Vindicate means to justify, prove, or reinforce an idea — or to absolve from guilt. If your family thinks you hogged the last piece of pie on Thanksgiving, you'll be vindicated when your younger brother fesses up.
Vindicate derives from the Latin vindicatus, which is the past tense of vindicare, meaning "lay claim to" or "avenge." When a physicist proves a theory that his colleagues derided, he vindicates it. When a lawyer clears her client's name in a trial, she vindicates him. Machiavelli argued that the results he got vindicated his tactics — in other words, the ends justified the means.