Distance Between a Point and a Line¶

Definition¶

The distance between a point $\boldsymbol{A}$ and a line $\boldsymbol{\ell}(t)$ is the shortest distance between $\boldsymbol{A}$ and all points on the line.

Interpretation¶

The distance from a point to a line is the length of the shortest route from the point to the line. That shortest route is the perpendicular segment: the closest point on the line is the foot $\boldsymbol{F}$ of the perpendicular from $\boldsymbol{A}$.

The line passes through $\boldsymbol{B}=\boldsymbol{\ell}(0)$ in the nonzero direction $\boldsymbol{v}$. The vector $\boldsymbol{A}-\boldsymbol{B}$ splits into a part along $\boldsymbol{v}$ (dashed, from $\boldsymbol{B}$ to $\boldsymbol{F}$) and a part perpendicular to $\boldsymbol{v}$ (green, from $\boldsymbol{F}$ to $\boldsymbol{A}$), and the distance is the length $d=\|\boldsymbol{A}-\boldsymbol{F}\|$ of the second part, where

The foot is the closest point because of Pythagoras. For any other point $\boldsymbol{P}$ on the line, $\boldsymbol{A}-\boldsymbol{F}$ is orthogonal to $\boldsymbol{F}-\boldsymbol{P}$$\mathbf{P}$, so

which is larger than $\|\boldsymbol{A}-\boldsymbol{F}\|^2$ when $\boldsymbol{P}\neq\boldsymbol{F}$$\mathbf{F}$. The same answer comes from minimizing the quadratic function $t\mapsto\|\boldsymbol{\ell}(t)-\boldsymbol{A}\|^2$ with a derivative. Both arguments work in every $\mathbb{R}^n$. In $\mathbb{R}^3$ there is also the formula $d=\dfrac{\|(\boldsymbol{A}-\boldsymbol{B})\times \boldsymbol{v}\|}{\|\boldsymbol{v}\|}$$\frac{\Vert (\mathbf{A}-\mathbf{B})\times \mathbf{v}\Vert }{\Vert \mathbf{v}\Vert }$, where $\boldsymbol{a}\times \boldsymbol{b}=(a_2b_3-a_3b_2,\,a_3b_1-a_1b_3,\,a_1b_2-a_2b_1)$ is the cross product.

For a plane in $\mathbb{R}^3$ through $\boldsymbol{B}$ the roles swap: the distance from $\boldsymbol{A}$ to the plane is the length of the part of $\boldsymbol{A}-\boldsymbol{B}$ along a normal vector, not the part perpendicular to a direction.

A common slip is to take $\|\boldsymbol{A}-\boldsymbol{B}\|$$\mathbf{B}\Vert$, the distance to the given point $\boldsymbol{B}$ of the line. It is only an upper bound for $d$, reached when $\boldsymbol{B}$ happens to be the foot.

Key Results¶

• The distance is zero if and only if the point lies on the line.
• In coordinates, the distance can be found by minimizing $\|\boldsymbol{\ell}(t)-\boldsymbol{A}\|$$\mathbf{A}\Vert$.

Examples¶

• For $\boldsymbol{\ell}(t)=(1+t,2t,-1,2-t)$$t)$, a point on the line gives minimum distance $0$.
• A point not on the line has positive distance.

Prerequisites¶

• Euclidean Distance
• Euclidean Norm
• Parametric Representation of a Line

Related Concepts¶

• Normal Vector
• Inner Product
• Point-Normal Equation of a Plane

Sources¶

• Math II: Linear Algebra · Week 5

Aliases: point-line distance, distance from a point to a line Symbols: real numbers

Last updated: 20 October 2026 03:02