Answer
Please see the figure attached.
Work Step by Step
The projection of ${\bf{r}}\left( t \right)$ onto the $xy$-plane is obtained by setting the $z$-component equal to zero. We denote it by ${{\bf{r}}_{xy}}\left( t \right)$. Thus ${{\bf{r}}_{xy}}\left( t \right) = \left( {t\cos t,t\sin t,0} \right)$. It is shown as blue dotted curve in the figure.
Similarly, the projection of ${\bf{r}}\left( t \right)$ onto the $xz$-plane is obtained by setting the $y$-component equal to zero. We denote it by ${{\bf{r}}_{xz}}\left( t \right)$. Thus ${{\bf{r}}_{xz}}\left( t \right) = \left( {t\cos t,0,t} \right)$. It is shown as black dotted curve in the figure.
Using a computer algebra system we plot the curve and its projections.
