Answer
$$r(t)=\lt t,3t,15t\gt.$$
Work Step by Step
Since the projection of the line on the xy-plane is a line of slope 3, then $y=3x$ and since its projection on the yz-plane is a line of slope 5, then $z=5y$.
Now, let $x=t$; then $y=3t$ and $z=5y=5(3t)=15t$ and we get the parametrization
$$r(t)=\lt t,3t,15t\gt.$$
(Where $t$ can be any real number.)
