The Cartesian Equation of a Line in Three Dimensions
The vector equation of a line takes the form
where
is the position vector of a point on the line relative to the origin and
is 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 have
and
so that
and
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 that
and t=-z-2.
Equating these gives
