Parametric Representations of Lines and Planes¶

Definition¶

A parametric representation of a line in $\mathbb{R}^n$ has the form

\boldsymbol{\ell}(t)=\boldsymbol{A}+t\boldsymbol{v},

where $\boldsymbol{A}$ is a point on the line, $\boldsymbol{v}\neq\boldsymbol{0}$ is a direction vector, and $t\in\mathbb{R}$ $R$ .

A parametric representation of a plane has the form

\boldsymbol{P}(s,t)=\boldsymbol{A}+s\boldsymbol{v}+t\boldsymbol{w},

where $\boldsymbol{A}$ is a point on the plane and $\boldsymbol{v},\boldsymbol{w}$ are linearly independent direction vectors.

Interpretation¶

Scalar multiplication turns the parameters into displacements $t\boldsymbol{v}$ , $s\boldsymbol{v}$ and $t\boldsymbol{w}$ . Their linear combinations, added to the starting point $\boldsymbol{A}$ , trace out the line or plane.

The parameters describe movement from a starting point along one direction for a line, or along two independent directions for a plane.

A line through a point A with direction vector v, with equally spaced points l(-1), l(0), l(1) and l(2) for equally spaced parameter values.

In $\boldsymbol{\ell}(t)=\boldsymbol{A}+t\boldsymbol{v}$ $t v$ , $t\in\mathbb{R}$ $R$ , the point $\boldsymbol{A}=\boldsymbol{\ell}(0)$ lies on the line and $\boldsymbol{v}\ne\boldsymbol{0}$ is its direction. Equally spaced parameter values give equally spaced points. Choosing another point on the line or a nonzero multiple of $\boldsymbol{v}$ gives another representation of the same line.

Move $t$ : $\boldsymbol{\ell}(t)=\boldsymbol{A}+t\boldsymbol{v}$ walks along the line, through $\boldsymbol{A}$ at $t=0$ and $\boldsymbol{A}+\boldsymbol{v}$ at $t=1$ $1$ . Drag $\boldsymbol{A}$ and the tip of $\boldsymbol{v}$ to move the line.

Key Results¶

A point lies on the line or plane exactly when it is obtained for some parameter value or parameter pair.
Two distinct points determine a line, while three non-collinear points determine a plane.

Examples¶

$\boldsymbol{\ell}(t)=(4,2)+t(1,1)$ represents a line through $(4,2)$ .
$\boldsymbol{P}(s,t)=(1,0,0)+s(1,1,0)+t(0,1,1)$ represents a plane in $\mathbb{R}^3$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 5

Aliases: vector equation of a line, vector equation of a plane Keywords: parametric equation of a line, parametric line, parametric representation of a plane, parametric equation of a plane, parametric plane Symbols: element of, real numbers

Last updated: 20 October 2026 03:02