Parametric equation of a line

The parametric is an alternate way to express a distinct line in R³. In R² 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: ax + (t)(m1)
{y: ay + (t)(m2)
{z: az + (t)(m3)

(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 [m₁,m₂,m₃] 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 R², the z expression would simply be omitted. If you wanted to express a line in R³ 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.)