Why did Vishnu take on an avatar that was half human and half lion?


The reason that Vishnu went through all that rigmarole is described in the Srimad Bhagavatam. To avenge the death of his brother Hiranyaksha at the hands of Vishnu in his Varaha (boar) incarnation, Hiranyakshupu undertook long and arduous Tapasya (deep meditation), until Brahma appeared before him. He asked Brahma for the following boon:

O my lord, O best of the givers of benediction, if you will kindly grant me the benediction I desire, please let me not meet death from any of the living entities created by you. Grant me that I not die within any residence or outside any residence, during the daytime or at night, nor on the ground or in the sky. Grant me that my death not be brought by any being other than those created by you, nor by any weapon, nor by any human being or animal. Grant me that I not meet death from any entity, living or nonliving. Grant me, further, that I not be killed by any god or demon or by any great snake from the lower planets. Since no one can kill you in the battlefield, you have no competitor. Therefore, grant me the benediction that I too may have no rival. Give me sole lordship over all the living entities and presiding deities, and give me all the glories obtained by that position. Furthermore, give me all the mystic powers attained by long austerities and the practice of yoga, for these cannot be lost at any time. [emphasis added]

That might seem like it covers everything, but Vishnu still found a way to kill him:

As a snake captures a mouse or Garuḍa captures a very venomous snake, Lord Nṛsiṁhadeva captured Hiraṇyakaśipu, who could not be pierced even by the thunderbolt of King Indra. As Hiraṇyakaśipu moved his limbs here, there and all around, very much afflicted at being captured, Lord Nṛsiṁhadeva placed the demon on His lap, supporting him with His thighs, and in the doorway of the assembly hall the Lord very easily tore the demon to pieces with the nails of His hand.

Let's review how it gets around the conditions of the boon:

  1. Vishnu assumes the form of Narasimha, a half-man half-lion creature, so he's neither a human being nor an animal.
  2. He kills him on the doorstep, so it's neither inside nor outside.
  3. He kills him at twilight, so it's neither day nor night.
  4. He puts him on his lap before he kills him, so it's neither on the earth nor in space
  5. He kills him with his fingernails, which is dead matter, so neither a living thing nor an inanimate object.

So, as always, Vishnu triumphed in the end.

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