The post is about vector arithmetic in R Language. In R, different mathematical operations can be performed on vectors, that is vectors can be used in arithmetic expressions. The vector arithmetic operations are performed element by element.

It is important to note that vectors occurring in the same mathematical expression need not be of the same length (size). The shorter vectors in the arithmetic expression are recycled until they match the length of the longest vector.

### Vector Arithmetic Operations

The vector arithmetic operations can be performed using arithmetic operators and vector functions. The +, -, *, /, and ^ are elementary arithmetic operators. The arithmetic functions are also available, such as, `log, exp, sin, cos, tan, sqrt`

, and so on. The `max()`

and `min()`

functions returns the largest and smallest elements of a vector, respectively. Similarly, the `range()`

function results in a vector of length two having minimum and maximum values from the vector, that is, `c(min(x), max(x))`

.

The length(x) function returns the number of elements (size or number of observations) in a vector say $x$, `sum(x)`

gives the total `(sum)`

of the elements in vector $x$, and `prod(x)`

returns the product of elements.

Instead of performing simple arithmetics (+, -, *, and /), we will use some functions for arithmetic that can be performed on a vector.

### Vector Arithmetic in R: Examples

The basic vector arithmetic in R can be performed just like adding numbers on a calculator.

x <- c(1, 2, 3, 4, 5) y <- c(4, 5, 6, 7, 8) # Addition x + y # Subtraction x - y # Multiplication x * y # Division x / y # Exponentiation x ^ y

One can compute the average (mean value) of a vector by performing arithmetics on a vector, such as

x <- c(5, 10, 5, 3, 5, 6, 7, 8, 4, 3, 10) sum(x)/ length(x) ## Output 6

The built-in function for the computation of the average value of a vector is `mean()`

, that is `mean(x)`

.

mean(x) ## output 6

The variance can also be computed by performing arithmetics on a vector say $x$.

sum((x - mean(x))^2)/ (length(x)-1) ## Output 6.2

The built-in function for sample variance is `var(x)`

. Note that if the argument `var()`

is a $n$-by-$p$ matrix, a $p$-by-$p$ matrix of the sample covariance matrix will return.

var(x) ## Output 6.2

The `sort(x)`

function returns a vector of the same size as $x$ with the elements arranged in increasing order.

sort(x) ## Output [1] 3 3 4 5 5 5 6 7 8 10 10

The `min()`

and `max()`

functions are used to select the smallest and largest values from the argument, even if the argument contains several vectors.

In summary, Vector arithmetic is a fundamental aspect of R programming, enabling efficient and concise mathematical operations on sequences of elements. By understanding the basic operations, vector recycling, and available functions, you can effectively leverage vectors to solve a wide range of problems in data analysis and scientific computing.