To answer your question, you just need to convert the low voltage to a high voltage by use of a transistor switch. Using a BJT transistor will work, but I recommend using a MOSFET. They are more efficient and can handle more power without getting hot due to the very low "on" resistance. Consider this circuit:



The P channel MOSFET gate is pulled HI by RA, so it is turned "OFF." Remember, P channel transistors (or PNP) work opposite to that of N channel transistors (OR NPN). Since the FET is OFF, the outputs signal is pulled LO by RB.

If a switch is closed/pressed, then the FET gate is LO, turning the FET "ON." Now, the transistor acts like a switch, connecting the output to the 12V power source. There is a bit of loss in the transistor, but not a lot. Using a "Power MOSFET" is necessary to handle the level of current required to operate an automotive relay - not a problem since many of these transistors can switch dozens of amps.

Since the MOSFET is a voltage controlled device, only a negligible amount of current would be needed to control the transistor (this is the current through your steering wheel wires). A similar circuit could be made using an N channel MOSFET instead. See this website on using a MOSFET as a switch:

http://www.electronics-tutorials.ws/transistor/tran_7.html

But realistically, this is all probably unnecessary. Have a look at the wire gauge table on this website:

http://www.powerstream.com/Wire_Size.htm

You will see that the few hundred mA needed to trip a relay are safely carried in wire size all the way down to 27 gauge (handles 288 mA). You should check those wires to see how big they actually are (usually printed on the side of the wire somewhere).

How can I build a switching circuit that scales several 12v signals down to <5v?
I was working on a problem where I had to calculate the period of a swinging incomplete hula-hoop given its center of mass and radius.  It only swings with very small amplitude so I considered the motion as harmonic so therefore I attempted to use the equation $$T=2\pi (\sqrt{(\frac{I support}{mgK})}$$ Where K is the distance to the point of rotation to the center of mass.  My problem with this equation is that I don't know the moment of inertia and I'm not exactly sure how to calculate it.

Is my approach to this problem correct?  

Thanks
