[Graphics:Images/dotprod_mma_gr_1.gif]

This demo ties together the concepts of dot product and a function—dot products are used to define a function that can be graphed in the plane. The ideas of orthogonality and parallel vectors are then connected to the concepts from calculus that are important in the study of curve sketching.

For a fixed vector v = {a,b} and w[t] = {Cos[t],Sin[t]} we observe the vector w[t] as t varies from 0 to 2 Pi.  We illustrate the moving vector w[t] as a clock hand moving about the origin.  Simultaneously, we observe the function f[t]=w[t].v (dot product) being traced in the plane as t varies from 0 to 2 Pi.  

The Mathematica code below generates the frames for the animation when v = {-1,2}.  To display the animation, select the closed cell below and from the toolbar, select Cell->Animate Selected Graphics. You can control speed of the playback using the playback tools.

To change v, edit the lines that define the values of a and b and Enter the commands.

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To view the animation as an animated gif, click HERE.

To download a (Windows) movie, click HERE.

To download a Quicktime movie, click HERE.

As the animation progresses, we want to observe the geometric relationship between the vectors w[t] and v as f[t] changes with 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?

The Calculus Connection (Optional)

Using techniques from Calculus, we can verify the relationships above and determine the max/min values of f[t] on the interval from 0 to 2 Pi.

We first locate (approximately) the critical numbers of f[t].

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The critical number t2 is negative so it is not in the interval from 0 to 2 Pi.  By adding 2 Pi, we obtain the value we need.

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We can classify the critical numbers by looking at the sign of the 2nd derivative.

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The maximum value occurs when t is approximately 2.03444; the minimum occurs when t is approximately 5.17604. To calculate the maximum and minimum values, compute the value of the function at t1 and t2.

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Looking at the graph of f[t], our max/min calculations are reasonable.

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We can observe the geometric relationship between vectors v and w(t) when t = t1 and t = t2.

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To download the Mathematica notebook, click HERE.


This Mathematica Notebook was created and submitted by

Elizabeth G. Carver and
Dr. Lila F. Roberts
Mathematics and Computer Science Department
Georgia Southern University
Statesboro, GA  30460
lroberts@gasou.edu

and is included in Demos with Positive Impact with their permission.

Acknowledgement:  We would like to give special thanks to Mr. James P. Braselton for his helpful suggestions.


Converted by Mathematica      February 11, 2001