# Direction vector

In mathematics, a direction vector that describes a line segment D is any vector

${\displaystyle {\overrightarrow {AB}}}$

where ${\displaystyle A}$ and ${\displaystyle B}$ are two distinct points on the line D. If v is a direction vector for D, so is kv for any nonzero scalar k; and these are in fact all of the direction vectors for the line D. Under some definitions, the direction vector is required to be a unit vector, in which case each line has exactly two direction vectors, which are negatives of each other (equal in magnitude, opposite in direction).

A direction vector indicates a directional quantity, such as Radiance.

## Direction vector for a line in R2

Any line in two-dimensional Euclidean space can be described as the set of solutions to an equation of the form

${\displaystyle ax+by+c=0}$

where a, b, c are real numbers. Then one direction vector of ${\displaystyle (D)}$ is ${\displaystyle (-b,a)}$. Any multiple of ${\displaystyle (-b,a)}$ is also a direction vector.

For example, suppose the equation of a line is ${\displaystyle 3x+2y+15=0}$. Then ${\displaystyle (-2,3)}$, ${\displaystyle (-4,6)}$, and ${\displaystyle (2,-3)}$ are all direction vectors for this line.

## Parametric equation for a line

In Euclidean space (any number of dimensions), given a point a and a nonzero vector v, a line is defined parametrically by (a+tv), where the parameter t varies between -∞ and +∞. This line has v as a direction vector.