### One Rotating Phasor

Here is a movie of a single phasor rotating. Notice that the phasor is blue and its projection on the real axis is red. The red real part is also plotted vs. time on the lower plot. The length of the red phasor traces out a cosine wave versus time. The accompanying movie shows the phasor rotating and the sinusoidal wave being traced out in time.

### Positive & Negative Frequency Phasors

This movie shows how a single frequency cosine is composed of the sum two phasors: one with positive frequency the other with negative frequency. The positive frequency phasor is blue (rotating counter-clockwise), and the negative frequency phasor is green (rotating clockwise). The vector sum of the two is the red phasor which is the real part.

### Two Phasors

This movie shows two phasors at different frequencies. One is four times the frequency of the other, but its amplitude is half as big. The green one is rotating faster than the blue one. The black vector is the sum of the two, and the red one its real part.

### Beats: Closely Spaced Frequencies

Here are two more phasors at different frequencies, but this time the ratio of the two frequencies is 1.15. Since the two frequencies are nearly the same, the resulting time waveform exhibits the characteristic shape of a "beat frequency." The blue and green vectors are rotating at nearly the same speed. Their sum is the erratically rotating black vector, whose projection onto the real axis is the red line. Hear the beats here. Use MATLAB to play with beats here.