Direction vector
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In mathematics, a direction vector that describes a line D is any vector
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).
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[edit] 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.
[edit] 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.
[edit] See also
[edit] 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
