Another, perhaps easier way, to determine which eye is dominant: Extend one arm (left or right doesn't matter); with both eyes open, point your first finger at an object across the room; close one eye and continue to look at your finger pointed at the object. If your finger still seems to be pointing at the object then your open eye is dominant; if your finger seems to be pointing off to the side of the object then your closed eye is dominant.
I am right handed and left eye dominant - discovered these facts between 55 and 60 years ago shooting a .22 rifle. I cannot write with my left hand but I shoot long guns and a bow left handed (and I prefer left hand bolt action rifles and compound bows). I shoot handguns right handed and usually attempt to keep both eyes open.
NRA Life Member - RMEF Life Member - Vietnam Vet