The University of Sydney - School of Mathematics and Statistics

The parametric vector equation	Page 1 of 3

Lines can be specified in a variety of ways. One way is described as follows.

Select a point P₀ on the line l, and a non-zero vector v parallel to the line.

The line l is then the unique line passing through P₀ and parallel to v.

Now P lies on l if and only if is parallel to v. As

this condition is equivalent to r - r₀ = tv for some scalar t. The equation

is called a parametric vector equation of the line l. (It is not unique, as a different point P₀ on the line could have been chosen, changing r₀, and v can be replaced by any other non-zero vector parallel to l.)

Each value of the parameter t determines a unique point P, with position vector r = r₀ + tv, on the line l. As t takes all possible values, P takes all possible positions on the line l.