The perpendicular distance of a point to a line.

Question

In the book Pattern Recognition and ML by Bishop, I am struggling to understand the formula $(4.6)$. Can somebody explain how it was derived? Here $y =w^{T}x+w_0$ where $x$ is the input vector, $w$ is called the weight vector and $w_0$ is the bias.

Answer 1

Note that

$w^Tx = \vec w \cdot \vec x = (||w||)(||x||)\cos \theta$

where $\theta$ is the angle between $\vec w$ and $\vec x$.

The surface $y=0$ is perpendicular to $\vec w$ and is a distance $\frac {-w_0}{||w||}$ from the origin. So to find the perpendicular distance from $x$ to the surface $y=0$ we must subtract $\frac {-w_0}{||w||}$ from $||x|| \cos \theta$, which gives us:

$\displaystyle ||x|| \cos \theta - \frac {-w_0}{||w||} = \frac {w^Tx}{||w||} + \frac {w_0}{||w||} = \frac {w^Tx + w_0}{||w||} = \frac {y(x)}{||w||}$