Both Uses of

vengeance
in
The Iliad by Homer - (translated by: Butler)

Then shall some braggart Trojan leap upon your tomb and say, 'Ever thus may Agamemnon wreak his vengeance; he brought his army in vain; he is gone home to his own land with empty ships, and has left Menelaus behind him.'†
Book 4 *

vengeance = the act of taking revenge

The other Achaeans sat where they were all silent and orderly to hear the king, and Agamemnon looked into the vault of heaven and prayed saying, "I call Jove the first and mightiest of all gods to witness, I call also Earth and Sun and the Erinyes who dwell below and take vengeance on him who shall swear falsely, that I have laid no hand upon the girl Briseis, neither to take her to my bed nor otherwise, but that she has remained in my tents inviolate.†
Book 19

Definitions:

(1) (vengeance as in: vengeance is mine)
the act of taking revenge

(Revenge means to harm someone to get them back for something harmful that they have done.)
(2) (with a vengeance as in: with a vengeance)
with intensity