Graph the curve $C$ traced out by the vector function $$\begin{equation*} r(t)=a\cos t\mathbf{i}+a\sin t\mathbf{j}+t\mathbf{k}\qquad t\geq 0 \end{equation*}$$

where $a$ is a positive constant.

Solution The parametric equations of the curve $C$ are $$\begin{equation*} x=x(t) =a\cos t\qquad y=y(t) =a\sin t\qquad z=z(t) =t \end{equation*}$$

Since $x^{2}+y^{2}=a^{2}\cos ^{2}t+a^{2}\sin ^{2}t=a^{2},$ for any real number $t$, any point $(x,y,z)$ on the curve $C$ will lie on the right circular cylinder $x^{2}+y^{2}=a^{2}$.

If $t=0$, then $\mathbf{r}( 0) =a\mathbf{i}$ so the point $(a,0,0)$ is on the curve. As $t$ increases, the vector $\mathbf{r}=\mathbf{r}(t)$ starts at $(a,0,0)$ and winds up and around the circular cylinder, one revolution for every increase of $2\pi$ in $t$. See Figure 6.

$r(t)=a\cos t\mathbf{i}\,{+}\,a\sin t\mathbf{j}\,{+}\,t\mathbf{k}, t\geq 0$