Making Connections: Functions from Dot Products
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 vectoras 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 vthe 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.

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
and

Dr. David R. Hill
Mathematics Department
Temple University

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.