All 3 Uses

vengeance
in
The Lost Hero
(Auto-generated)

The wind picked up with a vengeance.†
Chpt 1 *

with a vengeance = with intensity
We will not be denied our vengeance—not this time.†
Chpt 21 *

vengeance = the act of taking revenge
And she did—how Leo single-handedly defeated the Cyclopes family; how they freed Jason, then noticed the Cyclopes starting to re-form; how Leo had replaced the dragon's wiring and gotten them back in the air just as they'd started to hear the Cyclopes roaring for vengeance inside the factory.†
Chpt 25

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
(3) (meaning too rare to warrant focus)