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

  • #1

Homework Statement



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.

Homework Equations





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 .
 
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Answers and Replies

  • #2
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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.
 
  • #3
35,125
6,869
Much better! Thank you!
 
  • #4
Ray Vickson
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Homework Statement



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.

Homework Equations





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 .

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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