[Python] List of Vector Operations with NumPy: From Arithmetic to Dot and Cross Products

Using NumPy allows you to intuitively write everything from element-wise calculations (like vector addition and subtraction) to dot products and cross products, which are crucial in linear algebra.

Particular caution is required for “multiplication.” You need to be aware that Python’s * operator represents the “Hadamard product (element-wise multiplication)” and not the “dot product.”

Executable Sample Code

Here is an implementation example covering all patterns using vectors $\vec{a} = [1, 2, 3]$ and $\vec{b} = [4, 5, 6]$.

import numpy as np

def vector_operations_demo():
    # Definition of vectors
    a = np.array([1, 2, 3])
    b = np.array([4, 5, 6])
    scalar = 10

    print(f"Vector a: {a}")
    print(f"Vector b: {b}")
    print("-" * 30)

    # 1. Addition (Vector Addition)
    # Add elements together: [1+4, 2+5, 3+6]
    add = a + b
    print(f"1. Addition (a + b):         {add}")

    # 2. Subtraction (Vector Subtraction)
    # Subtract elements: [1-4, 2-5, 3-6]
    sub = a - b
    print(f"2. Subtraction (a - b):      {sub}")

    # 3. Scalar Multiplication
    # Multiply all elements by a constant: [1*10, 2*10, 3*10]
    mul_scalar = a * scalar
    print(f"3. Scalar Mult (a * 10):     {mul_scalar}")

    # 4. Scalar Division
    # Divide all elements by a constant
    div_scalar = a / scalar
    print(f"4. Scalar Div (a / 10):      {div_scalar}")

    # 5. Hadamard Product (Element-wise Multiplication)
    # Multiply elements at the same position: [1*4, 2*5, 3*6]
    # * Note: This is NOT matrix multiplication
    hadamard_mul = a * b
    print(f"5. Hadamard Product (a * b): {hadamard_mul}")

    # 6. Element-wise Division
    # Divide elements at the same position
    hadamard_div = a / b
    print(f"6. Element-wise Div (a / b): {hadamard_div}")

    # 7. Dot Product
    # Sum of the products of corresponding components: 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32
    # Use np.dot(a, b) or the @ operator
    dot_prod = np.dot(a, b)
    dot_prod_op = a @ b
    print(f"7. Dot Product (np.dot / @): {dot_prod}")

    # 8. Cross Product
    # Generate a vector orthogonal to two vectors (mainly used in 3D)
    cross_prod = np.cross(a, b)
    print(f"8. Cross Product (np.cross): {cross_prod}")

if __name__ == "__main__":
    vector_operations_demo()

Summary of Operation Methods

The meanings of operators in Python (NumPy) are as follows:

Supplement: About the @ Operator for Dot Products

In Python 3.5 and later, a dedicated operator @ was introduced to represent matrix multiplication and dot products.

Instead of writing np.dot(a, b), you can write a @ b, allowing for expressions closer to mathematical formulas.