We learned today in class about normal unit vectors of vector-valued functions. I don't understand why they aren't in the same direction as the acceleration vector.
The acceleration is the second derivative and isn't the normal unit vector just a double scaled version of the second derivative? The magnitude of $T(t)$ and $N(t)$ act as scalars when they are multiplied to make the vectors unit vectors, right? Therefore wouldn't the direction remain the same as the second derivative? What am I missing?