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 2x2 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...

In short: It's a shorthand notation for a mathematical hack.

Long explanation:

You can't do a cross product with vectors in 2D space. The operation is not defined there.

However, often it is interesting to evaluate the cross product of two vectors assuming that the 2D vectors are extended to 3D by setting their z-coordinate to zero. This is the same as working with 3D vectors on the xy-plane.

If you extend the vectors that way and calculate the cross product of such an extended vector pair you'll notice that only the z-component has a meaningful value: x and y will always be zero.

That's the reason why the z-component of the result is often simply returned as a scalar. This scalar can for example be used to find the winding of three points in 2D space.

From a pure mathematical point of view the cross product in 2D space does not exist, the scalar version is the hack and a 2D cross product that returns a 2D vector makes no sense at all.

Another useful property of the cross product is that its magnitude is related to the sine of the angle between the two vectors:

| a x b | = |a| . |b| . sine(theta)

or

sine(theta) = | a x b | / (|a| . |b|)

So, in implementation 1 above, if a and b are known in advance to be unit vectors then the result of that function is exactly that sine() value.