hat is the normal vector to the 2D-line $ \ mx+ny=l \ $?

Question

The normal vector to the plane $ \ ax+by+cz=d \ $ is $ \ (a,b,c) \ $.

What is the normal vector to the 2D-line $ \ mx+ny=l \ $ ?

Answer:

The line is a 2-dimensional line.

I think the normal vector to the line is $ \ (m,n) \ $

Am I right ?

Answer 1

Yes. Select $x_0$ and $y_0$ such that $\ell =m x_0 + n y_0$. Then the equation of the line can be written equivalently as $m(x-x_0) + n(y-y_0)=0$, which makes it clear that $(m,n)$ is a normal vector to the line.

Alternatively, note that this line is a level curve for the function $f(x,y) = mx+ny$. The gradient $\nabla f(x,y) = (m,n)$ is normal to the level curve through the point $(x,y)$.