Strong Side vs. Dominant Eye

If you are left handed, but right eye dominant, how do you shoot a pistol? 8O

When I took my riflery class, I was instructed to shoot a rifle on my right shoulder using my dominant eye and that has worked great. I've pretty much found it easier to shoot pistols right handed as well. In a static range situation it is not a problem. However, all the gun magazines use terms like "strong side" and "off hand." If I were to get into cowboy action shooting or if I decided to become a Brinks Guard in Atlanta for some crazy reason, should I be shooting a pistol with my right hand and strong right eye or my stronger left hand and weaker left eye? 8O

In my opinion, pistols are not very sensitive to eye dominance. Shotguns and archery.....now there are some big problems there, but I don't think rifles are that much of a deal, and pistols you can move slightly to align with either eye. It seems to me that you'd want to handle the pistol in your dominant hand, and I believe it will naturally be positioned in front of the dominant eye. Anyone have any experience with this subject (I'll ask my son when he gets home as he is right hand and left eye dominant).