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.


Here is a short code snippet which works whenever the cross product makes sense: the 3D version returns a vector and the 2D version returns a scalar. If you just want simple code that gives the right answer without pulling in an external library, this is all you need.

# Compute the vector cross product between x and y, and return the components
# indexed by i.
CrossProduct3D <- function(x, y, i=1:3) {
  # Project inputs into 3D, since the cross product only makes sense in 3D.
  To3D <- function(x) head(c(x, rep(0, 3)), 3)
  x <- To3D(x)
  y <- To3D(y)

  # Indices should be treated cyclically (i.e., index 4 is "really" index 1, and
  # so on).  Index3D() lets us do that using R's convention of 1-based (rather
  # than 0-based) arrays.
  Index3D <- function(i) (i - 1) %% 3 + 1

  # The i'th component of the cross product is:
  # (x[i + 1] * y[i + 2]) - (x[i + 2] * y[i + 1])
  # as long as we treat the indices cyclically.
  return (x[Index3D(i + 1)] * y[Index3D(i + 2)] -
          x[Index3D(i + 2)] * y[Index3D(i + 1)])
}

CrossProduct2D <- function(x, y) CrossProduct3D(x, y, i=3)

Does it work?

Let's check a random example I found online:

> CrossProduct3D(c(3, -3, 1), c(4, 9, 2)) == c(-15, -2, 39)
[1] TRUE TRUE TRUE

Looks pretty good!

Why is this better than previous answers?

  • It's 3D (Carl's was 2D-only).
  • It's simple and idiomatic.
  • Nicely commented and formatted; hence, easy to understand

The downside is that the number '3' is hardcoded several times. Actually, this isn't such a bad thing, since it highlights the fact that the vector cross product is purely a 3D construct. Personally, I'd recommend ditching cross products entirely and learning Geometric Algebra instead. :)


The help ?crossprod explains it quite clearly. Take linear regression for example, for a model y = XB + e you want to find X'X, the product of X transpose and X. To get that, a simple call will suffice: crossprod(X) is the same as crossprod(X,X) is the same as t(X) %*% X. Also, crossprod can be used to find the dot product of two vectors.

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