## The Vector Equation of a Line

The most commonly recognised equation of a line is the Cartesian equation y=mx + c is not the most general or the most useful. This equation is only for a line in two dimensions. We can derive Cartesian equations for lines in three or more dimensions, but the most useful form of the equation of a line is the vector form:

whereis the position vector of a point on the line

is the position vector of a fixed point on the line

is the direction vector of the line

Given the Cartesian form we can derive the vector form by putton one coordinate equal to a parameter and expressing all coorrdinates in terms of this parameter.

Example: Expressin vector form.

Putthenand

In 3D the method is the same. Express

Putthenand