Is there a circuit that can multiply the voltage of wire A by the voltage of wire B?
There are several possibilities, mainly depending on required accuracy and speed (signal frequency)


dual gate MOSFETs (e.g. BF961)
mixer ICs e.g. NE612
precision analog multiplier ICs e.g. AD633, MPY534
log-add-antilog OpAmp circuits (i.e. calculate logarithms of both inputs signals, add them and calculate anti-log (exponential function) of sum). The log and antilog functions can be accomplished by OpAmp circuits that exploit the exponential I-V characteristic of semiconductor diodes.


The first two solutions are good for RF and when scaling is not important (e.g. RF mixing).
The latter two are good when scale is important (e.g. in analog computer). 

For special cases when one of the input signals takes only values 0, 1 or 1, -1 you can use analog switches.