To answer the question, I guess most of us can only imagine up to what 3 dimension stuff is like.
phazan
I think only in 3D will you "take two vectors and get a vector." So in 4D for instance, an equivalent operation would have to either take three vectors and return a vector or take two vectors and return a bivector (?)
keenan
Right: it's simply the fact that in more than three dimensions, there's no longer a unique direction orthogonal to the two given vectors $\mathbf{u},\mathbf{v}$. (Likewise, in 3D there are many directions orthogonal to a single vector $\mathbf{u}$.)
As @phazan hints at, however, you can in any dimension still consider the orthogonal complement, i.e., the space of all vectors orthogonal to a given set. There is a nice language for manipulating collections of vectors and (roughly speaking) their orthogonal complements called exterior algebra, which is sometimes taught in advanced vector calculus classes.
To answer the question, I guess most of us can only imagine up to what 3 dimension stuff is like.
I think only in 3D will you "take two vectors and get a vector." So in 4D for instance, an equivalent operation would have to either take three vectors and return a vector or take two vectors and return a bivector (?)
Right: it's simply the fact that in more than three dimensions, there's no longer a unique direction orthogonal to the two given vectors $\mathbf{u},\mathbf{v}$. (Likewise, in 3D there are many directions orthogonal to a single vector $\mathbf{u}$.)
As @phazan hints at, however, you can in any dimension still consider the orthogonal complement, i.e., the space of all vectors orthogonal to a given set. There is a nice language for manipulating collections of vectors and (roughly speaking) their orthogonal complements called exterior algebra, which is sometimes taught in advanced vector calculus classes.