Newton's second law is a vector law. When when we resolve it in component form along the x, y and z axes we can conclude that force changes only the component of velocity along it, for example if the only force is along x axis then only the velocity along x changes but not other two. So why, in uniform circular motion, does the centripetal force changes the direction of velocity even though 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?