Final answer:
Achilles vows to avenge Patroclus's death by killing Hector and reentering the war with renewed ferocity as described in Homer's epic, 'The Iliad.'
Step-by-step explanation:
The student has asked about Achilles' vow before Patroclus's pyre in his wrath. As per the epic narrative in The Iliad by Homer, Achilles swears to avenge Patroclus's death by returning to battle and killing Hector. This act is driven by his deep friendship with Patroclus and his rage against those who caused Patroclus's death. Achilles is well known for his fierceness in battle, and his vow signals a significant turning point in the epic where he reenters the war with a renewed ferocity.