R - Compute Cross Product of Vectors (Physics)

What is R's crossproduct function?

According to the help function in R: crossprod (X,Y) = t(X)%*% Y is a faster implementation than the expression itself. It is a function of two matrices, and if you have two vectors corresponds to the dot product. @Hong-Ooi's comments explains why it is called crossproduct.

R - Compute Cross Product of Vectors (Physics)

Here is a generalized cross product:

xprod <- function(...) {
  args <- list(...)

  # Check for valid arguments

  if (length(args) == 0) {
    stop("No data supplied")
  }
  len <- unique(sapply(args, FUN=length))
  if (length(len) > 1) {
    stop("All vectors must be the same length")
  }
  if (len != length(args) + 1) {
    stop("Must supply N-1 vectors of length N")
  }

  # Compute generalized cross product by taking the determinant of sub-matricies

  m <- do.call(rbind, args)
  sapply(seq(len),
         FUN=function(i) {
           det(m[,-i,drop=FALSE]) * (-1)^(i+1)
         })
}

For your example:

> xprod(1:3, 4:6)
[1] -3  6 -3

This works for any dimension:

> xprod(c(0,1)) # 2d
[1] 1 0
> xprod(c(1,0,0), c(0,1,0)) # 3d
[1] 0 0 1
> xprod(c(1,0,0,0), c(0,1,0,0), c(0,0,1,0)) # 4d
[1]  0  0  0 -1

See https://en.wikipedia.org/wiki/Cross_product

R cross product matrix

Your desired matrix is the correlation matrix of myTab without centering the variables. Hence, we may obtain it as follows:

crossprod(myTab %*% diag(1 / sqrt(colSums(myTab^2))))
#           [,1]      [,2]      [,3]      [,4] [,5]      [,6]
# [1,] 1.0000000 0.4082483 0.0000000 0.3333333  0.0 0.5773503
# [2,] 0.4082483 1.0000000 0.0000000 0.4082483  0.0 0.3535534
# [3,] 0.0000000 0.0000000 1.0000000 0.5773503  0.0 0.0000000
# [4,] 0.3333333 0.4082483 0.5773503 1.0000000  0.0 0.5773503
# [5,] 0.0000000 0.0000000 0.0000000 0.0000000  1.0 0.5000000
# [6,] 0.5773503 0.3535534 0.0000000 0.5773503  0.5 1.0000000

Perform 'cross product' of two vectors, but with addition

expand.grid can answer your second question:

expand.grid(1:6,1:6)
expand.grid(1:6,1:6,1:4)

Add labels to calculated cross product matrix in R

We can use dimnames to assign the rownames and columnames. In this case, there is only colnames of 'myTab' is needed as dimnames

dimnames(CProd.Matrix) <-rep(list(colnames(myTab)), 2)
CProd.Matrix
#          111       112       113       114 115       116
#111 1.0000000 0.4082483 0.0000000 0.3333333 0.0 0.5773503
#112 0.4082483 1.0000000 0.0000000 0.4082483 0.0 0.3535534
#113 0.0000000 0.0000000 1.0000000 0.5773503 0.0 0.0000000
#114 0.3333333 0.4082483 0.5773503 1.0000000 0.0 0.5773503
#115 0.0000000 0.0000000 0.0000000 0.0000000 1.0 0.5000000
#116 0.5773503 0.3535534 0.0000000 0.5773503 0.5 1.0000000

Now, based on the rownames, it can be subsetted

CProd.Matrix ["111",, drop = FALSE]
#    111       112 113       114 115       116
#111   1 0.4082483   0 0.3333333   0 0.5773503

sort the values after subsetting ?

t(apply(CProd.Matrix ["111",, drop = FALSE], 1, sort, decreasing = TRUE))
#    111       116       112       114 113 115
#111   1 0.5773503 0.4082483 0.3333333   0   0

How to find the orientation of three points in a two dimensional space given coordinates of those points?

This formula is used to calculate cross product of vectors q-p and q-r. You can see in Geometric Meaning section that cross product value
C = A x B = |A|*|B|*Sin(Theta), where Theta is angle between these vectors (point-to-point directions). Sin(Theta) = 0 for parallel vectors, positive when Theta < 180, negative otherwise.

Example:

clockwise triplet ABC: cross product of AB and AC vectors is >0

anticlockwise triplet ACD: cross product of AC and AD is negative.

how to calculate the Euclidean norm of a vector in R?

This is a trivial function to write yourself:

norm_vec <- function(x) sqrt(sum(x^2))

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