A parametric position vector has the following definitions: r(t) = (a+sin(kt)) êp + hêz with (t) = wt, a, ẞ, k > 0, and a > ẞ. (1) The final condition ensures a positive radius. The following sequence of plots shows the space curve traced out by the position vector (black) for the specific values a = 3, ß = 1, k = 2, h = 4, and w = 1. At the selected snapshots for t, the velocity vector (blue) and acceleration vector (red) is also given. t= 2π 11 6 л t= 11 10πT t= 11 Note: in all questions except (v) do not use the specific values for the plots. Using the general definitions in equation (1) and the cylindrical vector definitions in the lecture notes, (i) determine all first and second order derivatives of the cylindrical coordinates p(t), (t), z(t); (ii) determine the velocity vector in cylindrical coordinates; (iii) determine the acceleration vector in cylindrical coordinates; and (iv) determine when the velocity vector is perpendicular to the position vector. (v) Simplify the answer in (iv) for the parameter values used to construct the space curve in the figure. Is the result consistent with the space curve? (vi) Rewrite the position vector in cartesian form - you may use whichever conversion technique you wish. (The plots are drawn using the cartesian form of the vectors).