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where and 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
where a, b, c are real numbers. Then one direction vector of is . Any multiple of is also a direction vector.
For example, suppose the equation of a line is . Then , , and 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.