Lines and Planes

Summary

Lines and planes are the basic flat objects of analytic geometry. Direction vectors describe lines; normal vectors describe planes. Cartesian, parametric, and point–direction forms convert geometry into solvable equations.

Prerequisites

Distance and Midpoint, Vectors and Dot Product (dot product for planes and angles). Hub: Analytic Geometry.

Object / Concept

• A line in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ is determined by a point and a nonzero direction vector.
• A plane in ${\mathbb{R}}^3$ is determined by a point and a nonzero normal vector (or by a point and two nonparallel direction vectors).

Coordinate System

Standard Cartesian coordinates in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ .

Notation

Conditions / Assumptions

• Direction vector $\mathbf{d}\ne \mathbf{0}$ .
• Normal vector $\mathbf{n}\ne \mathbf{0}$ .
• Two direction vectors spanning a plane must be linearly independent.

Equations

Line through ${\mathbf{r}}_0$ with direction $\mathbf{d}$

Parametric form (in ${\mathbb{R}}^3$ ):

Vector form:

Symmetric form (when $a,b,c\ne 0$ ):

Line in the plane ${\mathbb{R}}^2$

Slope–intercept (nonvertical): $y=mx+b$ .

Point–slope: $y-y_0=m(x-x_0)$ .

General linear: $Ax+By+C=0$ with $(A,B)\ne (0,0)$ .

Two distinct points $(x_1,y_1)$ , $(x_2,y_2)$ determine slope

when $x_2\ne x_1$ ; if $x_2=x_1$ the line is vertical: $x=x_1$ .

Plane through ${\mathbf{r}}_0$ with normal $\mathbf{n}=⟨A,B,C⟩$

or equivalently

A point $\mathbf{r}$ lies on the plane if and only if $\mathbf{n}\cdot (\mathbf{r}-{\mathbf{r}}_0)=0$ .

Worked Example

Line through $(1,2,3)$ parallel to $⟨2,-1,4⟩$ :

Plane through $(1,0,0)$ with normal $⟨1,1,1⟩$ :

Common Mistakes

• Using a normal vector as if it were a direction vector of a line in the plane (they are orthogonal, not parallel).
• Writing symmetric form when a component of $\mathbf{d}$ is zero.
• Confusing “parallel to a plane” (direction $\perp$ normal) with “perpendicular to a plane” (direction $\parallel$ normal).

Connections

• Related: Vectors and Dot Product, Distance and Midpoint
• Next: Conic Sections; linear systems in Systems of Linear Equations
• Later: subspaces and hyperplanes in Vector Spaces and Bases

References

Lines and planes in coordinates follow OpenStax Calculus Volume 3 (vectors and geometry of space).^[1]