Answer
$x=t; y=\dfrac{1}{3}t; z=t$
Work Step by Step
Since we know the velocity is:
$v(t)=r'(t)=\lt \dfrac{1}{t},(t^2+t+1)(t+2)^{-2},1+\ln t\gt$
and $v(1)=\lt 1,\dfrac{1}{3},1\gt$
The velocity components of $v$ are : $v_x=1,v_y=\dfrac{1}{3},v_y=1$
So, the parametric equations are:
$x=(1)t+0=t; y=(\dfrac{1}{3}) t+(0)=\dfrac{1}{3}t; z=(1)t+(0)=t$
Hence, $x=t; y=\dfrac{1}{3}t; z=t$
