Distance from a point to a plane

Builds on Euclidean distance Equation of a plane Normal vector

Statement

Let $a$ , $b$ , $c$ and $d$ be real numbers with $(a, b, c) \neq (0, 0, 0)$ , and let $P = \{(x, y, z) \in \mathbb{R}^3 : a x + b y + c z = d\}$ be a plane. Let $X = (x_0, y_0, z_0)$ be a point of $\mathbb{R}^3$ . The distance from $X$ to $P$ is the smallest Euclidean distance $d(X, Q)$ for $Q \in P$ . Then $\operatorname{dist}(X, P) = \frac{|a x_0 + b y_0 + c z_0 - d|}{\sqrt{a^2 + b^2 + c^2}}.$

Remarks

The smallest distance is reached at exactly one point of $P$ , the orthogonal projection of $X$ on $P$ . With $n = (a, b, c)$ this point is $X - \frac{a x_0 + b y_0 + c z_0 - d}{a^2 + b^2 + c^2} \, n$ . The number $(a x_0 + b y_0 + c z_0 - d) / \|n\|$ is the signed distance. Its sign tells on which side of the plane the point is. The same formula holds for a hyperplane $\{x : a \cdot x = d\}$ with $a \neq 0$ in $\mathbb{R}^n$ : the distance from $x_0$ is $|a \cdot x_0 - d| / \|a\|$ .

Reading path

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Equation of a plane concept
Euclidean distance concept
Normal vector concept
Distance from a point to a plane statement

Tag	`0K1E2S6Z69`
URI	`m2://0K1E2S6Z69`
Kind	statement, proposition
Writing	W1 stated
MSC	51N20
Status	This copy of the site has no build data. Run `m2 status 0K1E2S6Z69`.
Uses	Euclidean distance, Equation of a plane, Normal vector
Same as	Wikidata Q3030668 Wikipedia Distance from a point to a plane

Distance from a point to a plane

Statement

Remarks

Reading path

Further reading

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