Rotating Vectors

A circle has 360°. A sine wave can be represented as circular motion. Points along a sine wave thus correspond to angles, or positions, around a circle.

Rotating-vector representation of a sine wave. At A, at the start of the cycle; at B, onefourth of the way through the cycle; at C, halfway through the cycle; at D, threefourths of the way through the cycle.

Above figure shows the way a rotating vector can be used to represent a sine wave. A vector is a quantity with two independent properties, called magnitude (or amplitude) and direction. At A, the vector points east, and this is assigned the value of 0°, where the wave amplitude is zero and is increasing positively. At B, the vector points north; this is the 90° instant, where the wave has attained its maximum positive amplitude. At C, the vector points west. This is 180°, the instant where the wave has gone back to zero amplitude and is getting more negative. At D, the wave points south. This is 270°, and it represents the maximum negative amplitude. When a full circle (360°) has been completed, the vector once again points east.

The four points in above figure are shown on a sine wave graph in below figure. Think of the vector as revolving counterclockwise at a rate that corresponds to one revolution per cycle of the wave. If the wave has a frequency of 1 Hz, the vector goes around at a rate of 1 rps. If the wave has a frequency of 100 Hz, the speed of the vector is 100 rps, or a revolution every 0.01 s. If the wave is 1 MHz, then the speed of the vector is 1 million rps (106 rps), and it goes once around every 0.000001 s (10−6 s).

The peak amplitude of a pure ac sine wave corresponds to the length of its vector. In above figure, time is shown by the angle counterclockwise from due east. Amplitude is independent of time. The vector length never changes, but its direction does.

The four points for the vector model of above starting Figure, shown in the standard amplitudeversus time graphicalmanner.