Do this experiment: Extend one of your hands all the way out in front of you while holding up your thumb and sighting it on a target that's aways in front of you. Now, looking at your thumb (aligned on said target) with both eyes, close one of your eyes. Open your eye. Then, again looking at your thumb with both eyes (while sighted on the target), close the other eye.
You may now open the eye.
When your thumb appeared to have shifted off the target when you closed an eye, THAT'S your dominant side.
I'm (regrettably) left handed and so the position of my thumb appears to change when I close my left eye during said experiment.