A straight line with direction cosines $\{0,0,1\}$ is:

Direction cosines $(l,m,n)$ are defined as the cosines of the angles $\alpha ,\beta ,\gamma$ that a line makes with the $x,y,\text{ and }z$ axes respectively.Given: $\{l,m,n\}=\{0,0,1\}$ $
$l=\cos \alpha =0⟹\alpha =90^{\circ }$ $
$$m=\cos \beta =0⟹\beta =90^{\circ }$$
$$n=\cos \gamma =1⟹\gamma =0^{\circ }$$
Since the angle with the $z$ -axis is $0^{\circ }$, the line is parallel to the $z$ -axis.