Parametric Representations of Lines and Planes¶

Definition¶

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

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

A parametric representation of a plane has the form

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.

In $\boldsymbol{\ell}(t)=\boldsymbol{A}+t\boldsymbol{v}$$t\mathbf{v}$, $t\in\mathbb{R}$$\mathbb{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.

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¶

• vector
• linear combination
• Scalar Multiplication

Related Concepts¶

• Plane
• Affine Map

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: 27 October 2026 11:52