a. Show that the curve r(t) = (cos t)i + (sin t)j + (1 - cos t)k, 0 ≤ t ≤ 2π, is an ellipse by showing that it is the intersection of a right circular cylinder and a plane. Find equations for the cylinder and plane.

b. Sketch the ellipse on the cylinder. Add to your sketch the unit tangent vectors at t = 0, π/2, p, and 3π/2.

c. Show that the acceleration vector always lies parallel to the plane (orthogonal to a vector normal to the plane). Thus, if you draw the acceleration as a vector attached to the ellipse, it will lie in the plane of the ellipse. Add the acceleration vectors for t = 0, π/2, π, and 3π/2 to your sketch.

d. Write an integral for the length of the ellipse. Do not try to evaluate the integral; it is nonelementary.

Estimate the length of the ellipse to two decimal places.