Is the tangential velocity the same as finding the tangent vector? Sorry for stupid question. I am calculating the tangent vector for a vector function and the problem also asks for arc length, speed and unit tangent vector. I did OK but when I hear the term tangential velocity of an object in physics is that the same as the calculation I am making by finding the tangent vector?

The tangential velocity is usually the length of the tangent vector. It depends on the parametrization, if you, e.g., pass slower through the curve the length of the tangent vectors reduces.
