# Help with normal vectors to linear vector functions.

I am studying basic vector calculus and am on tangential and normal vectors. I understand why the derivative of a vector is tangential, and I also understand why the second derivative of a vector points perpendicularly to the tangent. But does this not apply to vectors defining linear functions? Because I can have a vector function $s(t)=(2t^3)i+(3t^3)j$ that defines a linear function. The second derivative is $s''(t)=(12t)i+(18t)j$ so neither the first nor second derivative are constant, yet the second derivative of vector $s(t)$ does not point perpendicularly to the tangent. Can someone explain why? This is my first post, so sorry for any formatting errors.

The second derivative is perpendicular to the first derivative if the latter has constant norm. This is because $$\langle s'(t),s'(t)\rangle = {\rm const.} \implies 2\langle s''(t),s'(t)\rangle = 0.$$ Your $s'(t)$ doesn't have constant norm. You'll be sure that the derivatives come out perpendicular if, for example, the curve is parametrized by arc-length (and you can always take such a reparametrization, it is an early result in most differential geometry book).