Vector Valued Functions (Parametric Equations part II)
A vector valued function is a 2‐D or 3‐D set of parametric curves which define a
set of vectors.
The function y = x 2
The parametric equations
x = t y = t 2 for −1 ≤ t ≤ 2
The vector valued function
r(t ) = t i + t 2 j for −1 ≤ t ≤ 2
Represents every vector ‘tailed’
at the origin with the vectors’
‘heads’ lying along the parametric
curve x = t y = t 2 for −1 ≤ t ≤ 2
The tiny arrows along the graph are used to indicate the direction of
increasing t values ... called the curve’s orientation.
It is perfectly healthy and logical to think of a vector valued function as being
nothing more than parametric equations.
x equation = i component
y equation = j component
z equation = k component
Now let’s add a third dimension to the curve.
ex) Match the following vector functions with their correct graphs. Each vector
function has the same i and j components AND each plot has the same projection
on the xy‐plane of x = t and y = t 2 graphed in red.
a) r(t ) = t i + t 2 j + t k
I.
III.
b) r(t ) = t i + t 2 j + et k
c) r(t ) = t i + t 2 j + sin(4t )k
II.
IV.
Vector Function (a.k.a. Parametric Equation) of a Circle
In 2‐D:
r(t ) = cos(t )i + sin(t ) j
... parametrizes the unit circle.
Any combination of sin(t ) and cos(t )
will produce a circle of radius 1.
ex) Sketch the vector function
r(t ) = −4sin(t )i + 4cos(t ) j for 0 ≤ t ≤ 2π
Make sure to indicate
the curve’s orientation.
Show the coordinate positions
of t = 0, π /2 and π .
ex) What is a parametrization f
or the curve shown here?
If you wanted to shift the circle’s center ...
ex) Sketch the vector curve
r(t ) = (2 + 2cos(t ))i + (−1 + 2sin(t )) j
In 3‐D:
The circle parametrization would still utilize cosine and sine with two of the 3
components of the vector function.
ex) Each of the blue curves graphed below have the same unit circle projection
onto the yz plane.
Match the vector function with its graph below.
a) r(t ) = t i + cos(t ) j + sin(t )k
b) r(t ) = 2 i + cos(t ) j + sin(t )k
c) r(t ) = 0.25sin(6t )i + cos(t ) j + sin(t )k
I.
II.
III.
IV.