Vector that realizes a given slope

Question

On a book I read the following:

the slope of the level set of a function G through the point$x_0,y_0$,is $-\frac{\frac{∂G}{∂x}(x_0,y_0)}{\frac{∂G}{∂y}(x_0,y_0)}$, so that the vector which realizes the slope is: $v=(1,-\frac{\frac{∂G}{∂x}(x_0,y_0)}{\frac{∂G}{∂y}(x_0,y_0)})$.

What is not clear to me is why is $v=(1,-\frac{\frac{∂G}{∂x}(x_0,y_0)}{\frac{∂G}{∂y}(x_0,y_0)})$ the vector that realizes the slope $-\frac{\frac{∂G}{∂x}(x_0,y_0)}{\frac{∂G}{∂y}(x_0,y_0)}$. Can you help me?

Answer 1

If the slope of a line is $m$, then $y$ changes by $m$ whenever $x$ changes by $1$. So a vector that goes $1$ in the $x$ direction and $m$ in the $y$ direction has slope $m.$

Answer 2

The slope of a vector $(a,b)$ is $\frac ba$. Therefore, the slope of the vector $v$ is indeed $-\frac{\frac{\partial G}{\partial x}(x_0,y_0)}{\frac{\partial G}{\partial y}(x_0,y_0)}$.