Almost,
F.G= f1g1+f2g2=0
In order for f1g1=-f2g2 g1 should equal f2 and g2 should equal -f1
Find a vector field F(x,y) in R^2 such that F is perpendicular to the field (x+y)i-(1+x^2)j at every point.
I'm thinking: Let F = ( f1, f2) be the components of the field where f1 and f2 are functions of x,y. Call (x+y)i-(1+x^2)j field G = (g1,g2) where g1 = (x+y), g2=(-1-x^2). Then F dot G = 0 = ||F|||G||cos(Theta). Will this work?