Vector Diagram

a graphical representation of the values of periodically varying quantities and the relationships between them by means of directed line segments called vectors.

Vector diagrams are widely used in electrical engineering, acoustics, optics, and other areas of science. The simple harmonic functions of one period, for example,

f₁ = B₁ sin ωt

f₂ = B₂ sin (α+ωt)

f₃ = B₃ sin (β+ωt)

can be represented graphically (see Figure 1) as projections on the O_y axis of vectors OA₁OA₂, and OA₃ rotating with constant angular velocity ω, while OA₂ and OA₃ are turned by the angles α and β with reference to OA₁. The length of the vectors corresponds to the amplitude of the oscillation:

The sum or difference of two or more oscillations is indicated on a vector diagram as the geometric sum or difference of the vectors of the component oscillations, obtained according to the parallelogram rule, and the instantaneous value of the unknown quantity is determined by the projection of the vector sum on the O_y axis.

Figure 1

For example, it is required to find the sum F of the oscillations f_l with amplitude OA₁ and f₂, with amplitude OA₂. By the geometric addition of vectors OA₁ and OA₂ on the vector diagram, we find that the amplitude of the summed oscillation F is equal to the length of vector OC = OA₁ + OA₂ and leads the oscillation f₁ in phase by the angle φ