We can use parametric equations in three dimensions as well.

1. We begin in two dimensions. Sketch the figure produced by $x\left(t\right) = 3 \cos t$ and $y\left(t\right) = 3 \sin t$. Describe the figure sketched on the interval $0 ≤ t ≤ 10$.

   Hint (a):

   $\frac{x}{3}=\cos t \; \; \; \frac{y}{3}=\sin t$

   $\sin^2 t + \cos^2 t = 1$

2. Now suppose we have $x\left(t\right) = 3 \cos t$, $y\left(t\right) = 3 \sin t$, and $z\left(t\right) = 2t$. Make a table with four columns: $t$, $x$, $y$, and $z$. Put in several values of $t$ to answer the question: what happens to the point as $t$ increases from $0$ to $10$?

   Hint (b):

   Imagine the $z$-axis as coming out of the paper on your desk vertically.What would it look like if you plotted the points of a circle on a graph, but moved vertically out of your paper while doing so?

3. How would the figure be different if the equations were $x\left(t\right) = 3 \cos t$, $y\left(t\right) = 3 \sin t$, and $z\left(t\right) = 5t$?