Wrong solution in the book? Calculating force projection.

[bardia sepehrnia]

The question asks to calculate the determine the projection of the resultant force of F1 and F2 onto b-axis. However instead, the solution is a projection of F1 on b axis plus F2. Shouldn't the solution involve the projection of R which is 163.4 on to be axis? and for answering that, don't we need to know the angle between R and the b axis? Or would that give us the same answer?

 

[mfb]

However instead, the solution is a projection of F1 on b axis plus F2. Shouldn't the solution involve the projection of R which is 163.4 on to be axis?

Same thing. A projection is linear, it doesn't matter if you first add the vectors and then take the projection or first take the projections and then add them. The second one is much easier to calculate as you know all relevant quantities already.

The projection of F2 onto b is just F2, of course.