Answer
(a) Since ${\bf{r}}\left( t \right) = \left( {t\cos t,t\sin t,t} \right)$ satisfies the equation of the cone ${x^2} + {y^2} = {z^2}$, it lies on the cone.
(b) Please see the figure attached.
Work Step by Step
(a) We have the curve given by ${\bf{r}}\left( t \right) = \left( {t\cos t,t\sin t,t} \right)$. The coordinate functions are
$x = t\cos t$, ${\ \ }$ $y = t\sin t$, ${\ \ }$ $z=t$.
So,
${x^2} + {y^2} = {t^2}{\cos ^2}t + {t^2}{\sin ^2}t = {t^2}$
Since ${z^2} = {t^2}$, so ${x^2} + {y^2} = {z^2}$. Therefore, the curve lies on the cone ${x^2} + {y^2} = {z^2}$.
(b) We sketch the cone and the curve that lies on the cone as is shown in the figure.
