R编程中向量的叉积

在数学中，叉积或也称为向量积是在三维空间中对两个向量进行的二元运算，用符号“ X ”表示。给定两个线性独立的向量 a 和 b，叉积， a × b是一个垂直于 a 和 b 的向量，因此垂直于包含它们的平面。

让我们给出两个向量，

$A = a_1i + a_2j + a_3k$

和，

$B = b_1i + b_2j + b_3k$

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

然后叉积计算为：

$Cross\, Product = (a_2 * b_3 - a_3 * b_2)i + (a_3 * b_1 - a_1 * b_3)j + (a_1 * b_2 - a_2 * b_1)k$

where,

$[(a_2 * b_3 - a_3 * b_2), (a_3 * b_1 - a_1 * b_3), (a_1 * b_2 - a_2 * b_1)]$

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

例子：

给定两个向量 A 和 B，

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

在 R 中计算叉积

R 语言提供了一种非常有效的方法来计算两个向量的叉积。通过使用pracma库中可用的cross()方法。此函数计算 3 维向量的叉积或向量积。在矩阵的情况下，它采用长度为 3 的第一个维度并计算相应列或行之间的叉积。

Syntax: cross(x, y)

Parameters:

x: numeric vector or matrix

y: numeric vector or matrix

# 将输入作为向量

示例 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))

输出：

[1] -3 -7 11

示例 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))

输出：

[1] -458  797  -29

# 以输入为矩阵

示例 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))

输出：

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

示例 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))

输出：

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