A vector notation is developed for describing complementary codes defined at finite apertures of particular relevance to measuring and imaging problems. The original complementary series of Golay defined at one-dimensional linear apertures, are extended for more general apertures, notably in two dimensions. Two generating rules of the codes are formulated, namely the doubling rule familiar from the one-dimensional Golay series, and the point symmetry rule for adding just one vector element to a code of a strict rotational symmetry. Both on-off-type and polarity-type modulation are included and decoding algorithms are presented based on the ideal delta-function-type autocorrelation function of the codes without any side structure. Statistical analysis of the application of the complementary vector codes to a modulated Poisson process is given and implementation of the codes to ultrasonics, powder diffraction and X-ray astronomy discussed.