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

 

[HalfLostAndSearching]

I would like to point out that both methods presented (using the parallelogram rule and using right triangles) are valid and can lead to the correct solution. The choice of which method to use may depend on personal preference or the specific problem at hand.

Using the parallelogram rule allows for a graphical representation of the problem, which can be helpful in visualizing the components of the force. However, using right triangles may be more straightforward and easier to understand for some individuals.

Ultimately, as long as the correct components of the force are identified and calculated, either method can be used. It is important to understand the underlying principles and concepts behind each method, rather than just blindly following a specific approach.