A vector modulator has two ports labeled "I" and "Q", where "I" represents "In - phase" and "Q" represents "Quadrature". Each of the ports can take a DC signal, and therefore together represent a complex DC signal that we would apply as $I+jQ$. The device will upconvert this complex DC signal to the RF carrier, which is the signal applied differentially to IN and /IN ports. Thus if we rotate our DC signal x degrees, x degrees will also be added the RF carrier's phase appearing at the RFOUT port. The way this works in practice is the multipliers shown serve as biphase attenuators based on the voltage applied to I or Q. It will control the amplitude of the incident RF signal, and if made negative, it will flip the RF signal 180°. The RF signal is split in quadrature, each leg is adjusted with the biphase attenuation, and then the two are combined resulting is a full 360° phase and amplitude control corresponding to $I + jQ$ applied. We can use one of these at each antenna input to an antenna array, and with a static phase and amplitude setting on each antenna, form a static beam pointing in a particular direction (no one would ever in practice refer to these as "DC batteries" controlling each antenna, but my point is they are indeed complex values that are constant with time-- so DC!).
The above implementation when shown in complex form is functionally equivalent to this block diagram below, which (once understood) demonstrates the significant simplifying convenience of the complex representations (the exponents simply sum in the product):. This is also why I consider single frequency tones as spinning phasors of the form $e^{j\omega t}$ and not sinusoids which complicate everything as consisting of two such tones.
The "Hilbert" block creates the "Analytic Signal" and is implemented in practice with a 90 degree quadrature splitter, which produces two outputs $\cos(\omega_c t)$ and $\sin(\omega_c t)$ which I represent above with the thicker line as a single complex datapath as $\cos(\omega_c t)+j\sin(\omega_c t)= e^{j\omega_c t}$. The multiplier shown above is a full complex multiplier, which would require four multipliers and two adders generally as a product of imaginary terms as $(I_1+Q_1)(I_2-jQ_2)= (I_1I_2 + Q_1Q_2) + j(I_2Q_1-I_1Q_2)$, where $I_1$, $Q_1$ are the real and imaginary ports from the output of the Hilbert, and $I_2$, $Q_2$ is our "DC phase". Since we in this case only need the real of the result (as a passband signal), we only need two of the multipliers representing the real terms of the complete product: $I_1I_2 + Q_1Q_2$, matching the implementation shown.
