What are the reasons for killing Hector in the book of Iliad?

Answer 1

In a nutshell, Achilles (or Achilleus) kills Hector because Hector kills Petroklos, Achilles' friend. As a result of Agamemnon dishonouring Achilles by expropriating his slave girl, Briseis, Achills laments to his mother, the nymph Thetis, of the indignity he had suffered at Agamemnon's hands. Thetis instructs her son to withdraw his aid from the Achaean forces, while she endeavours to call upon Zeus, who promises her a means by which Achilles' honour can be restored. Zeus choreographs the following days of the Trojan war to allow the Trojans to come to the brink of destroying the Achaean army, thus emphasizing their need of Achilles and his armies to succeed in the war. Agamemnon relents and sends envoys to plead with Achilles to return, but Achilles is adamant and continues to withhold his assistance. At the most dire moment, Patroclos, Achilles' dearest friend, assumes Achilles' armour and leads his armies into battle, momentarily driving off the Trojans, but ultimately leading to Patroclos' death by the great Trojan prince Hector's sword. Achilles returns to battle to avenge Patroclos' death. To allow Achilles to regain the honour he had lost at the beginning of the epic, Zeus allows Hector to be killed by Achilles, thus satisfying the promise Zeus had made to Thetis.