Definition
A vector is a mathematical entity commonly represented as a one-dimensional array of numbers. Vectors are essential tools in various fields such as mathematics, physics, engineering, computer science, and economics. Formally, a vector can be defined in different contexts, such as Euclidean spaces or abstract vector spaces.
Types of Vectors
- Geometric Vectors: Represented as directed line segments in geometry, having both magnitude and direction.
- Algebraic Vectors: Represented as tuples or arrays of numbers, typically in the form \(\mathbf{v} = (v_1, v_2, \ldots, v_n)\) where \(v_i\) are real numbers.
- Row and Column Vectors: In matrix algebra, these are distinguished by their orientation—row vectors are 1 x n matrices, whereas column vectors are n x 1 matrices.
Notation
Vectors are often denoted by boldface letters, such as v, or with an arrow overhead, \(\vec{v}\). In written text, vectors might also be represented by lowercase letters.
Mathematical Representation
Consider a vector \(\mathbf{v}\):
Operations on Vectors
- Addition: \(\mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, \ldots, u_n + v_n)\)
- Scalar Multiplication: \(c \mathbf{v} = (c v_1, c v_2, \ldots, c v_n)\)
- Dot Product: \(\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^{n} u_i v_i\)
- Cross Product: In 3D, \(\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \end{vmatrix}\)
Special Considerations
Vector Spaces
Vectors are elements of a vector space, which must satisfy certain axioms, such as closure under vector addition and scalar multiplication, existence of a zero vector, and others.
Norms and Magnitudes
The magnitude (or norm) of a vector \(\mathbf{v}\) is a measure of its length and is given by:
Applications
- Physics: Velocity, acceleration, and force are represented as vectors.
- Computer Graphics: Vectors are used to describe points and transformations in space.
- Economics: Vectors can represent quantities like resource allocation and consumption.
Historical Context
Vectors were formalized in the 19th century, with important contributions from mathematicians like Josiah Willard Gibbs and Oliver Heaviside, who developed vector notation and calculus for use in electromagnetism and physics.
Related Terms
- Matrix: A two-dimensional array of numbers, generalizing vectors and useful in linear transformations.
- Tensor: A more general object than a vector, capable of being multi-dimensional arrays.
- Linear Algebra: The branch of mathematics concerning vectors, vector spaces, and linear mappings.
FAQs
What is the difference between a vector and a scalar?
Can vectors be used in machine learning?
How do vectors relate to matrices?
Summary
Vectors are vital mathematical constructs widely used across many disciplines for representing quantities that have both magnitude and direction. They are integral to studies in physics, engineering, computer science, and beyond. Understanding vectors helps in grasping more complex concepts in fields like linear algebra and vector calculus.
References
- Lay, David C. (2012). Linear Algebra and Its Applications. Pearson.
- Strang, Gilbert (2009). Introduction to Linear Algebra. Wellesley-Cambridge Press.
- Kreyszig, Erwin (2010). Advanced Engineering Mathematics. John Wiley & Sons.
This structured and comprehensive entry aims to equip readers with thorough understanding and appreciation for the concept of vectors in various contexts.
