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We could continue with graphical solutions
We could continue this process for smaller and smaller angles between two vectors with the same or different magnitudes, and we wouldfind that
When the two vectors have the same magnitude
For the special case where the magnitudes of the two vectors being added and subtracted are the same,
This case will be very important in the modules on circular motion,
What happens to the angles?
Getting back to your graph board drawing of the most recent scenario, there is another very important characteristic that we might be able to recognize. When you addthe negative of vector A to vector B in order to subtract vector A from vector B, the direction of the resulting vector points at an angle that bisects the angle made by vectors B and -A.
An almost straight line
As the angle between the vectors B and A approaches zero, the angle between the vectors B and - Aapproaches 180 degrees. Therefore, those two vectors tend to describe a straight line when joined at their tails as the angle between B and A approaches 0.
Perpendicularity
The magnitude of the vector that is the sum of B and -A approaches 0, while the direction of that vector approaches perpendicularity with the (almost) straight line. That means that the difference vectorapproaches perpendicularity with each of the original vectors, B and A, as the angle between them approaches 0.
Please remember this when we discuss centripetal force in a future module.
It is difficult and time consuming for blind students to do vector addition and subtraction with the graph board. Therefore, I will make some minormodifications to the code in Listing 1 to cause the output to be more compact and then run the script for a series of decreasing angles between the vectors A and Bwhile keeping the magnitudes of the two vectors the same. The results are shown in Figure 2 .
| Figure 2 . Program output for smaller and smaller angles. |
|---|
Start Script
Bm = 10.00 Ba = 90.00 degAm = 10.00 Aa = 0.00 deg
Cm = 14.14 Ca = 45.00 degDm = 14.14 Da = 135.00 deg
Start ScriptBm = 10.00 Ba = 45.00 deg
Am = 10.00 Aa = 0.00 degCm = 18.48 Ca = 22.50 deg
Dm = 7.65 Da = 112.50 degStart Script
Bm = 10.00 Ba = 22.50 degAm = 10.00 Aa = 0.00 deg
Cm = 19.62 Ca = 11.25 degDm = 3.90 Da = 101.25 deg
Start ScriptBm = 10.00 Ba = 11.25 deg
Am = 10.00 Aa = 0.00 degCm = 19.90 Ca = 5.62 deg
Dm = 1.96 Da = 95.63 degStart Script
Bm = 10.00 Ba = 5.63 degAm = 10.00 Aa = 0.00 deg
Cm = 19.98 Ca = 2.81 degDm = 0.98 Da = 92.81 deg
Start ScriptBm = 10.00 Ba = 2.81 deg
Am = 10.00 Aa = 0.00 degCm = 19.99 Ca = 1.41 deg
Dm = 0.49 Da = 91.41 degStart Script
Bm = 10.00 Ba = 1.41 degAm = 10.00 Aa = 0.00 deg
Cm = 20.00 Ca = 0.70 degDm = 0.25 Da = 90.70 deg
Start ScriptBm = 10.00 Ba = 0.07 deg
Am = 10.00 Aa = 0.00 degCm = 20.00 Ca = 0.04 deg
Dm = 0.01 Da = 90.04 deg |
The results
Note in particular, the values of the magnitude of the difference vector (Dm) and the angle of the difference vector (Da) in Figure 2 as the angle between the vectors B and A approaches zero.
As you can see from Figure 2 , regardless of the angle between vectors B and A, the difference vector is always perpendicular to the sum vector.
As you also can also see from Figure 2 , for very small angles, the angle of the sum vector is very close to the angles of the other two vectors. (They almostoverlay one another.) Therefore, for very small angles, the difference vector is very close to being perpendicular toeach of the vectors being subtracted.
As I mentioned earlier, this conclusion will be very important in a future module dealing with circular motion.
I encourage you to run the script that I presented in this lesson to confirm that you get the same results. Confirm some of those results with yourgraph board.
Copy the code for the script into a text file with an extension of html. Then open that file in your browser. Experiment withthe code, making changes, and observing the results of your changes. Make certain that you can explain why your changes behave as they do.
I will publish a module containing consolidated links to resources on my Connexions web page and will update and add to the list as additional modulesin this collection are published.
This section contains a variety of miscellaneous information.
-end-
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