I am trying to build an intuition to understand SVMs. If we have a binary set of data which is linearly separable then we want to maximize the distance between the 2 planes:
$$\mathbf{wx} - b =1$$ $$\mathbf{wx} - b =-1$$ If we assumed we are in $\mathbb{R}^3$ the equations can be expanded as: $$\mathbf{A}\mathrm{x} + \mathbf{B}\mathrm{y} + \mathbf{C}\mathrm{z} - b =1$$ $$\mathbf{A}\mathrm{x} + \mathbf{B}\mathrm{y} + \mathbf{C}\mathrm{z} - b =-1$$ As far as I understand, the left hand of that equation should always evaluate to 0 because we are computing the scalar product of the normal of the plane with the vector $\mathbf{v = x-x_0}$ which lies on the plane. Does this mean that now the vectors that lie on those planes are of the form $\mathbf{v = x-(x_0-1)}$ and $\mathbf{v = x-(x_0+1)}$ for the first and second equation respectively? What I don't get is the intuition of the fixed point $\mathbf{x_0}$ increasing or decreasing one unit. Does it grow/shrink an unit in all its dimensions? Or what is really varying is the scalar product with the normal? If so, what does it mean to increase the scalar product $n$ units?