8e0e04…: “Into the maroon square”

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Consider the curve $C$ traced out by the vector-valued function $\vec {r}(t) = \vector {3\cos (t),5\sin (t),-4\cos (t)}, \quad 0 \leq t <2\pi$ and the surface $S = \left \{(x,y,z)\in \R ^3 : x^2+y^2+z^2 =25\right \}.$ Determine whether the curve lies on the surface, intersects the surface at finitely many points, or never intersects the surface. If it intersects the surface at finitely many points, give the $x$ , $y$ , and $z$ -coordinates of the intersection.

The curve lies on the surfaceintersects the surface at finitely many pointsdoes not intersect the surface .

(make sure to use the hint for a detailed walkthrough if you need it)

From the parameterization of $C$ , we find that $\begin{align*} x(t)&= \answer{3\cos(t)} \\ y(t)&= \answer{5\sin(t)} \\ z(t)&= \answer{-4\cos(t)} \end{align*}$

We now must substitute these into the equation that describes $S$ .

$x(t)^2+y(t)^2+z(t)^2 = \left (3\cos (t) \right )^2 + \left ( 5\sin (t)\right )^2 + \left ( -4\cos (t) \right )^2 =\answer {25} \cdot \cos ^2(t) + \answer {25} \cdot \sin ^2(t) = \answer {25}.$

Since the equation that describes $S$ holds for $t$ -values, the curve lies on the surfaceintersects the surface at finitely many pointsdoes not intersect the surface .