In a 2d plane, (-y, x) is also perpendicular to (x, y). Are we going to get the same results in terms of finding the area bounded by the triangle, as long as we are consistent with the "convention" we adopt?

I believe that if you were to flip how you calculated the norm vector, then you'd need to also flip your criteria for determining whether to light up a point, i.e. instead of L(x,y) < 0, you'd look for L(x,y) > 0. Otherwise you'd be looking at the area outside the triangle rather than inside. I suppose it also depends in what order you're calc'ing your points/lines in.

Not sure if that's exactly what you were asking but hope that helps!

@mm835 Yep just realized I forgot to mention the flipping inequality part, but I totally agree with you. Thanks for the comment!

So why is the normal vector chosen to be in this particular direction? Convention?

I think it is probably something like the right-hand rule vs the left-hand rule.