# True/False? If $\alpha(t)$ is NOT parametrized by arclength, then $T ' · T$ and $T · B$ need not be $0$.

True/False? If $\alpha(t)$ is NOT parametrized by arclength, then $T ' · T$ and $T · B$ need not be $0$.

Since $T' = k(s)N(s)$, then $T'·T$ should be zero since $N$ and $T$ are perpendicular.

However, how would this change if they are not parametrized by arc length. I though $T'$ still equaled $k(t)N(t)$ or am I wrong?

Thank you!

$T'(t) = |\alpha'(t)|K(t)N(t)$. So $T'T$ should still be zero for a curve not parametrized by arc length.