When do parametric equations constitute a line? The given equations specifically are  
$x=3t^3 + 7$
$y=2-t^3$
$z=5t^3 + 3$  
And  
$x=5t^2-1$
$y=2t^2 + 3$
$z=1-t^2$  

For the first case, $r=(7,2,3)+t^3(3,-1,5)$. As $t$ varies through $\mathbb R$, $t^3$ varies through $\mathbb R$, so we have a line.
For the second case, $r=(-1,3,1)+t^2(5,2,-1)$. As $t$ varies through $\mathbb R$, $t^2$ varies through the non-negative reals, so we have a ray.
