Yes, as long as you do not run it under the floor joists. New electrical code requires that you drill a hole through the floor joists and run the wire through those holes. It cannot be hanging under the floor joists. If you see wires are already running under the floor joists then go ahead and run the wire under the joists attaching it on every other joists with a wire staple. Make sure they wire does not make contact with any metal ductwork or any other metal objects. Match the wire with what is already in the wall outlet you are jumping off of. If it is 12/2 then use 12/2, if it is 14/2 then use 14/2. Never mix wire gauges.
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If you are talking about #6 building wire, it needs to have mechanical protection. This means that it has to be run in conduit. If you are talking about #6 cable set like Romex, then it only has to have mechanically protection below a five foot elevation from the garage's finished floor. If the wall that the cable set is running in is going to be finished with drywall or plywood then that is considered protected.
Yes, you can run electrical wires in a load bearing wall.
n you run thhn wire in a cable tray
Armoured cable is not rated for underground installations. You have to use a cable with a designation for direct burial or wet and damp locations.
10A