Cross product is not define in two space why?

[jas_fluidm]

why Cross product is not define in two space? 

Answer:
When performing the cross product of two vectors (vector A and vector B), one of the properties of the resultant vector C is that it is perpendicular to both vectors A & B. In two dimensional space, this is not possible, because the resultant vector will be perpendicular to the plane that A & B reside in. Using the (i,j,k) unit vector notation, you could add a 0*k to each vector when doing the cross product, and the resultant vector will have zeros for the i & j components, and only have k components.