Whenever I run into an expressions of the form $x_1y_2 - x_2y_1$, I usually have no problem finding some pertinent way to interpret it as the area of the parallelogram subtended by some two vectors.
But here's a case in which I can't figure out how to apply such an interpretation.
The line passing through points $[x_1, y_1]$ and $[x_2, y_2]$ consists of all those points $[x, y]$ that satisfy the equation
$$(y_1 - y_2)\,x + (x_2 - x_1)\,y + (x_1y_2 - x_2y_1) = 0$$
Now, in expressions of the form $A\,x + B\,y + C = 0$, I always interpret the $C$ term as a "location parameter", i.e. something "one-dimensional" somehow. And yet, if $C = (x_1y_2 - x_2y_1)$, as the above expression says, it seems to corresponds more to an area.
How can these two seemingly contradictory intuitions be reconciled?