Denote the sound-source velocity by where is time. Similarly, let denote the velocity of the listener, if any. The position of source and listener are denoted and , respectively, where is 3D position. We have velocity related to position by
The Doppler effect depends only on velocity components along the line connecting the source and listener [327, p. 453]. We may therefore orthogonally project the source and listener velocities onto the vector pointing from the source to the listener. (See Fig. 3.16 for a specific example.)
The orthogonal projection of a vector onto a vector is given by [422]