11.6 Vector Functions and Space Curves
A vector function is a function that takes one or more variables and returns a vector.
Let r(t) be a vector function whose range is a set of 3-dimensional vectors:
r(t) = hf (t), g(t), h(t)i = f (t)i + g(t)j + h(t)k
where f (t), g(t), h(t) are functions of one variable and they are called the component
functions.
The domain of r(t) consists of all values of t for which the expression for r(t) is
dened.
√
Example 1. Find the domain of the vector function r(t) = h 9 − t, ln(t − 2),
et
i.
t−5
The limit of a vector function r(t) = hf (t), g(t), h(t)i is dened by taking the limits
of its component functions as follows.
lim r(t) = hlim f (t), lim g(t), lim h(t)i
t→a
t→a
t→a
t→a
provided the limits of the component functions exist.
A vector function r(t) is continuous if and only if its component functions f (t), g(t),
h(t) are continuous.
Example 2. Find lim r(t) where r(t) = (1 + t3 )i + te−t j +
t→0
sin t
k.
t
Space Curves
Suppose that f (t), g(t) and h(t) are continuous functions on an interval I . Then
r(t) = hf (t), g(t), h(t)i denes a space curve C that is traced out by the tip of the
moving vector r(t).
The space curve C is given by the equations
x = f (t),
which are called parametric
y = g(t),
equations of
z = h(t)
C and t is called a
parameter
Example 3. Describe the curve dened by the following vector functions.
(a) r(t) = h1 − t, t, t − 2i
.
(b) r(t) = hcos(t), sin(t), 2i
(c) r(t) = hcos(t), sin(t), ti
Example 4. Find parametric equations for the curve of intersection of the cylinder
x2 + y 2 = 1 and the plane y + z = 2.
Derivatives
The derivative r0 (t) of a vector function r(t) is dened just as for a real-valued function:
r0 (t) =
dr
r(t + h) − r(t)
= lim
dt h→0
h
if the limit exists.
The vector r0 (t) is called the tangent vector to the curve dened by r(t) at the point
P , provided that r0 (t) exists and r0 (t) 6= 0. The tangent line to C at P is dened to
be the line through P parallel to the tangent vector r0 (t). The unit tangent vector is
T(t) =
Theorem
r0 (t)
|r0 (t)|
. If the functions f (t), g(t) and h(t) are dierentiable, then
r0 (t) = hf 0 (t), g 0 (t), h0 (t)i = f 0 (t)i + g 0 (t)j + h0 (t)k
Example 5. Given r(t) = (1 + t3 )i + et j + sin 3tk.
(a) Find r0 (t).
(b) Find the unit tangent vector to the curve at t = 0.
(c) Find the tangent line to the curve at t = 0.
(d) Find the tangent line to the curve at the point (1, 1, 0).
. Suppose u(t) and v(t) are dierentiable vector functions, c is a scalar, and
f (t) is a real-valued function. Then
Theorem
1.
2.
3.
4.
5.
6.
d
(u(t) + v(t)) = u0 (t) + v0 (t)
dt
d
(cu(t)) = cu0 (t)
dt
d
(f (t)u(t)) = f 0 (t)u(t) + f (t)u0 (t)
dt
d
(u(t) · v(t)) = u0 (t) · v(t) + u(t) · v0 (t)
dt
d
(u(t) × v(t)) = u0 (t) × v(t) + u(t) × v0 (t)
dt
d
[u(f (t))] = f 0 (t)u0 (f (t))
dt
Example 6. Show that if |r(t)| is a constant, then r0 (t) is orthogonal to r(t) for all t.
Note. Geometrically, this result says that if a curve lies on a sphere with center the
origin, then the tangent vector r0 (t) is always perpendicular to the position vector r(t).
Integrals
The denite integral of a continuous vector function r(t) = hf (t), g(t), h(t)i is dened
component-wise as
ˆ
a
b
ˆ b
ˆ b
ˆ b
r(t)dt =h f (t)dt,
g(t)dt,
h(t)dti
a
a
a
ˆ b
ˆ b
ˆ b
=
f (t)dt i +
g(t)dt j +
h(t)dt k
a
a
a
The Fundamental Theorem of Calculus for continuous vector functions says that
ˆ
b
r(t)dt = R(t)]ba = R(b) − R(a)
a
where is R(t) an antiderivative of r(t). We use the notation
integrals (antiderivatives).
Example 7. Find
ˆ
π/2
0
r(t)dt if r(t) = 2 cos ti + sin tj + 2tk.
´
r(t)dt for indenite