NumPy: Generate inner, outer, and cross products of matrices and vectors

Last update on August 29 2025 12:45:00 (UTC/GMT +8 hours)

14. Inner, Outer, and Cross Products

Write a NumPy program to generate inner, outer, and cross products of matrices and vectors.

Sample Solution:

Python Code:

# Importing the NumPy library
import numpy as np

# Creating NumPy arrays 'x' and 'y'
x = np.array([1, 4, 0], float)
y = np.array([2, 2, 1], float)

# Displaying matrices and vectors 'x' and 'y'
print("Matrices and vectors.")
print("x:")
print(x)
print("y:")
print(y)

# Computing the inner product of vectors 'x' and 'y'
print("Inner product of x and y:")
print(np.inner(x, y))

# Computing the outer product of vectors 'x' and 'y'
print("Outer product of x and y:")
print(np.outer(x, y))

# Computing the cross product of vectors 'x' and 'y'
print("Cross product of x and y:")
print(np.cross(x, y))

Sample Output:

Matrices and vectors.                                                  
x:                                                                     
[ 1.  4.  0.]                                                          
y:                                                                     
[ 2.  2.  1.]                                                          
Inner product of x and y:                                              
10.0                                                                   
Outer product of x and y:                                              
[[ 2.  2.  1.]                                                         
 [ 8.  8.  4.]                                                         
 [ 0.  0.  0.]]                                                        
Cross product of x and y:                                              
[ 4. -1. -6.]

Explanation:

np.inner(x, y) computes the inner product of two arrays. For 1-D arrays, it is the sum of the product of the corresponding elements. For higher dimensions, it is a sum product over the last axes.In the given code, x and y are 1-D arrays, and the output is the sum of the element-wise product of x and y, which is 10.

np.outer(x, y) computes the outer product of two arrays. For 1-D arrays, it returns the matrix of all possible products of the elements of x and the elements of y.In the given code, x and y are 1-D arrays, and the output is a 2-D array where each element is the product of the corresponding elements of x and y.

np.cross(x, y) computes the cross product of two arrays in a 3-dimensional space. The cross product of two 1-D arrays returns a vector perpendicular to both input vectors. In the given code, x and y are 1-D arrays, and the output is the cross product of x and y, which is a 1-D array [ 4., -1., -6.]. Since the input vectors are in a 3-dimensional space, the cross product is computed using the right-hand rule.

For more Practice: Solve these Related Problems:

Implement a function that computes the inner, outer, and cross products for two given vectors and returns all three.
Test the functions on 1D arrays and verify that the inner and outer products yield expected dimensions.
Compare the cross product output with manual computations on two 3D vectors.
Apply the outer product function on arrays with varying lengths to observe the resulting 2D matrix shape.

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