Let $\alpha$ be a curve parametrised by arc length $\alpha(s)=(x(s),y(s))$. So the tangent to this curve is a unit vector $t(s)=\bigl(\dot x(s),\dot y(s)\bigr)$.
My question is which one is the normal vector:
$n(s)=\bigl(-\dot y(s),\dot x(s)\bigr)$ OR $n(s)=\bigl(\dot y(s),-\dot x(s)\bigr)$ ?
Does it depend on the orientation? If the curve has a positive orientation, which normal vector does it have? How can we tell?
Many thanks for the help!
It depends on the orientation and on your convention. There isn't one "natural" choice of normal direction and so different authors might choose different conventions.
If I had to bet I would guess $(-\dot y, \dot x)$ so that the 90-degree rotation operator agrees with multiplication by $i$ in the complex plane, but I wouldn't place a large wager. Check your definition.