Find a vector function that represents the curve of intersection of the two surfaces.
The cylinder x^2 + y^2 = 4 and the surface z = xy.
What I do is make z = 0, then x^2 + y^2 = 4 is a circle and
xy = 0.
Then I make these equations equal each other.
x^2 + y^2 - 4 = xy and simplify
x^2 + y^2 - xy - 4 = 0
But now I can't seem to complete the square.
Any help would be appreciated.
I'm setting z = 0 to simplify the expression so I can complete the square. It appears that this is not the way to go. Could you write out in mathematical terms the rotation about the z - axis here for me please. I guess when you deal with surfaces its not the same as curves.
The curve is the set of points that satisfy and . Using parametric equations:
To satisfy , and (note: clearly other choices are possible).
Therefore, to satisfy it is required that .
Note that .
Therefore the vector equation of the curve is:
, where .