Answer
$x=t; y=-1; z=t+1$
Work Step by Step
We know that the velocity and acceleration is defined as: velocity is given as: $v(t)=r'(t)$ and acceleration is given as: $a(t)=v'(t)$
Now, $v(t)=\lt \cos (t), 2t+\sin (t), e^{(t)}\gt \\ \implies v(0)=\lt 1,0,1 \gt$
Thus, we have the velocity components $v_x=1,v_y=0,v_y=1$
Now, the parametric equations are as follows:
$x=(1)t+0=t; y=(0)( t)+(-1)=-1$ and $z=(1)t+1=t+1$
Thus, $x=t; y=-1; z=t+1$
