Hey everyone. I'm having a hard time getting a visual of being perpendicular to . The proof makes sense. Here it is:

Assume the vector function

Thus, we take the dot product of the same vector:

If we were to take the derivative of this equation, we would get:

Thus,

Now this make sense because we are talking about a position vector and its tangent. This is also easy to visualize: a point on a circle. The position vector is like the centripetal force, it points toward the center. So this makes perfect sense.

Now let's talk about and

so the proof above applies here too. The problem is that I can't visual this. I mean I can but it contradicts the proof. is not a position vector of . It is actually the unit vector of ; thus, we're basically saying is perpendicular to , which is not true. (ie. if f'(x) = x^2, f'"(x) = 2. They are not perpendicular.) HOWEVER, that is in 2D. idk what happens in 3D cuz I am having a hard time visualizing it.

So the question is, can someone please give me an example of where the tangent and the 2nd derivative of a function are perpendicular? Or somehow help me grasp this. Thanks in advance!