Objective: Calculus
is often a prerequisite course to linear algebra. As such, it is beneficial
for students to see connections between concepts from linear algebra and
more familiar topics from Calculus. This demo allows the instructor to
tie together the concepts of dot product and a function—we use dot products
to define a function that can be graphed in the plane. The ideas of orthogonality
and parallel vectors can then be connected to the concepts from calculus
that are important in the study of curve sketching.
Level: Any first course
in linear algebra, vector calculus, or any course that includes a study
of dot products.
Prerequisites:
Basic linear algebra, including dot products. An optional analysis
involves using a knowledge of calculus max/min procedures.
Platform: This
demo can be used in a browser by instructors for a classroom demonstration
of the topic and by students to carry out an investigation of the topic.
A MATLAB m-file, Mathematica notebook, and Mathcad worksheet may
be downloaded to provide routines for interactive experimentation beyond
the examples included as animations in this browser file.
Instructor's Notes:
Fix some vector 
.
Plot the dot product of the vector 
as
t varies from 
This dot product operation yields the function 
which traces out a curve as t varies from .
For fixed vector 
we
obtain the graph in Figure 1.
Figure 1.
Rather than look at the final curve as shown in Figure
1, it is helpful to see the curve generated by dot product computations.
To do this, we display the vector
as
a clock hand moving about the origin. To simulate this action, we
select a set of evenly spaced points between .
As the clock hand moves, each of the ordered pairs (t, f(t)) is then shown
on a separate graph as a red dot. The dots are connected by line
segments to obtain the graph in Figure 1. This generation is shown
in the following animation.
Begin by studying the values of f(t) for the
vector
as the animation above progresses. We want to investigate the relationship
between the dot product of the vector v and the set of vectors
w(t) = .
This relationship is shown geometrically in the graph of f(t). In
particular, we want to answer the following questions:
1. What is the geometric relationship
between
v and w(t) when f(t) is a maximum?
2. What is the geometric relationship
between
v and w(t) when f(t) is a minimum?
3. What is the geometric relationship
between
v and w(t) when f(t) = 0?
Use the animation above to investigate the
connection between v and w(t) for the stated properties
of f(t).
As an aid, by clicking on the box below
you may initiate an avi-file for the animation. You can stop the
execution of the avi-file at any point by clicking the mouse. Restart
by clicking the mouse again. (WARNING: This file is quite large
and takes a long time to load. If you have a Windows computer, you
may save the avi file to your local computer by right-clicking on the picture
below and "Save Target as..". If you have a Macintosh computer (or
wish to have a Quicktime 3.0 or higher version), click here.
If an instructor has used
the preceding material along with class discussion, the fundamental ideas
have been discussed. Optionally, the instructor can continue to use
the material below in class, or assign students the material to be used
as part of an individual or group investigation that leads to the
answers to the three questions stated below.
Further Discussion or
Student Investigation
As you change vector v
= the
graph of f(t) will change. For example, Figure 2 shows f(t) when
v
= 
and
Figure 3 shows f(t) when v = 
.
Click in the boxes below to see the animations
that generate Figure 2 and Figure 3. As you view the animations,
use these choices of the vector v to answer the three questions
posed above.
AVI (Windows) and MOV (Quicktime) movies
may be downloaded by clicking on the links below. Note that the files
are fairly large, so download time may be slow.
dpvideo2.avi,
dpvideo3.avi
dpvideo2.mov,
dpvideo3.mov
For the three cases investigated so far,
complete the following statements.
1. f(t)
has its maximum value when vectors v and w(t) are ______________.
2. f(t)
has its minimum value when vectors v and w(t) are _______________.
3. f(t)
= 0 when vectors v and w(t) are _____________________________.
Use the answers to the preceding statements
to answer the following. Let .
Then
1. For what
value of t is f(t) a maximum?___________________
2. For
what value of t is f(t) a minimum?___________________
3. For
what value of t is f(t) = 0?_________________________
By inspecting Figures 1-3 and keeping in
mind the vector v, conjecture the maximum and minimum values
of f(t). (Hint: Consider the
graphs that would be generated if v = 
or v = 
.
How are the maximum and minimum values related to vector v in these
special cases?)
Calculus Connection (Optional)
The function
is a linear combination of cos(t) and sin(t). Use max-min techniques
from calculus to determine the location and values of the maximum and minimums
in the interval 
.
Credits: This
demo was submitted by
Dr.
Gerald J. Porter
Mathematics Department
University of Pennsylvania
Philadelphia, Pa. 19104-6395
and
Dr. David R.
Hill
Mathematics Department
Temple University
Philadelphia, Pa 19122
and is included in Demos with
Positive Impact with their permission. A version of this demo
appears in their book Interactive Linear Algebra, A Laboratory Course
Using Mathcad, Springer-Verlag New York, 1996.
Interactive Files:
MATLAB:
MATLAB file dproddemo can
be downloaded and was created by David R. Hill. To see a sample screen
of the MATLAB routine click on the following box. 
Mathematica:
Mathematica notebook dproddemo.nb
for the above demo/investigation can be downloaded. It was created
by Elizabeth G. Carver and Lila F. Roberts.
Mathcad:
Mathcad worksheet dproddemo.mcd for
the above demo/investigation was adapted from Interactive
Linear Algebra, A Laboratory Course Using Mathcad (Porter/Hill) by
Lila F. Roberts.
DRH 2/6/01
Last updated 5/23/2006
