there is a very subtle difference..it might also depend on your regional accent..."peek" is pronounced with a slight stretching of the 'e' sound, ie pe..ek, where as "peak" is pronounced without...
now try piece,peace and piss :)
now try piece,peace and piss :)
Piece and peace are also homophones and pronounced identically \ˈpēs\ while piss is pronounced \'pis\.
Apparently you have trouble distinguishing between ē and i sounds but there are only two different sounds in those word triplets, not three.