Angle between two vectors in R

Solution 1:

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)) ) )

Solution 2:

My answer consists of two parts. Part 1 is the math - to give clarity to all readers of the thread and to make the R code that follows understandable. Part 2 is the R programming.

Part 1 - Math

The dot product of two vectors x and y can be defined as:

where ||x|| is the Euclidean norm (also known as the L₂ norm) of the vector x.

Manipulating the definition of the dot product, we can obtain:

where theta is the angle between the vectors x and y expressed in radians. Note that theta can take on a value that lies on the closed interval from 0 to pi.

Solving for theta itself, we get:

Part 2 - R Code

To translate the mathematics into R code, we need to know how to perform two matrix (vector) calculations; dot product and Euclidean norm (which is a specific type of norm, known as the L₂ norm). We also need to know the R equivalent of the inverse cosine function, cos^-1.

Starting from the top. By reference to ?"%*%", the dot product (also referred to as the inner product) can be calculated using the %*% operator. With reference to ?norm, the norm() function (base package) returns a norm of a vector. The norm of interest here is the L₂ norm or, in the parlance of the R help documentation, the "spectral" or "2"-norm. This means that the type argument of the norm() function ought to be set equal to "2". Lastly, the inverse cosine function in R is represented by the acos() function.

Solution

Equipped with both the mathematics and the relevant R functions, a prototype function (that is, not production standard) can be put together - using Base package functions - as shown below. If the above information makes sense then the angle() function that follows should be clear without further comment.

angle <- function(x,y){
  dot.prod <- x%*%y 
  norm.x <- norm(x,type="2")
  norm.y <- norm(y,type="2")
  theta <- acos(dot.prod / (norm.x * norm.y))
  as.numeric(theta)
}

Test the function

A test to verify that the function works. Let x = (2,1) and y = (1,2). Dot product between x and y is 4. Euclidean norm of x is sqrt(5). Euclidean norm of y is also sqrt(5). cos theta = 4/5. Theta is approximately 0.643 radians.

x <- as.matrix(c(2,1))
y <- as.matrix(c(1,2))
angle(t(x),y)          # Use of transpose to make vectors (matrices) conformable.
[1] 0.6435011

I hope this helps!