Rama did not kill Ravana with Vasishta's weapon.. As I discuss in this answer, Rama killed Ravana with a Brahmastra, and it was given to him by the sage Agastya, not his guru Vasishta. Here is how the Yuddha Kanda of the Ramayana describes it:
Then, the valiant Rama, who was reminded thus by Matali, took hold of a blazing arrow, which was given by Brahma and which in turn was given to him by the glorious sage, Agastya earlier in the battle-field and which looked like a hissing serpent. Having been made formerly for Indra, the lord of celestials by Brahma, the lord of creation of infinite strength, it was bestowed in the past on the ruler of gods, who was desirous of conquering the three worlds....
Making it sacred by a special formula as per the procedure specified in scriptures, Rama who was endowed with an extraordinary strength then fixed that arrow which was the foremost among the three worlds, capable of removing the fear of Ikshwaku dynasty, taking away the glory of the enemies and bestowing joy to one's own self on his bow.
While that excellent arrow was being fixed by Rama, all the beings were frightened and the earth trembled. That enraged Rama, stretching his bow well and with an attentive mind, hurled that arrow which can tear off the vitals, towards Ravana. That arrow, which was inviolable as a thunderbolt hurled by the arms of Indra and irresistible as Yama the lord of Death, fell upon Ravana's chest. That arrow, released with great speed and which was capable of destroying the body, tore off the heart of that evil-minded Ravana. That arrow, which was capable of causing death to the body, after taking away the life of Ravana and having been anointed with blood, penetrated the earth.... Having lost his life, that king of demons for his part, who was endowed with terrible swiftness and invested with great splendour, fell down from the chariot to the ground, like Vritra the demon when the latter was struck down by the thunderbolt.
That line at the end, by the way, is an allusion to story of Indra killing Veitrasura, which I discuss here.