Results 1 to 3 of 3

Math Help - sum of radius vectors

  1. #1
    Newbie
    Joined
    Dec 2008
    Posts
    12

    sum of radius vectors

    A circle with center at O is divided into n equal arcs, n≥ 2, by the
    points A₁A₂ A₃....A_{n} Find the sum of radius vectors i OA_{i}
    where
    , i=1,2,...,n.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    Let's use the circle whose radius is 1 in the complex plane. Assume that A_{0}=1.

    Your points are the n^{th} roots of the unity. Let \omega be a n^{th} primitive root of the unity (i.e. \omega^{n}=1 and \forall d<n, \omega^{d}\neq 1).
    Then your sum is 1+\omega +...+\omega^{n-1}=0

    If you want to generalize this to your problem, just apply a homothety and a rotation to the circle we used to obtain yours.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,706
    Thanks
    625
    Hello, makenqau!

    A circle with center at O is divided into n equal arcs, n \geq 2
    by the points: A_1, A_2, A_3, \hdots A_n.
    Find the sum of radius vectors: . \sum^n_{i=1} \overrightarrow{OA}_i

    This is a classic problem . . .

    I recall that the sum is zero, but the proof eludes me.

    With that in mind, you may be able to construct your own proof.

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: November 15th 2011, 05:10 PM
  2. Replies: 3
    Last Post: June 30th 2011, 08:05 PM
  3. Replies: 2
    Last Post: June 18th 2011, 10:31 AM
  4. degree radius-vectors
    Posted in the Geometry Forum
    Replies: 5
    Last Post: August 16th 2010, 05:28 AM
  5. Replies: 1
    Last Post: November 13th 2009, 06:42 AM

Search Tags


/mathhelpforum @mathhelpforum