By inspection, this vector is seen to be slightly greater in magnitude than 35.0 and at a small angle below the positive real axis. Again using a calculator to express the vector in polar form: 35.8Z-12.4°, an answer in agreement with what was anticipated. Figure III.4 shows roughly the same result using a graphical technique. Subtraction is accomplished in much the same way. Suppose that the vector 25Z-45° is to be subtracted from the vector 20Z30°.

To subtract

Fig. III.3 The vector A Z0 shown together with its rectangular components.

III.2.2 Addition and Subtraction of Vectors

When adding or subtracting vectors, it is most convenient to use the rectangular form. This is best demonstrated through an example. Suppose that we have two vectors, 20Z30° and 25Z-45°, and these vectors are to be added. The quickest way to accomplish this is to resolve each vector into its rectangular components, add the real components, then add the imaginary components, and, if needed, express the results in polar form:

first change sign of the subtrahend and then add:

The effect of changing the sign of the subtrahend is to push the vector back through the origin. as shown in Figure III.5.

The resulting vector appears to be about 28 units long and barely in the second quadrant. The calculator gives 27.7Z90.8°.

III.2.3 Multiplication and Division of Vectors

Vectors are expressed in polar form for multiplication and division. The magnitudes are multiplied or divided and the angles follow the rules governing

or consider

Fig. III.4 Use of the graphical parallelogram method for adding two vectors. The result or sum is the diagonal originating at the origin of the coordinate system.

Fig. III.5 Graphical solution to subtraction of vectors.

exponents, added when multiplying, subtracted when dividing. Consider

The magnitudes are multiplied and the angles are added. Consider

The magnitudes are divided and the angle of the divisor is subtracted from the angle of dividend.

Raising to powers is a special case of multiplication. The magnitude is raised to the power and the angle is multiplied by the power. Consider

III.2.4 Summary

Vector manipulation is straightforward and easy to do. This presentation is intended to refresh those techniques most commonly used by those working at a practical level with ac electrical circuits. It has been the author's intent to exclude material on dot and cross products in favor of techniques that tend to allow the user more of a feeling for what is going on.


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