How does centripetal force changes the direction of velocity isn't it a violation to Newton's second law of motion?

One can always express the position of some object with respect to some origin as a vector in as $mathbf r = r hat r$ where $mathbf r$ is the object's position vector, $r$ is the magnitude of that position vector, and $hat r$ is the unit vector parallel to $mathbf r$. Differentiating with respect to time yields $mathbf v equiv dot{mathbf r} = dot r hat r + r dot{hat r}$.

If the radial distance is constant, $dot r$ is identically zero. But what about $dot{hat r}$? This is a unit vector, which is a special case of a constant length vector. Consider a vector $mathbf x$ whose length is constant with respect to time: $||mathbf x||^2 = mathbf x cdot mathbf x= text{const}$. Differentiating with respect to time yields $mathbf x cdot dot{mathbf x} = 0$. In other words, the time derivative of a constant length vector is either zero or is normal to that constant length vector. A unit vector is obviously a special case of a constant length vector.