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.

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

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