So what does J-Hat mean? It's actually Ĵ, which is hard to type so "J-Hat" is a good compromise. Basically, it's a coordinate vector. If you want the nerdy explanation, you may remember from high school math that the three dimensions can be written in the Cartesian coordinates (x,y,z). If you think of the origin (0,0,0) and draw a vector of length 1 out from the origin along each axis, they become the coordinate vectors (î,ĵ,k). (k has a vector symbol too but I can't type it)

A few years ago in my classical mechanics class our prof, Dr. Hobill was teaching us rotational mechanics, which uses a lot of vectors. Dr. Hobill is the type of prof who likes to read out what he is writing on the board so for a whole 30 minutes we hear "e-theta-hat-dot plus e-phi-hat-double-dot minus one over theta r-hat-dot.." and so on. A girl in our class is a little confused so she raises her hand and asks, "Are all of those vectors?" to which he answers, "Yeah, anything with a hat is a vector." I usually wear my hat everyday and by this time I was falling asleep during the lecture (which is a bad habit of mine) with my hat pulled low. She points at me and says, "Does that mean John is a vector too?" Hearing my name mentioned jolts me awake but I have no idea what everyone is laughing at me for. I try to pretend that I've paying attention all along by faking a laugh, which must've been unconvincing since everyone including the prof is laughing at me. Afterwards, we were standing in the hallway outside of the classroom talking about the lecture when my friend Katie jokingly dubs me J-Hat. It's been my nickname ever since.