Why is the unit tangent vector $T$ always perpendicular to the unit normal vector $N$?
$T=\frac{r'(t)}{|r'(t)|}$ and $N=\frac{T'(t)}{|T'(t)|}$
Why is the unit tangent vector $T$ always perpendicular to the unit normal vector $N$?
$T=\frac{r'(t)}{|r'(t)|}$ and $N=\frac{T'(t)}{|T'(t)|}$
The unit normal vector $\bf N$ is by definition ${\bf T}'\over \Vert {\bf T}'\Vert$.
Since $\bf T$ has constant norm one, it is orthogonal to $\bf T'$: $${\bf T}\cdot {\bf T}'=0$$ (if $\bf X$ has constant norm, then $0={d\over dt}({\bf X}\cdot{\bf X })= {\bf X}'\cdot{\bf X }+{\bf X}\cdot{\bf X }' =2{\bf X}' \cdot{\bf X} $)$^\dagger$.
This implies ${\bf T}\cdot {{\bf T}'\over \Vert {\bf T}'\Vert}=0$; thus $\bf N$ is normal to $\bf T$.
Suppose ${\bf X}(t)$ has constant norm $a$ and consider it to be a position vector. Then as $t$ varies, ${\bf X}(t)$ describes a path on the surface of a sphere of radius $a$. We know that ${\bf X'}(t_0)$ is tangent to the path traced out by ${\bf X}(t)$ at the point ${\bf X}(t_0)$; so, ${\bf X'}(t_0)$ is tangent to the aforementioned sphere and, hence, orthogonal to ${\bf X}(t_0)$.