10.9#3

Shouldn't the 'j' component of the velocity for this be 3cos(3t)+0 since  v(0)=i+0j+k?

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No, because v(t)=∫_0^t a(s)ds+v(0), and ∫_0^t -9sin(3s)ds=3cos(3t)-3cos(3x0)