n-dimensional Vector Cross Product
Vector cross product of n-1 vectors in n-dimensional space
crossn(A)
A |
matrix of size |
The rows of the matrix A are taken as(n-1) vectors in
n-dimensional space. The cross product generates a vector in this
space that is orthogonal to all these rows in A and its length is
the volume of the geometric hypercube spanned by the vectors.
a vector of length n
The ‘scalar triple product’ in R^3 can be defined as
spatproduct <- function(a, b, c) dot(a, crossn(b, c))
It represents the volume of the parallelepiped spanned by the three vectors.
A <- matrix(c(1,0,0, 0,1,0), nrow=2, ncol=3, byrow=TRUE) crossn(A) #=> 0 0 1 x <- c(1.0, 0.0, 0.0) y <- c(1.0, 0.5, 0.0) z <- c(0.0, 0.0, 1.0) identical(dot(x, crossn(rbind(y, z))), det(rbind(x, y, z)))
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