# Calculating a 2D Vector’s Cross Product

Implementation 1 returns the magnitude of the vector that would result from a regular 3D cross product of the input vectors, taking their Z values implicitly as 0 (i.e. treating the 2D space as a plane in the 3D space). The 3D cross product will be perpendicular to that plane, and thus have 0 X & Y components (thus the scalar returned is the Z value of the 3D cross product vector).

Note that the magnitude of the vector resulting from 3D cross product is also equal to the area of the parallelogram between the two vectors, which gives Implementation 1 another purpose. In addition, this area is signed and can be used to determine whether rotating from V1 to V2 moves in an counter clockwise or clockwise direction. It should also be noted that implementation 1 is the determinant of the 2×2 matrix built from these two vectors.

Implementation 2 returns a vector perpendicular to the input vector still in the same 2D plane. Not a cross product in the classical sense but consistent in the “give me a perpendicular vector” sense.

Note that 3D euclidean space is closed under the cross product operation–that is, a cross product of two 3D vectors returns another 3D vector. Both of the above 2D implementations are inconsistent with that in one way or another.

Hope this helps…