Question

Find the unit tangent and unit normal vectors T(t) and N(t). r(t)=

Answer 1

Hint: the tangent vector will be parallel to the velocity.

Answer 2

First find the velocity vector. Assuming:\[\vec{r}(t)=<\sqrt{2}t,e^t,e^{-t}>\]Differentiate once to get the velocity vector. Since the velocity vector will be tangent to the curve r(t), it will be parallel to the unit tangent vector as yakeyglee pointed out. So, the only thing left to do to get the unit tangent vector from the velocity vector is to rescale the velocity vector to unit length. Recall the definition of the unit tangent vector:\[\hat{T}=\frac{\vec{r'}(t)}{\left| \vec{r'}(t) \right|}\]Let me know what you get.

Answer 3

Once you have the unit tangent vector, you can find the principal unit normal vector from:\[\vec{N}(t)=\frac{\vec{T'}(t)}{\left| \vec{T'}(t) \right|}\]

Answer 4

Thanks. I was on the right path, I messed up on the mechanics of deriving. I'll post my answer ASAP.