So the rate of change of the position vector is related to the parametrization of the curve? I think I get it. So if we have x = t, y = t^2, z = t^3, then the tangent vector would have a larger component in the direction of z, as opposed to x and y?
The unit tangent vector, T(t) = r'(t) / || r'(t) || always has length 1. Alright, so how do we get a sense of the length of the actual tangent vector itself? Its direction is easy to imagine, but I can't understand how its magnitude changes along the curve (does it have something to do with...