Getting into the details

Let us define the function $$f(x) = \boldsymbol{w}^T\boldsymbol{x}+b = 0,$$ as the function that determines the line $L$ that separates two classes (our two features), see the figure here.

Any point defined by $\boldsymbol{x}_i$ and $\boldsymbol{x}_2$ on the line $L$ will satisfy $\boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0$.

The signed distance $\delta$ from any point defined by a vector $\boldsymbol{x}$ and a point $\boldsymbol{x}_0$ on the line $L$ is then $$\delta = \frac{1}{\vert\vert \boldsymbol{w}\vert\vert}(\boldsymbol{w}^T\boldsymbol{x}+b).$$

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