The Cartesian Equation of a Line in Three Dimensions

The vector equation of a line takes the formwhereis the position vector of a point on the line relative to the origin andis the tangent vector or the direction vector of the line. We can change this into Cartesian form, containing x's, y's and z's by writeing writing the x, y and z components of the line in terms of the paramenter (in the above case), making the subject of each component equation and equation them all, since they all equal

Example: Find the Cartesian equation of the 2D line

We haveandso thatand

Equating these gives(which can be rearranged to give).

In 3D the method is exactly the same.

Example: Find the Cartesian equation of the line

We haveandand z=-2-tso thatand t=-z-2.

Equating these gives