Can't find the perpendicular vector of a hyperplane

Question

I have been reading this discussion here, and I tried to follow along the steps of the top response.

So, I chose $4x_1+x_2-3=0$ as my hyperplane. This means that the weight vector $w=(4,1)$ is perpendicular to the hyperplane, right? But, when I plot $x_2=-4x_1+3$, I don't see how $w$ is perpendicular at all.

Any thoughts?

Answer 1

The vector $w$ is perpendicular to vectors parallel to the hyperplane, not points in the hyperplane. For example $(0,3)$ and $(1,-1)$ are both points in the hyperplane. $u=(0,3)-(1,-1)=(-1,4)$ is a vector parallel to the hyperplane and indeed $\langle u,w\rangle=0$.