Answer
$x=t; y=-1; z=t+1$
Work Step by Step
Since we know the velocity is:
$v(t)=r'(t)=\lt \cos t, 2t+\sin t, e^{t}\gt$
and $v(0)=\lt 1,0,1 \gt$
The velocity components of $v$ are : $v_x=1,v_y=0,v_y=1$
So, the parametric equations are:
$x=(1)t+0=t; y=(0) t+(-1)=-1; z=(1)t+(1)=t+1$
Hence, $x=t; y=-1; z=t+1$
