The parametric is an alternate way to express a distinct line in R^{3}. In R^{2} there are easier ways of writing it.

Most often, the parametric equation of a line is formed from a corresponding vector equation of a line. If you aren't familiar with the form of the vector equation of a line, you may wish to review it before continuing.

The purpose of the parametric equation is to break the vector equation down into its x, y, and z components, and treat each seperatly with a common parameter. It is also often an intermediate step between a vector equation of a line, and a symmetric equation of a line.

The general form of the parametric equation of a line is:

{x: `a`_{x} + (`t`)(`m`_{1})
{y: `a`_{y} + (`t`)(`m`_{2})
{z: `a`_{z} + (`t`)(`m`_{3})

(Please note that the three { curly braces should actually be one large curly brace that encompasses all three lines.)

To get this, simply break the position vector [a_{x},a_{y},a_{z}] into three lines, and then break the parameter `t` and the direction vector [`m`_{1},`m`_{2},`m`_{3}] into the same three lines, so that there is a seperate definition for each variable.

It is *very* important that when the co-ordinates of positions on the line thus defined are calculated, that the value of `t` is constant. (i.e. you need to sub in the same value of `t` for each expression to actually get a point on the defined line.)

It is common at this point to continue on to express the line as a symmetric equation.

It is also possible to use this method of expressing a line in fewer or more dimensions. For example, if this form were to be used in R^{2}, the z expression would simply be omitted. If you wanted to express a line in R^{3} that was parallel to a coordinate plane the relevant expression would just be a constant term. (For example, if the line were parallel to the x-z plane, the expression for y would be expressed as a constant.)