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?
Best Answer
Geometrically it gives the (signed) area of the parallelogram defined by the two vectors.
If you multiply by the appropriate unit normal to the plane you get the normal three dimensional cross product. You don't get a vector in the plane though.
If you try to define a "cross product" in four dimensions, you might appreciate that the familiar situation in three dimensions is a happy coincidence which trips up people who try to generalise in the wrong way.