am i to understand that you are given vectors v and b and you want to express v as a sum of vectors parallel to b and then as a sum of vectors orthogonal to b?
if that is the case, why are you adding v to b and v to something parallel to b? it seems that you are not even attempting to do what the problem i think is asking. you want v = (sum of vectors parallel to b) and v = (sum of vectors orthogonal to b) right? how did you end up with v + b on the left side?