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Orthogonal vectors

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

  • Bm = 10 units
  • Ba = 90 degrees
  • Am = 10 units
  • Aa = 0 degrees

Use the graphical method and the trigonometric method to find Cs=B+A and Dd=B-A.

Graphical solution for Cs=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,

  • Cm = 14.14 units
  • Ca = 45 degrees

Graphical solution for Dd=B-A

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

  • Dm = 14.14 units
  • Dz = 135 degrees

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.

Trigonometric solution for Cs=B+A and Dd=B-A

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.
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.

Same magnitude, 45-degree angle

Now let's modify the problem and reduce the angle between the two vectors to 45 degrees. Assume that

  • Bm = 10 units
  • Ba = 45 degrees
  • Am = 10 units
  • Aa = 0 degrees

Compute B+A and B-A as before.

Graphical solutions for sum and difference vectors

When you draw your parallelograms, you should find that:

  • The angle for the sum vector is now 22.5 degrees (the sum vector falls half way between the two vectors being added).
  • The angle for the difference vector is still equal to the angle for the sum vector plus 90 degrees,or 112.5 degrees. (The difference vector is perpendicular to the sum vector.)
  • The magnitude of the sum vector is now longer than before.
  • The magnitude of the difference vector is now shorter than before.

Questions & Answers

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Source:  OpenStax, Accessible physics concepts for blind students. OpenStax CNX. Oct 02, 2015 Download for free at https://legacy.cnx.org/content/col11294/1.36
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