But I will once again add the mention that the only time one should rejoice in the fall of the enemy is when we see that the downfall of the enemies of Hashem, and not our own personal enemies we have created because of our own personal situation. The Torah clearly says that we should never take pleasure in the downfall of our personal enemies.
And the issue to always remember is that our enemies themselves are messengers from Hashem. Hashem himself sends us our enemies. He is the one who gave the Egyptians the power to enslave us. Torah teaches us that the Jews were destined to be enslaved, and the Egyptians provided a good place for them to be enslaved. So the Egyptians were only fulfilling the promise which Hashem made to Abraham, that his descendants would be strangers in a land of exile. But the Egyptians subjugated the children of Israel too much, and caused Hashem to smite them with plagues, signs and wonders.
The Jews surely sang at the sea because it was divine justice which was witnessed there. When they said, "This is my G-d and I will build him a sanctuary" they actually pointed with their fingers because they were able to virtually see Hashems divine providence. They were rejoicing because the sea spit the Egyptians onto the seashore. While the angels, according to the midrash, were admonished not to sing because Hashem was disappointed because the Egyptians were also his creations.
During the Purim story we know that the purim feast is to remind us of the party which the wicked King Achashveros held in order to get the Jews to desecrate the holy vessels of the Temple. We also remember the great military victory the Jews had because Hashem performed an open miracle. The Jews should not have won that war but with Hashems divine providence we saw our enemies fall before us. The Jews were on an incredible level, some say even higher than the level they were on when they accepted Torah at Sinai.
Hashem sends all of our adversaries against us for a reason. So that we should trust in him, and not in man. We should rejoice that Hashem is righteous, he is just, and he is faithful to those who trust in him. I rejoice only because I know that the wicked will be judged and they will receive justice. I also rejoice because I know this will send a sign that terrorists cannot kill innocent people without paying a price. I realize this is not the end of the war and it is not yet time to celebrate completely.
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EMOTION VERSUS INTELLECT
The Exodus experience was loaded with powerful spiritual impact. The miracles of the plagues and the parting of the waters of the Sea of Reeds had the capacity to bring the Divine Presence into such sharp focus that it became a part of perceived physical reality. The song that Israel sang as they crossed the dry seabed contained the following phrase: "This is my God and I will build Him a Sanctuary; the God of my father and I will exalt Him" (Exodus 15:2). Rashi, in the name of the Midrash, writes: God revealed Himself so openly to the Jews that they were able to point to His Presence and say to each other, "Look over there! You see, that is God!"