The official site is getting deep into the psychological analysis in the rivalry between Goku and Vegeta. In this chapter, Professor Ota from Chubu University discusses the relationship between the two Saiyans throughout the Dragon Ball series. The saiyans start out as enemies but their relationship changes into that of rivals, or even on sided rivals. Because Goku is always one step ahead of Vegeta, he uses this as motivation to get stronger through competition. Their thoughts on power and motivation are also different and you can read about the differences here!
