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Product representation for the harmonic series of a unit vector: A string application.

A representation is found for the finite Fourier series of a vector whereby any constant constraint on its magnitude is completely solved and automatically satisfied. The representation is a product of rotations, one set for each harmonic, such that the independent degrees of freedom are identified as rotational angles. Examples for the first few harmonics are presented. A recursive procedure is found whereby one can relate the standard Fourier coefficients to the angles.