Controlling a 3 wire AC 120v heater with relay

That's not 3-phase. It sounds like regular regular-American single-phase house wiring. You can just switch the hot or you can use a DPDT relay and switch the hot & neutral.

...A light switch just switches the hot (black) and the neutral (white) remains connected.

In the U.S. we don't have 3-phase coming into the house. 3-phase is only used industrially. We have "split phase", which is two 120V out-of-phase "hot" legs, a neutral, and a ground. Between these two wires you get 240V for certain high-power appliances. Otherwise one phase goes to half the house and the other phase goes to the other half of the house.

The neutral and ground are connected together at the outlet box. The "return" current is supposed to flow through the neutral and current only flows through ground in the case of certain faults.

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