \right\|^2 .
    v \cdot v_j  = \left( {\sum\limits_{k = 1}^n {\lambda _k v_k } } \right) \cdot v_j  = \left( {\sum\limits_{k = 1}^n {\lambda _k \left( {v_k  \cdot v_j } \right)} } \right) = \lambda _j \left( {v_j  \cdot v_j } \right) = \lambda _j \left\| {v_j } \right\|^2 .

    Now divide. \lambda _j  = \frac{{v \cdot v_j }}{{\left\| {v_j } \right\|^2 }}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782
    Maybe i'm just overcomplicating the question like I normally do.

    I understand what you've done up to:

    \left( {\sum\limits_{k = 1}^n {\lambda _k \left( {v_k \cdot v_j } \right)} } \right) = \lambda _j \left( {v_j \cdot v_j } \right)
    can you just explain this step?

    Is this because all the other dot product will be zero except for the one where i=j?

    Orthonormal means that both have a unit length and are orthagonal so  \lambda_i=v.v_i for the second part. I figured that out on my own! =D
    Last edited by Showcase_22; October 11th 2008 at 03:00 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1
    Quote Originally Posted by Showcase_22 View Post
    Is this because all the other dot product will be zero except for the one where i=j?
    Exactly!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782
    Right, I get it now!! I was super overcomplicating it. =S

    Thanks Plato.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: September 21st 2011, 08:21 PM
  2. Question regarding Vectors
    Posted in the Calculus Forum
    Replies: 2
    Last Post: July 24th 2009, 07:06 AM
  3. Another vectors question
    Posted in the Calculus Forum
    Replies: 1
    Last Post: July 5th 2009, 02:23 PM
  4. vectors question
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: February 9th 2009, 08:15 AM
  5. Need help with vectors question
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 21st 2008, 08:45 AM

Search Tags


/mathhelpforum @mathhelpforum