2d cross product definition

Question

In determine if line segment is inside polygon I noticed the accepted answer has an unusual 2d cross roduct definition of:

(u1, u2) x (v1, v2) := (u1 - v2)*(u2 -         


        

Answer 1

I suggest you to take a look at Exterior Algebra. It generalizes the notion of cross product and determinant. The "Motivation examples" section describing areas in the plane answers exactly your question.

It works in any dimension. 3D is a specific case where the result of the cross product of two vectors also has 3 components. However, in 2D, there is only one resulting component, and in 4D, there is 6. In 4D, you can apply a kind of cross product using 3 vectors, which give you also 4 components.

It is important to note that while the result of a cross product in 3D has 3 components, the units and the meaning are different. For example, the x-component has units of area, and represents the area in the YZ plane, as opposed to a "standard" vector where the x-component has unit of length and is a difference is coordinates. Using exterior algebra, these differences become clearer since the notation is also different (dx vs dy^dz).

Note: The answer that you referenced has a mistake. Instead of (u1, u2) x (v1, v2) := (u1 - v2)*(u2 - v1), it should be (u1, u2) x (v1, v2) := u1*v2 - u2*v1