Answer
$x=t; y=\dfrac{1}{3}t; z=t$
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,
$v(t)=\lt \dfrac{1}{t},(t^2+t+1)(t+2)^{-(2)},1+\ln t\gt \\ \implies v(1)=\lt 1,\dfrac{1}{3},1\gt$
Thus, we have the velocity components $v_x=1,v_y=\dfrac{1}{3},v_y=1$
Now, the parametric equations are as follows:
$x=(1)(t)+0=t; y=(\dfrac{1}{3}) t+(0)=\dfrac{1}{3}t; z=(1)(t)+(0)=t$
Thus, $x=t; y=\dfrac{1}{3}t; z=t$
