An old saiyan legend says that if someone makes a saiyan baby cry he will hate that person for ever. Based on forums, broly and goku were born on the same day, and their capsules were right by each other.
Broly had a high power level over 10,000, while goku only had a power level of 2 (just like a regular human power level). However, goku was crying so much which kept broly up and eventually made him cry. The other saiyans were laughing at how broly, a very strong infant was crying because of a very week infant such as goku.
As with Vegeta, sources suggest that Vegeta thought that he was the strongest Elite warrior sayian and in reality he was because Nappa and Raditz were weaker than him. Since hes the "Prince" he feels that he needs to protect that status by being the strongest since they are a warrior race. Then he gets beat by Goku who proves to be stronger.
For Cooler and Frieza, it wasn’t really hatred. Cooler was simply being critical of Frieza as he allowed Goku to escape Planet Vegeta's destruction. Coola's hatred was reserved for Goku and the Saiyajin as a whole for disgracing their royal bloodline. He's also jealous as King cold always spoiled Frieza.
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