Question

Consider the parabola r(t)=<t-t^2>, does the principal unit normal vector point in the + or - y-direction along the curve?

Answer 1

r(t)=t-t^2 is a parabola concave downwards.

The principal normal vector is defined as
<b>n</b> = dT/ds
where T is the tangent vector. This basically means that the direction of the normal vector is towards the centre of curvature, in this case, downwards, or towards -y.

For a more rigorous treatment, you can follow the link below:
http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node23.html