When we apply the parallelogram on two velocities acting on a single body we see that the velocity in any one of the two directions remains the same. ( the velocities act on different directions)
When one is velocity and the other acceleration or in other words accelerating a body in some direction other than the body's uniform velocity the situation gets complex. In projectile motion we assume the acceleration to act downwards (rather than the centre) in all the points of its trajectory so that the downward forces are parallel in all the points of its trajectory. This makes the problem easy. And it is legitimate considering the size of earth.
But when we consider circular motion, thats when i get confused. Because here we are dealing with bodies moving in a uniform velocity and at the same time accelerating toward the centre. SO i dont know how to manipulate such conditions. Here the parallelogram law is not applied. I have seen a proof of the equation of centripretal force using pure geometry. And it is intuitive than the vectorial proof. Can it be done for motion in a vertical circle? or is calculus necessary?