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?
 A: Consider $\vec r(t)$ with $t$ some parameter that we can call "time".
$\vec v=\dfrac{d\vec r}{dt}$ is the vector velocity, that is tangential. So, "tangential" applied to "velocity" is redundant. It seems a shorcut, and I see in the other answer it stands for the norm of the velocity, called sometimes as speed. 
The unit tangent vector is $\dfrac{\vec v}{\vert\vec v\vert}$.
A: From my experience with physics, velocity is a tangent, while "speed" is a number, a scalar. 
Speed measures how fast an object is moving, but velocity also says in which direction it is moving. So, $dr/dt$ as a vector will be the velocity, $\|dr/dt\|$, i.e. its norm, will be the speed. Finding the unit tangent, i.e. dividing the velocity vector by its norm, tell us the direction in which an object is moving regardless of how fast it is moving. 
A: 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.
