If someone hurts you, you might be inspired to plan an act of vengeance: some type of payback in order to settle the score. Be careful, though, because you might inspire someone to take vengeance on you in return.
Vengeance is a downward spiral of pain and betrayal, one person hurting another, who hurts another, and so on. You might demand vengeance if your brother throws your favorite jeans into a tree, and your brother might feel the same way after you spit gum in his hair. The phrase “with a vengeance” means to do something with a lot of intensity. Vengeance shares roots with the word revenge, and someone who wants revenge usually pursues it with a vengeance.