Answer
$$r(t)=\lt 3, 1+2\cos t, 5+3\sin t\gt.$$
Work Step by Step
We know that the equation of the given ellipse in Figure 15 B and centered at $(3,1,5)$ is $$
\left(\frac{y-1}{2}\right)^{2}+\left(\frac{z-5}{3}\right)^{2}=1
$$
To parametrize the ellipse, we put $\frac{y-1}{2}=\cos t$ and $\frac{z-5}{3}=\sin t$, so we have $y=1+2\cos t$ and $5=5+3\sin t.$ That is,
$$r(t)=\lt 3, 1+2\cos t, 5+3\sin t\gt.$$
