How Do You Calculate Net Force Components on a Tilted Surface?

[STEMucator]

Homework Statement

Three forces are exerted on an object placed on a tilted floor: http://gyazo.com/53e480ef2b77acc48de89d4cb4ba9e5c

a) What is the component of the net force parallel to the floor?
b) What is the component of the net force perpendicular to the floor?
c. What is the magnitude and direction of the net force?

Homework Equations

The Attempt at a Solution

I had a few questions about this problem.

I could take the usual route by defining North and East to be positive (the book always wants me to define +x and +y to be positive) and then tilting the vector $\vec{F}_3$ so that it matches up with with the axes. The y component of $\vec{F}_3$ would form a 30 degree angle in-between the two vectors:

http://gyazo.com/2446737a5ca979a1cd7c8d7d50a60682

I could then proceed to solve for the net forces in the x (parallel) and y (perpendicular) directions and then use them to find the magnitude and direction of the net force.

My question is what if I didn't take this route? What if I use component form to solve for the net force instead WITHOUT tilting the force $\vec{F}_3$. Would the angle between $\vec{F}_3$ and the positive x-axis simply be 60 degrees?

That would change my components from the other route I took. Would that not change the overall answer?

 

[haruspex]

I'm unclear on your alternative method. Please post the algebraic details for both.

 

[STEMucator]

I'm unclear on your alternative method. Please post the algebraic details for both.

Here's how I figured the problem at first:

http://gyazo.com/2446737a5ca979a1cd7c8d7d50a60682
http://gyazo.com/ab7d1f7521c310797a2158ee88228854
http://gyazo.com/920a635d7e9efae36cbb3883837b4988

The book provided this following example in the text, which uses a slightly different method by the looks of it. Though in the book's example they did something pretty weird, which in turn led to this confusion:

http://gyazo.com/09d55ca78facd6a85ea2682a9a6500cd

I understand the argument as to why the angle is 75 degrees and how they got the components. What I don't understand is why they did this instead of tilting the vector?

When I tried to apply this method to the question, I got confused as I would have to use a 60 degree angle, which looks like it's going to change the overall answer since it will change some components, namely $\vec{F_3}_{x}$ and $\vec{F_3}_{y}$.

EDIT : Never mind I figured out what was going on.