3 ms·
Well if you think of the cross product as returning a single vector that is orthogonal to all the others, then your question "what is AxB" for $\mathbb{R}^4$ se
by MiroF 6y ago
Well if you think of the cross product as returning a single vector that is orthogonal to all the others, then your question "what is AxB" for $\mathbb{R}^4$ seems ill-specified.
You would instead need to do CrossProduct(A, B, C) and calculate the determinant of the 4x4 matrix with the last row being the $\mathbb{R}^4$ basis.
You could also create a function by lazy evaluation of CrossProduct(A, B) : $\mathbb{R}^4 \rightarrow \mathbb{R}^4$
- a1369209993 6y ago> "what is AxB" for $\mathbb{R}^4$ seems ill-specified. Thats also true for R^3: does <1,0,0>x<0,1,0> give <0,0,1>, or <0,0,-1>?
- MiroF 6y agothe determinant of the matrix {{1,0,0},{0,1,0},{e_1,e_2,e_3}} is e_3 so <0,0,1>. it's completely specified by the order of A and B - there are two possible orders and two possible vectors.
- a1369209993 6y agoIt's not specified at all because there is no such matrix - the e_i are vectors, not scalars. At best it's a convenient mnemonic abbreviation for something decided on some other basis; that's like saying that the gradient is 'specified' by the 'vector' ∇ = <∂/∂x,∂/∂y,∂/∂z>.