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Why is this true? If $\alpha$ is a time dependent curve, $T$ is the unit tangent and $N$ is a normal field along $\alpha$, then $$\langle \partial_s N, T \rangle = -\langle N, \partial_s T \rangle$$ where $s$ is the arc length parametrisation. I assume this inner product thing is a dot product but I don't understand why this holds.

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Differentiate $\langle N, T\rangle \equiv 0 $ with respect to $s$.

(yes, $\langle, \rangle$ is presumably a scalar product, and in the setting I assume you are working in an equality $$\frac{d}{ds}\langle X,Y\rangle = \langle\partial_s X,Y\rangle + \langle X, \partial_s Y \rangle$$ almost certainly holds true. Since you did not mention the source of this question I can only guess this. Check your source for this kind of formulae.)

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