How is Abigail Williams from The Crucible vengeful?

Abigail Williams demonstrates her vengeful personality by setting up Elizabeth Proctor and accusing her of witchcraft in order to be with Elizabeth's husband, John. Abigail Williams seeks revenge on Elizabeth Proctor for casting her out of their home and giving her a bad reputation throughout Salem's community. In act 1, Abigail expresses her negative feelings for Elizabeth by telling John

She [Elizabeth] is blackening my name in the village! She is telling lies about me! She is a cold, sniveling woman, and you bend to her!

Abigail also wishes to be with John and blames Elizabeth's spirit for attacking her. In act 2, Abigail stabs herself with a needle knowing that Mary Warren has a poppet with a needle in its stomach that is in Elizabeth's home. The authorities find the poppet and believe that Elizabeth used it as some sort of a voodoo doll to harm Abigail. In act 3, Abigail once again reveals her vengeful personality by accusing Mary Warren of sending her spirit to attack the girls. Abigail views Mary Warren as a threat and seeks revenge after Mary tells Salem's authorities that the girls are frauds.

Almost everything that Abigail Williams does in The Crucible is motivated by revenge. She's the ultimate example of a woman scorned, furious at being dumped by John Proctor after he ends their brief affair. So she sets out to destroy not just him but his whole family. Abigail harbors a particularly deep resentment towards John's wife, Elizabeth. She uses her newfound power as the main accuser in the rapidly developing witch craze to make patently false accusations of witchcraft against Elizabeth. Abigail knows exactly what she's doing. She knows that, given the tense, hysterical atmosphere in Salem, an accusation of witchcraft, however ridiculous, is tantamount to a death sentence. And that's exactly what she wants to happen to Elizabeth; she wants her to be burned as a witch. Abigail also goes out of her way to make sure that John ends up being executed as well. If she can't have him, she's determined that no one else can.