Angle between two vectors in R

According to page 5 of this PDF, sum(a*b) is the R command to find the dot product of vectors a and b, and sqrt(sum(a * a)) is the R command to find the norm of vector a, and acos(x) is the R command for the arc-cosine. It follows that the R code to calculate the angle between the two vectors is

theta <- acos( sum(a*b) / ( sqrt(sum(a * a)) * sqrt(sum(b * b)) ) )

You should use the dot product. Say you have V₁ = (x₁, y₁, z₁) and V₂ = (x₂, y₂, z₂), then the dot product, which I'll denote by V₁·V₂, is calculated as

V₁·V₂ = x₁·x₂ + y₁·y₂ + z₁·z₂ = |V₁| · |V₂| · cos(θ);

What this means is that that sum shown on the left is equal to the product of the absolute values of the vectors times the cosine of the angle between the vectors. the absolute value of the vectors V₁ and V₂ are calculated as

|V₁| = √(x₁² + y₁² + z₁²), and
|V₂| = √(x₂² + y₂² + z₂²),

So, if you rearrange the first equation above, you get

cos(θ) = (x₁·x₂ + y₁·y₂ + z₁·z₂) ÷ (|V₁|·|V₂|),

and you just need the arccos function (or inverse cosine) applied to cos(θ) to get the angle.

Depending on your arccos function, the angle may be in degrees or radians.

(For two dimensional vectors, just forget the z-coordinates and do the same calculations.)

Good luck,

John Doner

if you install/upload the library(matlib): there is a function called angle(x, y, degree = TRUE) where x and y are vectors. Note: if you have x and y in matrix form, use as.vector(x) and as.vector(y):

library(matlib)
matA <- matrix(c(3, 1), nrow = 2)  ##column vectors
matB <- matrix(c(5, 5), nrow = 2)
angle(as.vector(matA), as.vector(matB))  
##default in degrees, use degree = FALSE for radians

I think what you need is an inner product. For two vectors v,u (in R^n or any other inner-product spaces) <v,u>/|v||u|= cos(alpha). (were alpha is the angle between the vectors)

for more details see:

http://en.wikipedia.org/wiki/Inner_product_space

If you want to calculate the angle among multiple variables, you can use the following function, which is an extension of the solution provided by @Graeme Walsh.

angles <- function(matrix){

  ## Calculate the cross-product of the matrix
  cross.product <- t(matrix)%*%matrix

  ## the lower and the upper triangle of the cross-product is the dot products among vectors 
  dot.products<- cross.product[lower.tri(cross.product)]

  ## Calculate the L2 norms
  temp <- suppressWarnings(diag(sqrt(cross.product)))
  temp <- temp%*%t(temp)
  L2.norms <- temp[lower.tri(temp)]

  ## Arccosine values for each pair of variables
  lower.t <- acos(dot.products/L2.norms)

  ## Create an empty matrix to present the results
  result.matrix <- matrix(NA,ncol = dim(matrix)[2],nrow=dim(matrix)[2])

  ## Fill the matrix with arccosine values and assign the diagonal values as zero “0”
  result.matrix[lower.tri(result.matrix)] <- lower.t
  diag(result.matrix) <- 0
  result.matrix[upper.tri(result.matrix)] <- t(result.matrix)[upper.tri(t(result.matrix))]

  ## Get the result matrix
  return(result.matrix)
}

In addition, if you mean-centered your input variables and get the cosine values of the result matrix provided above, you will get the exact correlation matrix of the variables.

Here is an application of the function.

set.seed(123)
n <- 100
m <- 5

# Generate a set of random variables 

mt <- matrix(rnorm(n*m),nrow = n,ncol = m)

# Mean-centered matrix
mt.c <- scale(mt,scale = F)

# Cosine angles 
cosine.angles <- angles(matrix = mt)

> cosine.angles
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,] 0.000000 1.630819 1.686037 1.618119 1.751859
[2,] 1.630819 0.000000 1.554695 1.523353 1.712214
[3,] 1.686037 1.554695 0.000000 1.619723 1.581786
[4,] 1.618119 1.523353 1.619723 0.000000 1.593681
[5,] 1.751859 1.712214 1.581786 1.593681 0.000000



# Centered-data cosine angles 
centered.cosine.angles <- angles(matrix = mt.c)

> centered.cosine.angles
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,] 0.000000 1.620349 1.700334 1.614890 1.764721
[2,] 1.620349 0.000000 1.540213 1.526950 1.701793
[3,] 1.700334 1.540213 0.000000 1.615677 1.595647
[4,] 1.614890 1.526950 1.615677 0.000000 1.590057
[5,] 1.764721 1.701793 1.595647 1.590057 0.000000

# This will give you correlation matrix 
cos(angles(matrix = mt.c))

            [,1]        [,2]        [,3]        [,4]        [,5]
[1,]  1.00000000 -0.04953215 -0.12917601 -0.04407900 -0.19271110
[2,] -0.04953215  1.00000000  0.03057903  0.04383271 -0.13062219
[3,] -0.12917601  0.03057903  1.00000000 -0.04486571 -0.02484838
[4,] -0.04407900  0.04383271 -0.04486571  1.00000000 -0.01925986
[5,] -0.19271110 -0.13062219 -0.02484838 -0.01925986  1.00000000

# Orginal correlation matrix
cor(mt)

            [,1]        [,2]        [,3]        [,4]        [,5]
[1,]  1.00000000 -0.04953215 -0.12917601 -0.04407900 -0.19271110
[2,] -0.04953215  1.00000000  0.03057903  0.04383271 -0.13062219
[3,] -0.12917601  0.03057903  1.00000000 -0.04486571 -0.02484838
[4,] -0.04407900  0.04383271 -0.04486571  1.00000000 -0.01925986
[5,] -0.19271110 -0.13062219 -0.02484838 -0.01925986  1.00000000

# Check whether they are equal
all.equal(cos(angles(matrix = mt.c)),cor(mt))
[1] TRUE

The traditional approach to obtaining an angle between two vectors (i.e. acos(sum(a*b) / (sqrt(sum(a*a)) * sqrt(sum(b*b)))), as presented in some of the other answers) suffers from numerical instability in several corner cases. The following code works for n-dimensions and in all corner cases (it doesn't check for zero length vectors, but that's easy to add). See notes below.

# Get angle between two n-dimensional vectors
angle_btw <- function(v1, v2) {

  signbit <- function(x) {
    x < 0
  }

  u1 <- v1 / norm(v1, "2")
  u2 <- v2 / norm(v2, "2")

  y <- u1 - u2
  x <- u1 + u2

  a0 <- 2 * atan(norm(y, "2") / norm(x, "2"))

  if (!(signbit(a0) || signbit(pi - a0))) {
    a <- a0
  } else if (signbit(a0)) {
    a <- 0.0
  } else {
    a <- pi
  }

  a
}

This code is adapted from a Julia implementation by Jeffrey Sarnoff (MIT license), in turn based on these notes by Prof. W. Kahan (page 15).