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The Poynting vector is a 3-tuple of real or complex numbers (depending on the respective formulation of electrodynamics) times a unit. It may be pictured as an arrow with some length and direction in IR^3 or IC^3. But is it a "vector" in the strict mathematical sense, i.e. an element of a vector space? If yes, what exactly is that vector space? As far as I know, Poynting vectors of two sources can only be added if the sources are completely decoherent, i.e. if there's no (time-averaged?) interference.