In 3D the cross product can be understood as follows:<p>-You define the cross product as the area of the parallelogram formed by two vectors,<p>-Draw a normal to it,<p>-Use some convention to decide on which side the normal
should stick out such that when you reverse the two vectors being multiplied, the resultant vectors is multiplied by -1.<p>In more (3+n) dimensions there is a problem with this approach. The resulting normal vector can point in any one of (n+1) dimensions. The intuition here is try to draw of normal vector to a line in 3D space. You can do it in 2D space, but in 3D space there is a <i>plane</i> that it normal to the line.<p>So we need to decide what direction should the vector point in inside this 1+n dimensional space. It seems like any convention will do. We could solve the orientation problem in 3D after all. But it seems like any convention you try has the property that when you break vectors <i>a</i> and <i>b</i> into parts and perform the cross product operation on all pairs of parts (one part from each vector <i>a_i x b_j</i>) and then sum it up the result isn't equal to <i>a x b</i>.<p>This article is saying a working convention can be given in 7 dimensions (n=4), and no other dimensions. Which is nuts. If anyone has any insight as to why I'd love to hear it.