# True/False: The derivative of the unit binormal with respect to arclength is always parallel to the unit normal.

True/False: If $\alpha(t)$ is a regular parametrized curve such that $\alpha'(t) \neq 0$ for any t, then the derivative of the unit binormal with respect to arclength is always parallel to the unit normal.

I think this is false, since $T$ and $N$ are perpendicular, and $B = T \times N$ and $B' = \tau*N$, then $B'$ should not be parallel to $N$.

Could anyone better explain this to me?

True. If $B' = \tau N$ (you said it yourself!), then $B'$ is parallel to $N$ - they're multiples of each other. What happens is that $B$ is not parallel to $N$, but orthogonal to $N$, in view of $B = T \times N$. Perhaps you mistook $B$ for $B'$ while reading.