In mathematics, the **cross**** product** or also known as the **vector product** is a binary operation on two vectors in **three-dimensional** space and is denoted by the symbol ‘**X**‘. Given two linearly independent vectors a and b, the cross product, **a × b** is a vector that is perpendicular to both a and b and thus normal to the plane containing them.

Let we have given two vectors,

and,

where,

i:the unit vector along the x directions

j:the unit vector along the y directions

k:the unit vector along the z directions

Then the cross product is calculated as:

where,are the coefficient of unit vector along i, j and k directions.

**Example:**

Given two vectors A and B as,

A = 3i + 5j + 4k,

and

B = 2i + 7j + 5k

Cross Product = (5 ? 5 – 4 ? 7)i + (4 ? 2 – 3 ? 5)j + (3 ? 7 – 5 ? 2)k

= (?3)i + (?7)j + (11)k

#### Computing Cross Product in R

R language provides a very efficient method to calculate the cross product of two vectors. By using **cross()** method which is available in the **pracma** library. This function computes the cross or vector product of vectors in 3 dimensions. In the case of matrices, it takes the first dimension of length 3 and computes the cross product between corresponding columns or rows.

Syntax:cross(x, y)

Parameters:

x:numeric vector or matrix

y:numeric vector or matrix

**# Taking Input as Vectors**

**Example 1:**

## R

`# R Program illustrating` `# cross product of two vectors` ` ` `# Import the required library` `library` `(pracma)` ` ` `# Taking two vectors` `a = ` `c` `(3, 5, 4)` `b = ` `c` `(2, 7, 5)` ` ` `# Calculating cross product using cross()` `print` `(` `cross` `(a, b))` |

**Output:**

[1] -3 -7 11

**Example 2:**

## R

`# R Program illustrating` `# cross product of two vectors` ` ` `# Import the required library` `library` `(pracma)` ` ` `# Taking two vectors` `a = ` `c` `(23, 15, 49)` `b = ` `c` `(28, 17, 25)` ` ` `# Calculating cross product using cross()` `print` `(` `cross` `(a, b))` |

**Output:**

[1] -458 797 -29

**# Taking Input as Matrix**

**Example 1:**

## R

`# R Program illustrating` `# cross product of two vectors` ` ` `# Import the required library` `library` `(pracma)` ` ` `# Taking two matrices` `a = ` `matrix` `( ` ` ` `c` `(1, 2, 3, 4, 5, 6, 7, 8, 9), ` ` ` `nrow = 3, ` ` ` `ncol = 3, ` ` ` `byrow = ` `TRUE` `) ` `b = ` `matrix` `( ` ` ` `c` `(5, 2, 1, 4, 6, 6, 3, 2, 9), ` ` ` `nrow = 3, ` ` ` `ncol = 3, ` ` ` `byrow = ` `TRUE` `) ` ` ` `# Calculating cross product using cross()` `print` `(` `cross` `(a, b))` |

**Output:**

[, 1] [, 2] [, 3] [1, ] -4 14 -8 [2, ] -6 0 4 [3, ] 54 -36 -10

**Example 2:**

## R

`# R Program illustrating` `# cross product of two vectors` ` ` `# Import the required library` `library` `(pracma)` ` ` `# Taking two matrices` `a = ` `matrix` `( ` ` ` `c` `(11, 2, 31, 4, 52, 64, 7, 8, 9), ` ` ` `nrow = 3, ` ` ` `ncol = 3, ` ` ` `byrow = ` `TRUE` `) ` `b = ` `matrix` `( ` ` ` `c` `(85, 21, 1, 4, 61, 6, 32, 2, 9), ` ` ` `nrow = 3, ` ` ` `ncol = 3, ` ` ` `byrow = ` `TRUE` `) ` ` ` `# Calculating cross product using cross()` `print` `(` `cross` `(a, b))` |

**Output:**

[, 1] [, 2] [, 3] [1, ] -649 2624 61 [2, ] -3592 232 36 [3, ] 54 225 -242