You're already given [tex]x(t)=6\cos t[/tex] and [tex]y(t)=6\sin t[/tex]. To find the [tex]z[/tex] component, plug these into the second equation:
[tex]z=xy=36\sin t\cos t=18\sin(2t)[/tex]
so we have
[tex]\vec r(t)=6\cos t\,\vec\imath+6\sin t\,\vec\jmath+18\sin(2t)\,\vec k[/tex]
