Question 9.1.1: Graph the curve traced by the vector function r(t) = 2 cos t......
Graph the curve traced by the vector function
\mathrm{r}(t)=2 \cos t \mathrm{i}+2 \sin t \mathrm{j}+t \mathrm{k}, \quad t \geq 0.
Step-by-Step
The parametric equations of the curve are x=2 \cos t, y=2 \sin t, z=t. By eliminating the parameter t from the first two equations:
x^{2}+y^{2}=(2 \cos t)^{2}+(2 \sin t)^{2}=2^{2}
we see that a point on the curve lies on the circular cylinder x^{2}+y^{2}=4. As seen in FIGURE 9.1.2 and the accompanying table, as the value of t increases, the curve winds upward in a spiral or circular helix.
\begin{array}{|lrrr|}\hline t & x & y & z \\\hline 0 & 2 & 0 & 0 \\\pi / 2 & 0 & 2 & \pi / 2 \\\pi & -2 & 0 & \pi \\3 \pi / 2 & 0 & -2 & 3 \pi / 2 \\2 \pi & 2 & 0 & 2 \pi \\5 \pi / 2 & 0 & 2 & 5 \pi / 2 \\3 \pi & -2 & 0 & 3 \pi \\7 \pi / 2 & 0 & -2 & 7 \pi / 2 \\4 \pi & 2 & 0 & 4 \pi \\9 \pi / 2 & 0 & 2 & 9 \pi / 2 \\\hline\end{array}
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