The scalar projection (component) of b onto a is: $comp_a$(b) = a.b / |a|. This gives the signed length of the shadow of b on a. The vector projection of b onto a is: $proj_a$(b) = (a.b / |a|^{2}) a. The perpendicular component is b - $proj_a$(b). Work done by a force F over displacement d equals F.d = |F||d|cos(theta). In JEE, projection problems often appear disguised as "find the component of one vector along another."