This answer on StackOverflow answers a question about intersection of two segments. Right at the beginning, it introduces a “vector cross product”.
Define the 2-dimensional vector cross product $\vec v \times \vec w$ to be $v_x w_y − v_y w_x$.
However, this doesn't seem to be a regular cross product (nor does it even produce a vector). I realize that the formula is a simple determinant of the two vectors, but I cannot understand its meaning or relation to the rest of the post.
Does it have a meaning (motivation) or is it just a “lucky-guess” operation in order to transform the equation $\vec p + t\vec r = \vec q + u \vec s$ into a solvable state? In other words, how does this operation (intuitively) relate to the described algorithm?