Results 1 to 3 of 3

Math Help - Decompose a Vector

  1. #1
    Junior Member
    Joined
    Aug 2009
    Posts
    37

    Decompose a Vector

    Let n = (-2, 1). Decompose the vector g = (0, -9.8) into the sum of two orthogonal vectors, one parallel to n and the other orthogonal to n.

    I'm not sure how to go about doing this.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by RedKMan View Post
    Let n = (-2, 1). Decompose the vector g = (0, -9.8) into the sum of two orthogonal vectors, one parallel to n and the other orthogonal to n.

    I'm not sure how to go about doing this.
    From \vec n = (-2,1) you'll get the orthogonal vector as \vec o = (1,2) since \vec n \cdot \vec o = 0.

    Now you are asked to find values of s, t \in \mathbb{R} such that

    s \cdot (-2,1) + t \cdot (1,2) = (0, -9.8)

    Can you handle it from here?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,740
    Thanks
    645
    Hello, RedKMan!

    Let \vec n \,=\,\langle-2, 1\rangle
    Decompose the vector \vec g \,=\, \langle0, -9.8\rangle into the sum of two orthogonal vectors,
    one parallel to \vec n, the other orthogonal to \vec n
    Are you familiar with "projections"?


    Given vectors \vec u, \vec v

    The projection of u onto v is given by: . \overrightarrow{w_1} \;=\;\frac{\vec u \cdot\vec v}{|\vec v|^2}\,\vec v

    . . This is the component of \vec u that is parallel to \vec v.


    And: . \overrightarrow{w_2} \;=\;\vec u -\overrightarrow w_1

    . . This is the component of \vec u that is orthogonal to \vec v.



    We have: . \vec u \:=\:\langle 0,-9.8\rangle,\;\vec v \:=\:\langle-2,1\rangle

    . . \overrightarrow{w_1} \;=\;\frac{\langle0,-9.8\rangle\cdot\langle-2,1\rangle}{(\sqrt{(-2)^2 + 1^2})^2}\,\langle-2,1\rangle \;=\;\frac{-9.8}{5}\langle-2,1\rangle \;=\;\langle3.92,-1.96\rangle

    . . \overrightarrow{w_2} \;=\;\langle 0.-9.8\rangle - \langle3.92,-1.96\rangle \;=\;\langle-3.92,-7.84\rangle



    Therefore: . \boxed{\;\vec g \;\;=\;\;\langle 3.92,-1.96\rangle + \langle -3.92,-7.84\rangle\;}

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] decompose the difference g(y)-g(y') help!
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: October 18th 2010, 11:29 AM
  2. Decompose Gram Matrix
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: April 1st 2009, 06:22 PM
  3. another fraction to decompose
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: May 29th 2008, 06:56 PM
  4. decompose fraction
    Posted in the Algebra Forum
    Replies: 2
    Last Post: February 27th 2007, 02:41 PM
  5. Decompose
    Posted in the Algebra Forum
    Replies: 2
    Last Post: July 7th 2006, 06:40 AM

Search Tags


/mathhelpforum @mathhelpforum