Vector Functions
We first saw vector functions back when we were looking at
the
Equation of Lines. In that section we talked about them because we wrote
down the equation of a line in
 in
terms of a vector function (sometimes called a vector-valued function).
In this section we want to look a little closer at them and we also want to look
at some vector functions in
 other
than lines.
A vector function is a function that takes one or more
variables and returns a vector. We�ll spend most of this section looking at
vector functions of a single variable as most of the places where vector
functions show up here will be vector functions of single variables. We will
however briefly look at vector functions of two variables at the end of this
section.
A vector functions of a single variable in
 and
 have
the form,
respectively, where
 ,
 and
 are
called the component functions.
The main idea that we want to discuss in this section is
that of graphing and identifying the graph given by a vector function. Before
we do that however, we should talk briefly about the domain of a vector
function. The domain of a vector function is the set of all t�s
for which all the component functions are defined.
Example 1
Determine the domain of the following function.
Solution
The first component is defined for all t�s. The
second component is only defined for
 .
The third component is only defined for
 .
Putting all of these together gives the following domain.
This is the largest possible interval for which all three
components are defined.
Let�s now move into looking at the graph of vector
functions. In order to graph a vector function all we do is think of the vector
returned by the vector function as a position vector for points on the graph.
Recall that a position vector, say
 ,
is a vector that starts at the origin and ends at the point
 .
So, 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 2
Sketch the graph of each of the following vector functions.
(a)
(b)
Solution
(a)
Okay, the first thing that we need to do is plug in a few
values of t and get some position vectors. Here are a few,
So, what this tells us is that the following points are all
on the graph of this vector function.
Here is a sketch of this vector function.
In this sketch we�ve included many more evaluations that
just those above. Also note that we�ve put in the position vectors (in gray and
dashed) so you can see how all this is working. Note however, that in practice
the position vectors are generally not included in the sketch.
In this case it looks like we�ve got the graph of the line
 .
(b)
Here are a couple of evaluations for this vector function.
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.
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