Parameterizing by Arc Length:

Parameterize the following function by arc length: $$\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle$$

More Interesting:

Find a vector function, $\mathbf{r}(t)$, that represents the intersection of the cone $z = \sqrt{x^2 + y^2}$ with the plane $z = 4+y$.

What is the length of this curve?

Parameterize this curve in terms of arc length.

More Theoretical:

Suppose the curve $C$ is parameterized with respect to arc length by $\mathbf{r}(t)$, that is, $|\mathbf{r}'(t)| = 1$ for all $t$. What is the distance along $C$ between $\mathbf{r}(3)$ and $\mathbf{r}(10)$?