Equation of Straight Line in Plane/Homogeneous Cartesian Coordinates

Theorem
A straight line $\LL$ is the set of all points $P$ in $\R^2$, where $P$ is described in homogeneous Cartesian coordinates as:
 * $l X + m Y + n Z = 0$

where $l, m, n \in \R$ are given, and not both $l$ and $m$ are zero.

Proof
Let $P = \tuple {X, Y, Z}$ be a point on $L$ defined in homogeneous Cartesian coordinates.

Then by definition:

where $P = \tuple {x, y}$ described in conventional Cartesian coordinates.

Hence:

Hence the result.