The Polar Form of Vector Representation

Author: E.E. Kimberly

The rectangular vector complex form a+jb is convenient to use in addition and subtraction of vectors, but is somewhat less convenient for multiplication, division, and the finding of powers and roots. The polar form A/θ describes a vector of magnitude A at an angle θ from the axis of reference. A vector which would be E = 8.66+j5 would be expressed in polar form as E = 10/30°. Also, the complete specification of the voltage E in Example 4-3 would be E = 96/4°34' volts. The vector length is called the absolute value, and the angle is called the argument.

Vectors may be multiplied in polar form by multiplying their absolute values and adding their arguments. Thus,

(4-1)

Vectors may be divided in polar form by dividing their absolute values and subtracting their arguments. Thus,

(4-2)

Vectors may be raised to powers by raising the absolute value to the required power and multiplying the argument by the order of the power.

Thus,

(4-3)

Also, roots may be obtained by taking the root of the absolute value and dividing the argument by the order of the root. Thus,

(4-4)

If the sign of θ in the result of a calculation is positive, the absolute value lies θ degrees counter-clockwise from the positive axis of reference. If the sign of θ is negative, the absolute value lies θ degrees clockwise.