Equation of a line in homogenous coordinates given 2 points in affine coordinates

In planar geometry the line passing through two points A and B in homogeneous coordinates is found from their vector cross product:

$$ \begin{pmatrix} C_0 \\ C_1 \\ C_2 \end{pmatrix} = \begin{pmatrix} A_0 \\ A_1 \\ A_2 \end{pmatrix} \times \begin{pmatrix} B_0 \\ B_1 \\ B_2 \end{pmatrix} = \begin{pmatrix} A_1 B_2 - A_2 B_1 \\ A_2 B_0 - A_0 B_2 \\ A_0 B_1 - A_1 B_0 \end{pmatrix} $$

The equation for the line is

$$ C_0 + C_1 x + C_2 y = 0 $$

Example

Two points A and B have coordinates $A=(1,7,2)$ and $B=(1,3,-2)$. The line that joints them is

$$ C = \begin{pmatrix} 1 \\ 7 \\ 2 \end{pmatrix} \times \begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix} = \begin{pmatrix} -20 \\ 4 \\ -4 \end{pmatrix}$$

with the equation $$ (-20)+(4)x+(-4)y=0 \\ x-y-5 = 0 $$