Perpendicular Distance Between a Hyperplane and a Point

Hyperplane

A hyperplane in two dimensions: a line
e.g. $\beta_0 + \beta_1 X_1 + \beta_2 X_2 = 0$

A hyperplane in two dimensions: a plane
e.g. $\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 = 0$

Perpendicular Distance Between a Hyperplnae and a Point in Two Dimensions

Let a point be $(x_1, x_2)$, and the hyperplane be $\beta_0 + \beta_1 X_1 + \beta_2 X_2 = 0$ (line). Then the shortest distance from the point to the line is the perpendicular distance $d$.

Then, the perpendicular distance is defined as:

$$ d = \frac{| \beta_0 + \beta_1 x_1 + \beta_2 x_2 |}{\sqrt{ \beta_1^2 + \beta_2^2 }} $$

Perpendicular Distance Between a Hyperplnae and a Point in $p$ Dimensions

The above equation can easily be expanded for the $p$ dimensions. A point is defined as $(x_1, x_2, \cdots, x_p)$ and a hyperplane is defined as $\beta_0 + \beta_1 X_1 + \cdots + \beta_p X_p$. Then, the perpendicular distance $d$ is:

$$ d = \frac{| \beta_0 + \beta_1 x_1 + \cdots + \beta_p x_p |}{\sqrt{ \beta_1^2 + \cdots + \beta_p^2 }} $$