A $570$ kg rocket in deep space moves with a velocity of $(5530$ m$/$s$)$ i $+(2580$ m$/$s$)$ j before impacting an $8400$ kg asteroid initially at rest. Upon impact, the asteroid breaks into three pieces, with the first piece $(5600$ kg plus the mass of the rocket imbedded in it$)$ moving at $(45$ m$/$s$)$ i $+(11$ m$/$s$)$ j and the second $(1600$ kg$)$ moving at $-(3\mathrm{.}1$ m$/$s$)$ i $+(107$ m$/$s$)$ j$.$ Find the velocity of the third piece.