What Causes the First Plot to Be Positive and the Second Negative?

[asi123]

Homework Statement

I took 2 plots.
Can someone explain to me why the first one comes out positive and second negative?

Homework Equations

The Attempt at a Solution

 

[Dick]

If (a,b,c) is normal to a surface, so is (-a,-b,-c). There are two normals to a surface at a given point. For example in the first one you could also have written the plane -x-y-z=0. It's the same plane.

 

[asi123]

If (a,b,c) is normal to a surface, so is (-a,-b,-c). There are two normals to a surface at a given point. For example in the first one you could also have written the plane -x-y-z=0. It's the same plane.

Yeah, but why did it switch it on the second one, and the first one stayed positive?
If I learned anything, it is that there is no random in math...

 

[Defennder]

Um, what do you mean by "switch"? I can't make out the coefficient of y in the equation of plane for the second one. You could use the right-hand screw rule along with r_u X r_v to see which direction the normal vector points. It all depends on how you parametrise it. If you had done u=y, v=x for the second one, it would have been the same as the normal vector read off the equation of the plane.