(Engineering Vectors) Why is this answer correct?

[engineering810]

Hello,
I am in engineering 2 at my school and have been trying to figure out why a homework problem has the answer it has. The question and answer is posted below. My question is, how should I have known the angle between the two vectors (when using parallelogram law) should be a 90 degree angle? I understand the length of a vector corresponds to it's magnitude so why shouldn't the angle from Fa to the vector connecting Fa to Fr be 60 degrees since that would've caused the shortest distance to the resultant force? I do not want to continue studying until I understand this so any help would be greatly appreciated. Thanks in advance!

(Problem and answer)
http://www.flickr.com/photos/102827963@N02/9992475615/
http://www.flickr.com/photos/102827963@N02/9992475665/

 

[voko]

The problem is of hoisting, which means the resultant force $F_R$ must be strictly vertical. Now, $F_A$ is directed at an angle to the vertical, so $F_B$ must necessarily compensate for the non-vertical component of $F_A$. Clearly, $F_B$ is smallest when it does nothing else but compensate. Think what direction it must have in this case.

 

[engineering810]

I understand completely, this is exactly what logic I was using and is why I am so confused. The problem is, if Fb was only compensating it would have been a 180 degree angle from the x-axis but instead its a 150 degree angle. Why wouldn't it only compensate for the vertical angle by pulling the force to the left and evening out the force in the y direction?

 

[voko]

Actually now that I think about it more, this logic is flawed. It would have been correct had $F_A$ been fixed. But it is not. It is the sum of $F_A$ and $F_B$ that is fixed, so it is not clear a priori that the magnitude of $F_B$ is smallest when it only compensates.

I think the simplest approach is to assume nothing and express what $F_B$ must be when $\theta$ is arbitrary, then minimize.