The following problem was confusing me so any help would be greatly appreciated!
Find the unit vectors T and N for the given curve at the indicated point.
r(t) = <t, t^2, t^3> at (1, 1, 1)
T and N are tangent and normal vectors, yes?
I can get you T, but I don't have a specific definition for N: the best I can do is tell you how to find one of an infinite number of normal vectors.
Anyway, for T:
At the point (1, 1, 1) we have for an unnormalized T = <1, 2, 3>. This vector has a length of , so your answer for the unit vector in the direction of T is:
Someone else is going to have to show you how to get N. (I can find a vector perpendicular to T easily enough using , but there will be an infinite number of possible N that can do this. You obviously have a specific property defined that picks out a specific N.)
-Dan