My book says that a vector normal to a tangent plane at point $(a,b)$ is $$<f_x(a,b), f_y(a,b), -1>$$ If I understand vectors correctly, wouldn't this mean that the vector always moves $1$ unit down in the $z$ coordinates? But this should be obviously untrue, as the normal vector to a point in a tangent plane can be going in any direction, depending on the plane. And my books gives an image where it seems to do just that as an example!
Obviously over here the normal vector is not moving $1$ unit down in the z-coordinates. So what does it mean that a vector normal to a tangent plane at point $(a,b)$ is of the form $<f_x(a,b), f_y(a,b), -1>$?
Also, my book gives the derivation of Stokes Theorem and says that the normal vector to the surface is $<-f_x(x,y), -f_y(x,y), 1>$. Why are the signs switched? Also , when my book discusses the flux of a vector field, it does the same thing and says "For the normal vector to point upward, we need a positive z-component." But in the image I pasted, it certainly looks like the normal vector is pointing upward, and yet it is of the form $<f_x(x,y), f_y(x,y), -1>$