Vector-Valued Functions
MATH 311, Calculus III
J. Robert Buchanan
Department of Mathematics
Fall 2011
J. Robert Buchanan
Vector-Valued Functions
Vector-Valued Functions
We can follow the path of an object moving in R3 using a vector
with initial point at the origin.
Definition
A vector-valued function r(t) is a mapping from its domain
D ⊂ R to its range R ⊂ V3 , so that for each t in D, r(t) = v for
exactly one vector v in V3 . We can write a vector-valued
function as
r(t) = f (t)i + g(t)j + h(t)k
where the scalar functions f , g, and h are called the
component functions of r(t).
Remark: we can make a similar definition for vectors in two
dimensions.
J. Robert Buchanan
Vector-Valued Functions
Example
t
r(t) = (2 cos t)i + (3 sin t)j +
k
2
for 0 ≤ t ≤ 4π.
2
y
0
-2
6
4
z
2
0
-2
-1
0
x
1
2
J. Robert Buchanan
Vector-Valued Functions
Example
t
r(t) = (2 cos t)i + (3 sin t)j +
k
2
for 0 ≤ t ≤ 4π.
2
y
0
-2
6
4
z
2
0
-2
-1
0
x
1
2
J. Robert Buchanan
Vector-Valued Functions
Example
t
r(t) = (2 cos t)i + (3 sin t)j +
k
2
for 0 ≤ t ≤ 4π.
2
y
0
-2
6
4
z
2
0
-2
-1
0
x
1
2
J. Robert Buchanan
Vector-Valued Functions
Example
t
k
r(t) = (2 cos t)i + (3 sin t)j +
2
for 0 ≤ t ≤ 4π.
2
y
0
-2
6
4
z
2
0
-2
-1
0
x
1
2
J. Robert Buchanan
Vector-Valued Functions
Example
4
4
r(t) = (1 + t)3/2 i + (1 − t)3/2 j +
9
9
t
k
3
for −1 ≤ t ≤ 1.
0.2
z 0.0
1.0
-0.2
0.0
0.5
y
0.5
x
1.0
0.0
J. Robert Buchanan
Vector-Valued Functions
Example
r(t) = (e−t )i + (2 cos 3t)j + (2 sin 3t)k
for −π/2 ≤ t ≤ π.
2
y
1
0
-1
-2
2
1
z
0
-1
-2
0
2
x
4
J. Robert Buchanan
Vector-Valued Functions
Arc Length in R2
Recall: for a parametrically defined curve in the plane given by
(x, y ) = (f (t), g(t)) for a ≤ t ≤ b, the arc length of the curve is
given by the definite integral
Z
s=
b
q
[f 0 (t)]2 + [g 0 (t)]2 dt
a
provided f 0 and g 0 are continuous on [a, b].
J. Robert Buchanan
Vector-Valued Functions
Arc Length in R2
Recall: for a parametrically defined curve in the plane given by
(x, y ) = (f (t), g(t)) for a ≤ t ≤ b, the arc length of the curve is
given by the definite integral
Z
s=
b
q
[f 0 (t)]2 + [g 0 (t)]2 dt
a
provided f 0 and g 0 are continuous on [a, b].
The definite integral also gives us the arc length of the path
traced out by the vector-valued function r(t) = f (t)i + g(t)j.
J. Robert Buchanan
Vector-Valued Functions
Arc Length in R3
We can generalize this definite integral to curves in three
dimensions.
The arc length of the path traced out by the vector-valued
function
r(t) = f (t)i + g(t)j + h(t)k
for a ≤ t ≤ b is
Z
s=
b
q
[f 0 (t)]2 + [g 0 (t)]2 + [h0 (t)]2 dt
a
provided f 0 , g 0 , and h0 are continuous on [a, b].
J. Robert Buchanan
Vector-Valued Functions
Example
Find the arc length of the curve traced out by the√endpoint of
the vector-valued function r(t) = h2 cos t, 2 sin t, 5ti for
0 ≤ t ≤ π.
J. Robert Buchanan
Vector-Valued Functions
Example
Find the arc length of the curve traced out by the√endpoint of
the vector-valued function r(t) = h2 cos t, 2 sin t, 5ti for
0 ≤ t ≤ π.
q
√
(−2 sin t)2 + (2 cos t)2 + ( 5)2 dt
Z0 π q
=
4 sin2 t + 4 cos2 t + 5 dt
0
Z π
=
3 dt
Z
π
s =
0
= 3π
J. Robert Buchanan
Vector-Valued Functions
Example
Find the arc length of the curve traced out by the endpoint
of
√
the vector-valued function r(t) = ht cos t, t sin t, (2 2/3)t 3/2 i
for 0 ≤ t ≤ π.
J. Robert Buchanan
Vector-Valued Functions
Example
Find the arc length of the curve traced out by the endpoint
of
√
the vector-valued function r(t) = ht cos t, t sin t, (2 2/3)t 3/2 i
for 0 ≤ t ≤ π.
Z
s =
π
q
√
(cos t − t sin t)2 + (sin t + t cos t)2 + ( 2t)2 dt
0
J. Robert Buchanan
Vector-Valued Functions
Example
Find the arc length of the curve traced out by the endpoint
of
√
the vector-valued function r(t) = ht cos t, t sin t, (2 2/3)t 3/2 i
for 0 ≤ t ≤ π.
Z
π
q
s =
=
Z0 π p
√
(cos t − t sin t)2 + (sin t + t cos t)2 + ( 2t)2 dt
t 2 + 2t + 1 dt
0
J. Robert Buchanan
Vector-Valued Functions
Example
Find the arc length of the curve traced out by the endpoint
of
√
the vector-valued function r(t) = ht cos t, t sin t, (2 2/3)t 3/2 i
for 0 ≤ t ≤ π.
Z
π
q
s =
=
Z0 π p
√
(cos t − t sin t)2 + (sin t + t cos t)2 + ( 2t)2 dt
t 2 + 2t + 1 dt
0
Z
π
(t + 1) dt
=
=
0
π2
2
+π
J. Robert Buchanan
Vector-Valued Functions
Homework
Read Section 11.1.
Exercises: 1–49 odd.
J. Robert Buchanan
Vector-Valued Functions