Can the Cross Product Exist in Two Dimensions?

It is not possible to find the cross product for two-dimensional vectors. The usual cross product in multivariable calculus is only defined for three-dimensional vectors.

Given two three-dimensional vectors, the cross product computes a third vector, which is orthogonal, or perpendicular, to the original two vectors. Two vectors, in general, define a plane in a three-dimensional space, and the cross product of those two vectors points straight up from that plane. When visualizing this, it helps to remember that the unit vector along the z-axis is the cross product of the unit vector along the x-axis and the unit vector along the y-axis. In two dimensions, there cannot be a third vector that is independent of the first two.

Be careful not to try to apply the cross product to two-dimensional vectors. It simply does not work. While it is not possible to define a cross product in two dimensions, it is possible to define a cross product in seven dimensions. It is an interesting and surprising mathematical fact that three and seven dimensions are the only cases where a cross product can defined. Only the three-dimensional cross product is discussed in the typical multivariable calculus class, so that is the one that is more familiar.