Question

Find a vector function, r(t), that represents the curve of intersection of the two surfaces. The cylinder x² + y² = 36 and the surface z = xy?

Answer 1

The vector function that represents the curve of intersection of the two surfaces is r(t) = 6cos(t)i + 6sin(t)j + 6cos(t)sin(t)k.

Step-by-step explanation:

To find a vector function r(t) that represents the curve of intersection of the two surfaces x^2 + y^2 = 36 and z = xy, we need to find the parametric equations for x, y, and z and combine them into a vector function r(t). Let's start by solving x^2 + y^2 = 36 for x and y, which gives us x = 6cos(t) and y = 6sin(t), where t represents the parameter. Next, we substitute these expressions into z = xy, giving us z = 6cos(t)sin(t). Finally, we combine x, y, and z into the vector function r(t) = 6cos(t)i + 6sin(t) j + 6cos(t)sin(t) k.