Direction vector

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

$\overrightarrow{AB}$

where A and 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).

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

ax + by + c = 0

where a, b, c are real numbers. Then one direction vector of (D) is ( − b,a). Any multiple of ( − b,a) is also a direction vector.

For example, suppose the equation of a line is 3x − 2y + 15 = 0. Then (2,3), (4,6), and ( − 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.

See also

• Unit vector

External links

• Weisstein, Eric W. "Direction." From MathWorld--A Wolfram Web Resource. [1]
• Glossary, Nipissing University
• Finding the vector equation of a line
• Lines in a plane - Orthogonality, Distances, MATH-tutorial
• Coordinate Systems, Points, Lines and Planes