In Othello, why do Iago, Roderigo and Brabantio hate the man they are discussing in Act 1, scene i?
Each of these characters has a specific situation to deal with in the play as revealed in the opening scene, and these situations drive their opinions about Othello:
- Iago, it appears, feels slighted at being passed over for a promotion by Othello, it could be that a sort of revenge is driving his hatred;
- Roderigo potentially hates Othello, because he has "stolen" the girl that Roderigo wanted to marry -- Desdemona;
- Brabantio potentially hates Othello, because he believes that Othello bewitched his daughter to make her run off and elope with him.
However, the only character who comes straight out and uses the word "hate" is Iago. Since he is the villain of the play, it is important that his bad feelings seem larger and more destructive than other characters. He mentions his hate quite a few times, and Roderigo even begins the scene (and the play) with the line: "Thou told'st me, thou did'st hold him in thy hate." So, if you believe what Iago says, then he hates Othello.
The thing that makes Roderigo appear to hate Othello (and he might, in fact, hate Othello; it just isn't stated blatantly in this opening scene) is that he refers to him with racial slurs. He calls Othello "thicklips" and "a lascivious Moor." Referring to Othello by his lips and his heritage rather than as his individual self is a bigoted thing to do, but cannot confirm that he hates Othello.
As for Brabantio, we know from Othello's speech later in the Act that he had often been Othello's kind host, that, in fact, Brabantio's home is where Othello and Desdemona met and fell in love. But he also refers to Othello as simply "Moor" and assumes that Othello has used some voodoo to bewitch his daughter to marry him. Again, a bigoted point of view, but not proof of hatred.
If you are looking for proof in Act I, scene i of hatred towards Othello, you will only find direct textual confirmation of this in the lines of Iago, the other feelings can only be inferred.