Vectorvalued Functions
John E. Gilbert, Heather Van Ligten, and Benni Goetz
A
vectorvalued
function
r
(
t
) =
x
(
t
)
i
+
y
(
t
)
j
=
⟨
x
(
t
)
, y
(
t
)
⟩
assigns the
position
vector
r
(
t
)
to each value of
t
in the domain of
r
i.e.
,
r
(
t
)
is the vector with tail at
(0
,
0)
and head at the point
(
x
(
t
)
, y
(
t
))
. The graph of
r
is the
path
the head of the vector
r
(
t
)
sweeps out as
t
varies.
For example, to any function
y
=
f
(
x
)
corresponds the
vectorvalued function
r
(
x
) =
⟨
x, f
(
x
)
⟩
=
x
i
+
f
(
x
)
j
as shown to the right. The graph of
r
is then exactly the
same as the graph of
y
=
f
(
x
)
.
Thus all we did before with
explicitly
deﬁned functions
carries over to vector functions.
(
x, f
(
x
))
y
=
f
(
x
)
x
i
+
f
(
x
)
j
But vector functions deal equally well with
implicitly
deﬁned functions
f
(
x, y
) = 0
.
Example 1:
if
r
(
t
) =
⟨
a
cos
t, a
sin
t
⟩
,
a >
0
ﬁxed
,
then
x
(
t
) =
a
cos
t, y
(
t
) =
a
sin
t
. So by using the trig
identity
cos
2
t
+ sin
2
t
= 1
and eliminating the variable
t
, we get
x
2
+
y
2
=
a
2
. The
graph of
r
is thus the circle to the right.
x
y
r
(
t
)
r
(0)
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View Full DocumentExample 2:
What vector function
r
(
t
)
would start at
(2
,
3)
when
t
= 0
, and then trace clockwise the
circle of radius
4
centered at
(2
,
3)
as
t
increases
?
Hint:
the equation of this circle is
(
x

2)
2
+ (
y

3)
2
= 16
.
Examples 1 and 2 show the many ways vector functions are utilized:
•
given
r
(
t
) =
⟨
x
(
t
)
, y
(
t
)
⟩
, the head of the vector
r
(
t
)
traces out a
plane curve
in
2
space;
•
by identifying the head of
r
(
t
)
with the point
(
x
(
t
)
, y
(
t
))
we get a curve in the plane given
parametrically
as the
parameter
t
varies. We could even forget about vectors and simply think of the
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 Fall '07
 Gilbert
 Multivariable Calculus, vector function, Conic section

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