Question

find a vector function, r(t), that represents the curve of intersection of the two surfaces. the cylinder x^2 + y^2 = 9 and the surface z = xy. r(t) =

Explanation:

Step1: Parametrize the cylinder

We know that the equation of the cylinder $x^{2}+y^{2}=9$ can be parametrized using the trigonometric identities $x = 3\cos t$ and $y=3\sin t$, where $t\in[0, 2\pi]$.

Step2: Find the $z -$ component

Since $z = xy$, substitute $x = 3\cos t$ and $y = 3\sin t$ into the equation for $z$. Then $z=(3\cos t)(3\sin t)=9\cos t\sin t$.

Step3: Write the vector - valued function

A vector - valued function $\mathbf{r}(t)$ is of the form $\mathbf{r}(t)=\langle x(t),y(t),z(t)
angle$. Substituting $x = 3\cos t$, $y = 3\sin t$ and $z = 9\cos t\sin t$ into it, we get $\mathbf{r}(t)=\langle3\cos t,3\sin t,9\cos t\sin t
angle$, where $t\in[0,2\pi]$.

Answer:

$\langle3\cos t,3\sin t,9\cos t\sin t
angle$