Question

Find a vector function, r(t), that represents the curve of intersection of the two surfaces. The cylinder x2 + y2 = 64 and the surface z = xy

Answer 1

we are given

equations as

`x^2+y^2=64`

`z=xy`

Let's assume

general form of r(t)

`r(t)=x(t)i+y(t)j+z(t)k`

We can take

`x(t)=8cos(t)`

`y(t)=8sin(t)`

`(x(t))^2+(y(t))^2=(8cos(t))^2+(8sin(t))^2`

`(x(t))^2+(y(t))^2=64((cos(t))^2+(sin(t))^2)`

`(x(t))^2+(y(t))^2=64`

so, x(t) and y(t) satisfies the given equation

now, we can find z

`z=x(t)y(t)`

`z=8cos(t)*8sin(t)`

`z=64cos(t)sin(t)`

so, we will get vector as

`r(t)=(8cos(t))i+(8sin(t))j+(64cos(t)sin(t))k`................Answer