Vector Valued Functions:
A function whose domain is the set of real numbers and
range is a set of vectors.
In
2
, r (t )  f (t ), g (t )
In
3
, r (t )  f (t ), g (t ), h(t )
where f(t), g(t), and h(t) are called the component functions.
A. Domain of Vector Valued Functions:
The domain of r (t ) consists of all values of t for which r (t ) is defined.
This is the largest possible interval for which all three components are defined.
Ex 1: Determine the domain of the following function. r (t )  cos t i  ln(4  t ) j  t  1 k
Ex 2: Find the domain of the vector valued function r(t)  t 2  2t  8i 
8t
j  ln  25  t 2  k .
t 9
2
B. Graphs of Vector Valued Functions:
In order to sketch the graph of a vector function all we need to do is plug in some values of t and
then plot points that correspond to the resulting position vector we get out of the vector function.
Because it is a little easier to visualize things we’ll start off by looking at graphs of vector
functions in
.
Example 3 Sketch the graph of each of the following vector functions.
(a)
Here is a sketch of this vector function.
which appears to be the line y=1.
Ex 4: Graph
So, we’ve got a few points on the graph of this function. However, unlike the first part this isn’t really
going to be enough points to get a good idea of this graph. In general, it can take quite a few function
evaluations to get an idea of what the graph is and it’s usually easier to use a computer to do the
graphing.
Here is a sketch of this graph. We’ve put in a few vectors/evaluations to illustrate them, but the reality
is that we did have to use a computer to get a good sketch here.
Ex 5: Graph
Ex 6: Graph
C. Limits of Vector Valued Functions:
Find the limit of each component function.
 1 t3 
sin 3t
Ex 7: Find lim r (t ) if r (t )  
i  tet j 
k

3
t 0
t
 2  5t 
Ex 8: Find lim r (t ) if r(t)= et ,
t 
6t 2  5
, t 1 ln t
2  4t 2
D. Derivatives of Vector Valued Functions
r (t )  f  (t ), g (t ), h(t )  f (t )i  g (t ) j  h(t ) k
Ex 9: Compute r (t ) for r (t )  t 6 i  sin(2t ) j  ln(3t  1)k
Ex 10: Find r (t ) if r (t )  cos(t 3 )i  arctan t j  e
t 3 1
t
k
Ex 11: Find the velocity vector for the position vector r(t)  sin 1 (3t) , e5t , csc(4t) .
3
E.
Unit Tangent Vector T (t ) 
r (t )
r (t )
2
t
Ex 12: Find the unit tangent vector of r(t)  3t , (1  t)e , sin(2t)
at t = 0.
Ex 13: r (t )  t i  (2  t ) j . Find r (t ) , and T (t ) . Then sketch r (1) and r (1) .
.

Ex 14: Find the speed of a particle at time t that moves along the curve
r  (3cos t )iˆ  (3sin t ) ˆj  t 2 kˆ .
Then find the speed at t =2.
F.
Integration of Vector Valued Functions.
Indefinite Integration:
Ex 15: Compute
Ex 16: Find

Ex17: Find
 r (t )dt
 r (t )dt
2
 r (t )dt
0
if
for r (t )  sin t , 5,
if r (t )  t 2et i 
3
Definite Integration.
6e3t .
ln t
j  t csc t 2 cot t 2 k
t
t
r (t )  2 cos i  sin t j  2tk
2
Ex 18: Find r (t ) if r(t )  sin 2t i  sec2 t j  e3t k and r (0)  3i  j  2k .




Ex 19: Find the distance traveled by particle traveling along the curve r  3e2t i  e2t j  6 k , from
t = 2 to 5.