Using the parameter s = t3 we get:
t
3, −7t
3, t3
= s, −7s,s = s 1, −7, 1
This is a parametrization of the line through the origin, with the direction vector v = −1, 7, 1.
(c) The parametrization
8 − 4t3, 2 + 5t2, 9t3
does not parametrize a line. In particular, the points (8, 2, 0) (at t = 0),
(4, 7, 9) (at t = 1), and (−24, 22, 72) (at t = 2) are not collinear.
2. What is the projection of r(t) = ti + t4j + etk onto the xz-plane?
solution The projection of the path onto the xz-plane is the curve traced by ti + etk =
t, 0, et
. This is the curve
z = ex in the xz-plane.
3. Which projection of cost, cos 2t,sin t is a circle?
solution The parametric equations are
x = cost, y = cos 2
