Vector Equations of Lines and Planes — Mathematics Knowledge System

Overview

A vector equation describes a line or a plane not by an algebraic relation between coordinates, but by 'walking' from a known point along one or more direction vectors. A line is traced out by a point plus a single direction vector, scaled by a free parameter t; a plane is traced out by a point plus two independent direction vectors, scaled by two free parameters s and t. This vector viewpoint is what makes lines and planes easy to describe uniformly in any dimension — the same r(t) = r₀ + tv formula works in the plane, in 3-space, or beyond — and it is the natural language for computer graphics, robotics path-planning, and physics trajectories.

Intuition

Think of r(t) = r₀ + tv as a set of driving directions: start at the point r₀, then drive in the direction v, and t tells you how far along that direction you've traveled (negative t means you drive backward). As t ranges over all real numbers, the tip of r(t) sweeps out the entire line. A plane needs two independent directions to sweep out a full two-dimensional sheet — one parameter alone only gives a line lying in the plane, but two independent directions u and v, combined with two free parameters s and t, let you reach every point in the plane by mixing the two directions in different proportions.

Formal Definition

Definition

Let r₀ be the position vector of a known point, v (and w, for a plane) be direction vectors, and t, s be scalar parameters ranging over all real numbers:

\mathbf{r}(t) = \mathbf{r_0} + t\mathbf{v}, \qquad t \in \mathbb{R}

Vector equation of a line

\mathbf{r}(s,t) = \mathbf{r_0} + s\mathbf{u} + t\mathbf{v}, \qquad s,t \in \mathbb{R}

u, v are two non-parallel direction vectors lying in the plane

Vector equation of a plane

\langle x,y\rangle = \langle x_0,y_0\rangle + t\langle a,b\rangle

Component form in the plane

\langle x,y,z\rangle = \langle x_0,y_0,z_0\rangle + t\langle a,b,c\rangle

Component form in space

Derivation

The vector equation of a line through two known points P₀ and P₁ falls right out of vector subtraction: the direction from P₀ to P₁ is a valid direction vector for the line.

\mathbf{v} = \mathbf{r_1} - \mathbf{r_0}

Direction vector between two known points on the line

\mathbf{r}(t) = \mathbf{r_0} + t(\mathbf{r_1}-\mathbf{r_0})

Line through r₀ and r₁; t=0 gives r₀ and t=1 gives r₁

\mathbf{r}(t) = (1-t)\mathbf{r_0} + t\mathbf{r_1}

Equivalent affine-combination form, useful for t \in [0,1] to get the segment between the two points

Properties

Parameter meaning

t=0 \text{ gives the base point } \mathbf{r_0}; \text{ every other real } t \text{ gives a different point on the line}

Segment restriction

\text{Restricting } t \in [0,1] \text{ in } (1-t)\mathbf{r_0}+t\mathbf{r_1} \text{ traces exactly the segment from } \mathbf{r_0} \text{ to } \mathbf{r_1}

Non-uniqueness of direction

\text{Any nonzero scalar multiple of } \mathbf{v} \text{ describes the same line, just re-parametrized}

Plane needs two directions

\text{A single direction vector can only sweep out a line lying in the plane; two independent directions are required to reach every point of the plane}

Applications

Ray tracing represents every camera ray as a vector equation r(t) = r₀ + tv, and rendering reduces to finding the smallest t>0 where the ray meets a surface.

Worked Examples

Compute the direction vector v = r₁ − r₀.
$\mathbf{v} = \langle 4-1,\ 6-2\rangle = \langle 3, 4\rangle$
Write the vector equation using r₀ = (1,2).
$\mathbf{r}(t) = \langle 1,2\rangle + t\langle 3,4\rangle$
Substitute t = 2.
$\mathbf{r}(2) = \langle 1+3(2),\ 2+4(2)\rangle = \langle 7, 10\rangle$

Answer: r(t) = ⟨1,2⟩ + t⟨3,4⟩; at t = 2 the point is (7, 10).

Practice Problems

Difficulty 4/10

Write the vector equation of the line through P₀ = (2, -1) and P₁ = (5, 3), and find the point at t = 1/3.

Difficulty 5/10

A line has vector equation r(t) = ⟨-1, 4, 2⟩ + t⟨2, -1, 3⟩. Find the point where t = -2, and verify it satisfies the equation.

Difficulty 6/10

A drone starts at position (0, 0, 10) meters and flies with constant velocity vector ⟨2, 3, -1⟩ meters/second. Using r(t) = r₀ + tv, find its position after 4 seconds, and how many seconds until it reaches altitude z = 2.

Common Mistakes

Common Mistake

Believing the direction vector v in r(t) = r₀ + tv is uniquely determined by the line.

Any nonzero scalar multiple of v gives an equally valid vector equation for the same line — only the line's direction (up to scaling) matters, not the specific vector chosen.

Common Mistake

Assuming one parameter is enough to describe a plane, by analogy with a line.

A single parameter t only reaches a one-dimensional line within the plane. A full plane, being two-dimensional, requires two independent direction vectors and two independent parameters s and t.

Quiz

Summary

A line's vector equation is r(t) = r₀ + tv: a base point plus a scaled direction vector, t ranging over all reals.
A plane's vector equation is r(s,t) = r₀ + su + tv, using two independent direction vectors and two parameters.
The line through two known points r₀, r₁ is r(t) = (1-t)r₀ + t r₁; restricting t to [0,1] gives exactly the connecting segment.
This vector formulation generalizes uniformly across dimensions and underlies ray tracing, robotics path planning, and constant-velocity motion.

References

WebsiteWikipedia — Line (geometry)
WebsiteWikipedia — Plane (geometry)