[Math] Show that $T(t)$ and $N(t)$ are Orthogonal

Question

If $r(t)$ is the smooth parametrization of a curve $C$ in 3-space, then the unit tangent and unit normal vectors are denoted as $T$ and $N$, and are given by:
$$T(t)=\frac{r'(t)}{||r'(t)||},N(t)=\frac{T'(t)}{||T'(t)||}$$
How do I show that $T(t)$ and $N(t)$ are orthogonal for all $t$ at which they are defined. I cannot assume anything about the binormal vector $B$, so I want to know how to do this without assuming so.

Answer 1

$$ T(t)\cdot N(t)={1\over\|T'(t)\|}T(t)\cdot T'(t)={1\over2\|T'(t)\|}{d\over dt}\left[ T(t)\cdot T(t)\right]=0$$

Because $T(t)\cdot T(t)={r'(t)\cdot r'(t)\over \|r'(t)\|^2}\equiv 1$