Calculus 2nd Edition Rogawski

SECTION 14.1 Vector-Valued Functions 733
Thus, the circle of radius 3 centered at (0, 0, 0) has parametrization ⟨3 cos t,3 sin t,0⟩.
To move this circle in a parallel fashion so that its center lies at P = (2, 6, 8), we translate
by the vector ⟨2, 6, 8⟩:
r 1 (t) = ⟨2, 6, 8⟩ + ⟨3 cos t,3 sin t,0⟩ = ⟨2 + 3 cos t,6 + 3 sin t,8⟩
(b) The parametrization ⟨3 cos t,0, 3 sin t⟩ gives us a circle of radius 3 centered at the
origin in the xz-plane. To move the circle in a parallel fashion so that its center lies at
(2, 6, 8), we translate by the vector ⟨2, 6, 8⟩:
r 2 (t) = ⟨2, 6, 8⟩ + ⟨3 cos t,0, 3 sin t⟩ = ⟨2 + 3 cos t,6, 8 + 3 sin t⟩
These two circles are shown in Figure 7.
z
z
8
P
〈2, 6, 8〉 〈2, 6, 8〉
8
P
FIGURE 7 Horizontal and vertical circles of
radius 3 and center P = (2, 6, 8) obtained
by translation.
x
2
(A)
6
y
x
2
(B)
6
y
14.1 SUMMARY
• A vector-valued function is a function of the form
r(t) = ⟨x(t), y(t), z(t)⟩ = x(t)i + y(t)j + z(t)k
• We often think of t as time and r(t) as a “moving vector” whose terminal point traces
out a path as a function of time. We refer to r(t) as a vector parametrization of the path,
or simply as a “path.”
• The underlying curve C traced by r(t) is the set of all points (x(t), y(t), z(t)) in R 3 for
t in the domain of r(t). A curve in R 3 is also called a space curve.
• Every curve C can be parametrized in infinitely many ways.
• The projection of r(t) onto the xy-plane is the curve traced by ⟨x(t), y(t), 0⟩. The
projection onto the xz-plane is ⟨x(t), 0, z(t)⟩, and the projection onto the yz-plane is
⟨0, y(t), z(t)⟩.
14.1 EXERCISES
Preliminary Questions
1. Which one of the following does not parametrize a line?
(a) r 1 (t) = ⟨8 − t,2t,3t⟩
(b) r 2 (t) = t 3 i − 7t 3 j + t 3 k
(c) r 3 (t) = 〈 8 − 4t 3 , 2 + 5t 2 , 9t 3〉
2. What is the projection of r(t) = ti + t 4 j + e t k onto the xz-plane?
3. Which projection of ⟨cos t,cos 2t,sin t⟩ is a circle?
4. What is the center of the circle with parametrization
r(t) = (−2 + cos t)i + 2j + (3 − sin t)k?