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.
Cylinders are discussed in Section 10.7, pp. 748–750.