Hi http://img686.imageshack.us/img686/2237/img0001zy.jpg In the 3rd figure on the right... I believe the x Force component (the sine term) should also be negative since it would point to the left, but the author has it as positive. Does it possibly have to do with even/odd function of sine? I wouldn't think so because sin(-x)= -sin(x) and I don't see any of that going on here. Agree? Thanks in advance.
No, the x component points in the positive x direction, exactly as shown. A way to mentally picture this is to imagine that force being applied to a frictionless object sitting on the x axis. Then ask yourself which way will it move if you apply the indicated force. Note that physically, it's essentially the same situation as the picture above it -- only the coordinate system has changed (the force is in a slightly different direction, but ignore that qualitatively). The coordinate system above it has been rotated by π/2 counterclockwise about the origin from the coordinate system in the third picture; you can use the equations of a coordinate rotation to see how the force vector's components change under this rotation. You need to get comfortable with the idea that you can resolve a vector in the plane into two components that add vectorially to give the original vector -- and the only requirement on the two components' directions is that they are not parallel. A few minutes of playing with some straightedges, a pencil, and some paper will convince you of this. It's a nifty concept that extends to higher dimensions too. As you go on in science/engineering/math, it's a concept you will use a lot.
I think I summed up my confusion in the following edited version: http://img251.imageshack.us/img251/2237/img0001zy.jpg So is the purple version or the red version correct? Surely they both can't be right (sorry it's kind of messy) *Also, if the red version is wrong -- why is it wrong?