Welcome to the vector operations guide! Vectors are fundamental in fields like physics, computer graphics, and machine learning. Let's explore their core concepts and applications.

📌 Basic Concepts

A vector is a mathematical object with both magnitude and direction. In programming, it's often represented as an array of numbers. Key properties include:

  • Magnitude: Length of the vector (e.g., √(x² + y² + z²))
  • Direction: Angle relative to coordinate axes
  • Dimension: Number of components (2D, 3D, etc.)

🧮 Fundamental Operations

1. Vector Addition

To add two vectors, sum their corresponding components:

A = [a1, a2, a3]
B = [b1, b2, b3]
A + B = [a1+b1, a2+b2, a3+b3]
Vector_Addition

2. Scalar Multiplication

Multiply each component by a scalar (single number):

k * A = [k*a1, k*a2, k*a3]
Scalar_Multiplication

3. Dot Product

Calculates the cosine similarity between vectors:

A · B = a1*b1 + a2*b2 + a3*b3
Dot_Product

4. Cross Product

Produces a vector perpendicular to both inputs (3D only):

A × B = |i  j  k|
        |a1 a2 a3|
        |b1 b2 b3|
Cross_Product

🔍 Practical Applications

  • Game Development: Physics simulations (e.g., velocity vectors)
  • Machine Learning: Feature vectors in algorithms
  • Data Science: Vectorized operations for efficient computations

📘 Further Reading

For a deeper dive into linear algebra, check out our Linear Algebra Basics tutorial. It covers matrices, determinants, and more!