Multiplying surface equation by -1 gives different gradient

Question

So while trying to find the gradient of the surface given by the equation

x+y=5

The gradient was <1,1,0>

And then I multiplied both sides of the equation by negative 1 giving me

-x-y=-5

The gradient became <-1,-1,0>

So the gradient changed although it is the equation of the same surface... so why is that or do I have some misconception?

Answer 1

The gradient will give you the direction of greatest increase of the objective function, which in this case is $$f(x,y,z)=x+y-5$$ and your surface is $f=0$. (You could define $f$ in many ways to give this same surface, this is just an example way).

The second case you mention is when the objective function is now $$g(x,y,z)=-x-y+5,$$ so naturally the gradient of $g$ will point in the opposite direction to that of $f$, as we would now need to decrease $x$ and $y$ to increase the value of $g$.

So the gradient of a surface can be slightly ambiguous, and some care is needed to define what we mean by 'increase'.

Answer 2

You are changing you coordinate system when you multiply by $1$. For example, the direction of increasing $x$ reverses. So in the new coordinate system the gradient will have opposite sign.

The transformation $$ x\to -x \qquad y \to -y $$ is a coordinate system inversion, a reflection through the origin. Your image is now upside down.