Vector

A vector is a fundamental mathematical object that represents both magnitude (size/length) and direction, making it essential for describing quantities that cannot be fully captured by a single number (scalar).

Mathematical Definition

A vector can be represented in several ways:

• As an arrow in space, defined by its length and orientation
• As an ordered list of numbers (components)
• As a mathematical object that follows specific algebraic rules

In an n-dimensional space, a vector is typically written as:

v = (v₁, v₂, ..., vₙ)

Key Operations

Basic Operations

• Vector addition: Combining two vectors (tip-to-tail method)
• Scalar multiplication: Scaling a vector's magnitude
• Dot Product: Resulting in a scalar value
• Cross Product: Producing a perpendicular vector (in 3D space)

Properties

• Direction: Indicated by the vector's orientation
• Magnitude: The length or size of the vector
• Normalization: Process of creating a unit vector

Applications

Physics

• Velocity and acceleration
• Force and momentum
• Electromagnetic Field

Computer Graphics

• 3D Modeling
• Animation
• Computer Vision

Data Science

• Feature Vector
• Machine Learning
• Dimensionality Reduction

Historical Development

The concept of vectors emerged from multiple mathematical traditions:

1. Geometric Algebra
2. Complex Numbers
3. Linear Algebra

Vector Spaces

Vectors exist within Vector Space, which provide a formal mathematical framework for:

• Linear combinations
• Basis vectors
• Linear Independence
• Span and Dimension

Programming Implementation

Modern programming languages often include vector operations:

# Example in Python using NumPy
import numpy as np
v1 = np.array([1, 2, 3])
v2 = np.array([4, 5, 6])
dot_product = np.dot(v1, v2)

See Also

• Matrix
• Tensor
• Coordinate System
• Vector Calculus

The concept of vectors forms a crucial bridge between abstract mathematics and practical applications, serving as a foundational tool across multiple disciplines.