To find the requested vectors and accelerations, differentiate the position vector r(t) to get velocity and acceleration vectors, and then normalize and decompose as needed to find the unit tangent and normal vectors, as well as tangential and normal accelerations at t = 4π.
To find the unit tangent vector T(t), unit normal vector N(t), normal acceleration aN, and tangential acceleration aT for the curve defined by r(t) = ⟨e−tcos(t), etsin(t)⟩, we first need to compute the velocity vector v(t) by differentiating r(t). The unit tangent vector is v(t) normalized. The acceleration vector a(t) is the derivative of v(t), and this can be decomposed into the tangential and normal components aT and aN.
These components are in the direction of T(t) and the unit normal vector N(t), respectively. At t = 4π, each of these vectors and accelerations can be calculated through differentiation and normalization steps.
To get the answer, one would calculate each of these values specifically at t = 4π. Since the explicit solution is not provided, we can only outline the steps necessary for the student to follow to achieve their final answer.