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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?

Thank you in advance!

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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.

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