Newton's second law is a vector law. When when we resolve it in component form along x,y and z axes we can conclude that force changes only the component of velocity along it ie for example if only force is along x axis the velocity along x axis only changes but not other two. So why in a uniform circular motion the centripetal force changes the direction of velocity even if it is perpendicular to velocity.
My reasoning is that at any instant say at $t = 0$ the force is along radius and perpendicular to velocity, at t=dt the velocity perpendicular to force is unchanged both in magnitude and direction but a new velocity is gained dv in dt time which is along the radius and now the resultant velocity has the same magnitude as before approximately but a different direction.
Is my reasoning correct or is there some a other explanation?