Sketch the curve r(t) = (cos t e^3t , sin t e^3t ) in R^2 and compute its arc length for 0 t 8. [For the sketch, use of software is acceptable, but the graph should be drawn by hand and the right features should be present.] [4 points]

(b) The vector v makes an angle of 5/6 with the positive x-axis. Write the vector v in component form. Furthermore, write the equation of the line l(t') passing through the origin with direction vector v. [2 points]

(c) Find the points of intersection between the curve r(t) and the line l(t') in the second quadrant. [3 points]

(d) Show that the acute angle between r(t) and l(s) at their points of intersection from (c) is a constant. What is this constant? [4 points]

(e) Explain how to do each part of this problem using polar coordinates. [5 points]