Assuming we have vector A with elements (A₁, A₂, A₃) and vector B with elements (B₁, B₂, B₃), we can calculate the cross product of these two vectors as:

Cross Product = [(A₂*B₃) – (A₃*B₂), (A₃*B₁) – (A₁*B₃), (A₁*B₂) – (A₂*B₁)]

For example, suppose we have the following vectors:

• Vector A: (1, 2, 3)
• Vector B: (4, 5, 6)

We could calculate the cross product of these vectors as:

• Cross Product = [(A₂*B₃) – (A₃*B₂), (A₃*B₁) – (A₁*B₃), (A₁*B₂) – (A₂*B₁)]
• Cross Product = [(2*6) – (3*5), (3*4) – (1*6), (1*5) – (2*4)]
• Cross Product = (-3, 6, -3)

You can use one of the following two methods to calculate the cross product of two vectors in R:

Method 1: Use cross() function from pracma package

library(pracma)
  
#calculate cross product of vectors A and B
cross(A, B)

Method 2: Define your own function

#define function to calculate cross product 
cross <- function(x, y, i=1:3) {
  create3D <- function(x) head(c(x, rep(0, 3)), 3)
  x <- create3D(x)
  y <- create3D(y)
  j <- function(i) (i-1) %% 3+1
  return (x[j(i+1)]*y[j(i+2)] - x[j(i+2)]*y[j(i+1)])
}

#calculate cross product
cross(A, B)

The following examples show how to use each method in practice.

Example 1: Use cross() function from pracma package

The following code shows how to use the cross() function from the package to calculate the cross product between two vectors:

library(pracma)
  
#define vectors
A <- c(1, 2, 3)
B <- c(4, 5, 6)
  
#calculate cross product
cross(A, B)

[1] -3  6 -3

The cross product turns out to be (-3, 6, -3).

This matches the cross product that we calculated earlier by hand.

Example 2: Define your own function

The following code shows how to define your own function to calculate the cross product between two vectors:

#define function to calculate cross product 
cross <- function(x, y, i=1:3) {
  create3D <- function(x) head(c(x, rep(0, 3)), 3)
  x <- create3D(x)
  y <- create3D(y)
  j <- function(i) (i-1) %% 3+1
  return (x[j(i+1)]*y[j(i+2)] - x[j(i+2)]*y[j(i+1)])
}

#define vectors
A <- c(1, 2, 3)
B <- c(4, 5, 6)

#calculate cross product
cross(A, B)

[1] -3 6 -3

The cross product turns out to be (-3, 6, -3).

This matches the cross product that we calculated in the previous example.

Additional Resources

The following tutorials explain how to perform other common tasks in R: