Homework 5
Math 352, Fall 2014
Due Date: Friday, October 10
Note: When parameterizing curves, feel free to leave your answers in vector form, possibly
including addition and scalar multiplication. For example,
~x(t) =
cos t
sin t
1
(1, 1, 1) +
(2, −1, −1) + √ (0, 1, −1).
3
3
3
is a perfectly good final answer.
1. The sphere x2 + y 2 + z 2 = 25 and the plane x + 2y + 2z = 9 intersect along a circle C.
Find parametric equations for C.
2. In HelixPedal.gif, the black curve is the helix ~x(t) = cos t, sin t, 12 t , the black line is
tangent to this helix, the red point is (0, 0, 0), and the red and black lines are always
perpendicular. Find parametric equations for the blue curve.
3. Let P be the parabola y = x2 in the xy-plane. Find parametric equations for the
parabola that results from reflecting P across the plane x + y + z = 1.
4. (a) A vertical circle with unit radius rolls along the circle ~x(t) = (3 cos t, 3 sin t, 0), as
shown in RollingOnACircle.gif. A point P~ (t) lies on the perimeter of the rolling
circle, with P~ (0) = (3, 0, 0). Find a formula for P~ (t).
(b) Suppose instead that the rolling circle tilts inwards at a 45◦ angle, as shown in
RollingOnACircle2.gif. Find a formula for P~ (t) in this case.