Distance from a point to a line

Builds on Euclidean distance Line Normal vector

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 $\|(P - A) \times v\| / \|v\|$ .

Reading path

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Euclidean distance concept
Line concept
Normal vector concept
Distance from a point to a line statement

Where this leads

4 pages build on this one.

Tag	`01AAYNSBN6`
URI	`m2://01AAYNSBN6`
Kind	statement, proposition
Writing	W0 stub
MSC	51N20
Status	This copy of the site has no build data. Run `m2 status 01AAYNSBN6`.
Uses	Euclidean distance, Line, Normal vector
Same as	Wikidata Q2485106 Wikipedia Distance from a point to a line

Distance from a point to a line

Statement

Remarks

Reading path

Where this leads

Further reading

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