Distance from a point to a line

Statement

Let $a$, $b$ and $c$ be real numbers with $(a, b) \neq (0, 0)$, and let $L = \{(x, y) \in \mathbb{R}^2 : a x + b y + c = 0\}$ be a line in the plane. Let $P = (x_0, y_0)$ be a point of $\mathbb{R}^2$. The distance from $P$ to $L$ is the smallest Euclidean distance $d(P, Q)$ for $Q \in L$. Then $$\operatorname{dist}(P, L) = \frac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}.$$

Remarks

The smallest distance is reached at exactly one point $Q$ of $L$, the foot of the perpendicular from $P$. It is $Q = P - \frac{a x_0 + b y_0 + c}{a^2 + b^2} (a, b)$. In $\mathbb{R}^n$, let $L$ be the line through the point $A$ with the direction vector $v$. The distance from $P$ to $L$ is the length of the part of $P - A$ that is perpendicular to $v$: the length of $(P - A) - \frac{(P - A) \cdot v}{v \cdot v} v$. In $\mathbb{R}^3$ this distance is also $\|(P - A) \times v\| / \|v\|$.

Explanations

L1 human

The shortest way from a point to a line is the path that meets the line at a right angle. The formula measures this path. For the line $y = 0$, thus $a = 0$, $b = 1$, $c = 0$, it gives $|y_0|$: the height of the point above or below the horizontal axis.