Vector power in alternating-current circuits

It has long been known that in any simple alternating-current circuit, the current and voltage may be conveniently regarded as rotatable vector quantities.¹ It is also known that the power in such circuits is not to be regarded as the vector product of the rotating vector voltage and rotating vector current.² It does not seem to have been pointed out, however, that, under certain restrictions, it is proper to regard the power in an alternating-current circuit as a non-rotating vector quantity. Moreover, it does not appear to be generally known, although the fact has not escaped notice, that the imaginary component of vector power, or so-called “wattless power” is, in a restricted sense, just as much power, and just as “wattful” as the real component.³