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Find the amount of a vector acting in the direction of another vector.

  1. Sep 8, 2014 #1
    1. The problem statement, all variables and given/known data

    I have a problem in a statics class that asks to find the component of a force acting along an axis seen here http://i.imgur.com/aZ1vMIu.jpg?1.

    2. Relevant equations



    3. The attempt at a solution
    The book and my professor say to use the parallelogram rule and then use sine law to find the solution like this http://i.imgur.com/dipBfjq.jpg?1 , but I do not see why it should be done this way; I understand that the axis are tilted slightly but I would think in order to solve this problem you would still use a right triangle rather than the parallelogram rule like this http://i.imgur.com/3Pazp0l.jpg?1 and I can turn the force into a vector and then find the component of the force onto the axis and I get the same answer that I would if I used right triangles like this http://i.imgur.com/naGp3us.jpg?1 . Now I'm assuming my professor and the book are correct(even though I would love to prove them wrong) so why would I use a parallelogram instead of a right triangle?
    Also here is the answer in the back of the book http://i.imgur.com/5k4UaLC.jpg?1 .
     
    Last edited: Sep 8, 2014
  2. jcsd
  3. Sep 8, 2014 #2

    Mark44

    Staff: Mentor

    Your images are way too large. Please resize them to about 800 X 600 and repost them so that people will be able to see them without having to scroll across and from top to bottom.
     
  4. Sep 8, 2014 #3

    Mark44

    Staff: Mentor

    Much better! Thank you!
     
  5. Sep 9, 2014 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    If you have vectors ##\vec{u}## and ##\vec{v}##, you can decompose ##\vec{u}## into a component ##\vec{u}_{||}## that is parallel to ##\vec{v}## and a component ##\vec{u}_{\perp}## that is perpendicular to ##\vec{v}##. That is,
    [tex] \vec{u} = \vec{u}_{||} + \vec{u}_{\perp}[/tex]
    [tex]\vec{u}_{||} = \left(\frac{\vec{u} \cdot \vec{v} }{\vec{v} \cdot \vec{v}} \right) \vec{v} [/tex]
    [tex]\vec{u}_{\perp} = \vec{u} - \vec{u}_{||}
    = \vec{u}- \left( \frac{\vec{u} \cdot \vec{v} }{\vec{v} \cdot \vec{v}} \right) \vec{v} [/tex]
    So, if you can compute the inner product of ##\vec{u}## and ##\vec{v}## you are almost done. This works in any number of dimensions.
     
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