What is the Frénet-frame of a streamline at a given point?

[Jonmundsson]

Homework Statement

Find the Frénet-frame of the streamline \textbf{r}(t) = \left(\frac{1}{2} \cosh t, e^t, \frac{1}{2} \cosh t\right) at the point (1,1,1)

Homework Equations

\textbf{T}(t) = \frac{\textbf{r}'(t)}{||\textbf{r}'||}
\textbf{B}(t) = \frac{\textbf{r}'(t) \times \textbf{r}''(t)}{||\textbf{r}'(t) \times \textbf{r}''(t)||}
\textbf{N}(t) = \textbf{B}(t) \times \textbf{T}(t)

The Attempt at a Solution

This is pretty straightforward. The only thing that is confusing me is what to do with (1,1,1). Do I find T,B,N and plug (1,1,1) into that?

Thanks

 

[Office_Shredder]

Pretty much

 

[Jonmundsson]

To be on the safe side here is how I calculated T.

\textbf{r}'(t) = \left(\frac{1}{2} \sinh t, e^t, \frac{1}{2} \sinh t\right)

||\textbf{r}'(t)|| = \displaystyle \sqrt{(\frac{1}{2} \sinh t)^2 + (e^t)^2 + (\frac{1}{2} \sinh t)^2} = \sqrt{\frac{1}{2} \sinh ^2 t + e^{2t}}

So

T(t) = \displaystyle \frac{\left(\frac{1}{2} \sinh t, e^t, \frac{1}{2} \sinh t\right)}{\sqrt{\frac{1}{2} \sinh ^2 t + e^{2t}}}

and

T(1,1,1) = \displaystyle \frac{\left(\frac{1}{2} \sinh 1, e, \frac{1}{2} \sinh 1\right)}{\sqrt{\frac{1}{2} \sinh ^2 1 + e^{2}}}

 

[Dick]

Looks fine to me, so far.