Unit tangent vector understanding

Question

So I have this question here. I did part a) . That was not too bad and I got it correct.

I'm not really sure what part (b) is asking though. I know how to calculate the unit tangent vector using:

but I'm not really sure how to incorporate the arc length parameter into this. What does that mean exactly? It's been almost 4 years since I did Calculus II haha xD .

Answer 1

The calculation of the unit tangent vector can be found by using the formula

                                    v(t) 
           T(t)  =                       
                           ||v(t)||

That is to say that the vector is divided by its norm to arrive at the unit vector. An example is given in the link.

The calculation of the derivative of unit tangent vector T, with respect to the arc length, ds, can be found by using the so=called Frenet formulas. dT/ds = k N, where N is the unit normal vector, and k is the so-called curvature.