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Orthogonal vectors are vectors that form a right angle when placed tail-to-tail. In other words, the angle between them is 90 degrees. Since thisis a fairly easy case to work with, lets begin with a couple of examples of this sort.
Specification for two vectors
Use the graphical method and the trigonometric method to find Cs=B+A and Dd=B-A.
Draw the two vectors on your graph board and construct the parallelogram as described earlier.
In this case, the parallelogram simply becomes a square. You should find that the angle for the vector C is 45 degrees. Using the Pythagorean theorem, you should findthat the magnitude of the vector C is 14.14. Thus,
To subtract the vector A from the vector B, flip vector A over and draw it pointing in exactly the opposite direction. Stated differently, add 180 degreesto the angle for A and draw it. Then add the modified vector A to the original vector B.
Once again, the parallelogram is a square. Now you should find that the magnitude for vector D is 14.14 units, and the angle for vector D is 135degrees. Thus
Sum and difference magnitudes are the same
Note that the magnitude of the difference vector is the same as the magnitude of the sum vector in this case. As you will see later, that is not the case ingeneral.
Difference vector is perpendicular to the sum vector
Also, the angle of the difference vector is 90 degrees greater than the angle of the sum vector. In other words, the twovectors are perpendicular. As you will see later, that is the case in general.
Figure 1 shows the program output for the sum and difference of a pair of orthogonal vectors. These are the same vectors for which you estimated themagnitudes and angles of the sum and difference vectors earlier.
Figure 1 . Program output for orthogonal vectors. |
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Start Script
Bm = 10.00Ba = 90.00 deg
Am = 10.00Aa = 0.00 deg
Bx = 0.00By = 10.00
Ax = 10.00Ay = 0.00
Cx = 10.00Cy = 10.00
Ca = 45.00 degCm = 14.14
Da = 135.00 degDm = 14.14
End Script |
The last five lines of output text in Figure 1 show the same results that you got using graphical methods to add and subtract the vectors.
Now let's modify the problem and reduce the angle between the two vectors to 45 degrees. Assume that
Compute B+A and B-A as before.
When you draw your parallelograms, you should find that:
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