Question

Find the unit tangent and normal vectors T(t) and N(t) for the curve C determined by r(t)=t^3i+3tj.

Answer 1

I'm not sure I'm doing this right, but this is how I would do something like this:
Take the derivative:\[r'(t) = 3t^{2}i+3j\]
That will give you the tangent at any point t. To make it a unit vector, divide by its magnitude:\[T(t)=\frac{3t^{2}i+3j}{\sqrt{(3t^{2})^{2}+(3)^{2}}}\]

I'm not sure about the normal vector. Instinctively I want to say that it would just be the opposite reciprocal of T(t) but I think that may be wrong.