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A number of reasons. They share the same father, but Sesshomaru resents that his father fell in love with a human and have birth to a half demon. He blames Inuyasha for their father's demise because he died protecting Izayoi. Their father also left Inuyasha the Tetsusaiga, a very powerful sword that Sesshomaru wanted, but was left with Tensaiga, a sword which can not cut beings in the living world that he deems useless in the beginning of the show.

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