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

[Slartibartfest]

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 .

 

[Mark44]

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.

 

[Mark44]

Much better! Thank you!

 

[Ray Vickson]

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,
\vec{u} = \vec{u}_{||} + \vec{u}_{\perp}
\vec{u}_{||} = \left(\frac{\vec{u} \cdot \vec{v} }{\vec{v} \cdot \vec{v}} \right) \vec{v}
\vec{u}_{\perp} = \vec{u} - \vec{u}_{||}<br /> = \vec{u}- \left( \frac{\vec{u} \cdot \vec{v} }{\vec{v} \cdot \vec{v}} \right) \vec{v}
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.