The **Euclidean distance** between two vectors, A and B, is calculated as:

**Euclidean distance = √Σ(A _{i}-B_{i})^{2}**

To calculate the Euclidean distance between two vectors in R, we can define the following function:

euclidean <- function(a, b) sqrt(sum((a - b)^2))

We can then use this function to find the Euclidean distance between any two vectors:

#define two vectors a <- c(2, 6, 7, 7, 5, 13, 14, 17, 11, 8) b <- c(3, 5, 5, 3, 7, 12, 13, 19, 22, 7) #calculate Euclidean distance between vectors euclidean(a, b) [1] 12.40967

The Euclidean distance between the two vectors turns out to be **12.40967**.

Note that we can also use this function to calculate the Euclidean distance between two columns of a data frame:

#define data frame df <- data.frame(a=c(3, 4, 4, 6, 7, 14, 15), b=c(4, 8, 8, 9, 14, 13, 7), c=c(7, 7, 8, 5, 15, 11, 8), d=c(9, 6, 6, 7, 6, 15, 19)) #calculate Euclidean distance between columnsaanddeuclidean(df$a, df$d) [1] 7.937254

Note that this function will produce a warning message if the two vectors are not of equal length:

#define two vectors of unequal length a <- c(2, 6, 7, 7, 5, 13, 14) b <- c(3, 5, 5, 3, 7, 12, 13, 19, 22, 7) #attempt to calculate Euclidean distance between vectors euclidean(a, b) [1] 23.93742 Warning message: In a - b : longer object length is not a multiple of shorter object length

*You can refer to this Wikipedia page to learn more details about Euclidean distance.*