Question

Find a vector function, r(t), that represents the curve of intersection of the two surfaces. the cylinder x2 y2 = 9 and the surface z = xy

Answer 1

Let $x = 3\cos(t)$ and $y=3\sin(t)$. Then

$z = xy = 9\cos(t)\sin(t) = \dfrac92 \sin(2t)$

So we can parameterize the intersection by

$\vec r(t) = 3\cos(t)\,\vec\imath + 3\sin(t)\,\vec\jmath + \dfrac92 \sin(2t)\,\vec k$

with $0\le t\le2\pi$.