unit speed curves and frenet serret

Question

Let us assume that $\alpha(s)$ is a unit speed curve with $\kappa > 0$. I'm trying to find the vector function $w(s)$ such that

$$T' = w \times T,\quad N' = w \times N,\quad B' = w \times B.$$

I see that this is an application of Frenet Serret with $T' = \kappa\times N, N' = \frac{T'}{|T|}$, and $B' = \langle-\tau, N\rangle$ and I am just not seeing how to group these guys to get what I want, i.e., vector $w$. Do I need to use the def of an osculating plane or the right hand rule? Thanks

Answer 1

If you do the procedure in my comment you find that $w=\kappa B -\tau T$ where $\kappa$ is the curvature and $\tau$ is the torsion of $\alpha$.