Final answer:
To find T, N, and κ for a helix, calculate the first derivative to get the velocity, normalize it for T, derive T with respect to arc length to find κ, and the derivative of T with respect to t normalized for N.
Step-by-step explanation:
To find the unit tangent vector T, normal vector N, and curvature κ for a helix r(t) given by r(t) = 3sin(t)i + 3cos(t)j + 4tk, we start by calculating the first derivative of r(t), which gives us the velocity vector v(t). The unit tangent vector T is the normalized velocity vector. The curvature κ can be found by taking the magnitude of the derivative of T with respect to arc length and then dividing by the magnitude of the velocity vector. The normal vector N can be found by taking the derivative of T with respect to t and normalizing the result.