Vector Space Calculator

Perform vector operations, analyze vector spaces, and visualize vectors in 2D to 5D space

Vector 1
Vector 2
Vector 1 Magnitude
3.74
||v₁||
Vector 2 Magnitude
8.77
||v₂||
Dot Product
32
v₁ · v₂
Angle Between Vectors
12.6°
θ
Cross Product (v₁ × v₂)
[-3, 6, -3]
Projection of v₁ onto v₂
[1.66, 2.08, 2.49]
Note: 4D and 5D vectors are visualized using their first 3 components
Vector Space Properties
  • Linear Independence: Linearly independent
  • Span: R³
  • Basis: Basis for R³
  • Dimension: 3
  • Orthogonality: Not orthogonal
Vector Formulas
Magnitude: ||v|| = √(x₁² + x₂² + ... + xₙ²)
Dot Product: v₁ · v₂ = x₁x₂ + y₁y₂ + z₁z₂ + ...
Cross Product: v₁ × v₂ = [y₁z₂ - z₁y₂, z₁x₂ - x₁z₂, x₁y₂ - y₁x₂] (3D only)
Projection: projᵤv = (u·v / ||u||²) u
Common Vector Examples
Vector Type Vector Properties
Zero Vector [0, 0, 0, 0, 0] Magnitude: 0
Unit Vector [1, 0, 0, 0, 0] Magnitude: 1
Orthogonal Vectors [1, 0, 0] and [0, 1, 0] Dot product: 0
Parallel Vectors [1, 2, 3] and [2, 4, 6] Cross product: [0, 0, 0]
Standard Basis [1,0,0], [0,1,0], [0,0,1] Basis for R³
Higher-Dimensional Vector Facts
  • 5D vectors are used in physics to describe spacetime with extra dimensions
  • Higher-dimensional vectors are essential in machine learning and data science
  • Vector spaces can have infinite dimensions (function spaces)
  • Dot product generalizes to any number of dimensions
  • Cross product is only defined for 3D and 7D vectors

Vector Space Fundamentals

A vector space is a collection of vectors that can be added together and multiplied by scalars (real numbers) while satisfying certain axioms.

Vector Space Axioms

For a set V to be a vector space, it must satisfy:

  • Closure under addition: u + v ∈ V for all u, v ∈ V
  • Closure under scalar multiplication: c·u ∈ V for all u ∈ V, c ∈ ℝ
  • Commutativity of addition: u + v = v + u
  • Associativity of addition: (u + v) + w = u + (v + w)
  • Additive identity: There exists 0 ∈ V such that u + 0 = u
  • Additive inverse: For each u ∈ V, there exists -u such that u + (-u) = 0
  • Distributivity: c(u + v) = cu + cv and (c + d)u = cu + du
  • Scalar multiplication identity: 1·u = u

Common Vector Spaces

  • ℝⁿ: n-dimensional Euclidean space
  • Function spaces: Spaces of polynomials or continuous functions
  • Matrix spaces: Spaces of m×n matrices
  • Sequence spaces: Spaces of infinite sequences

Key Concepts

  • Basis: A linearly independent set that spans the vector space
  • Dimension: The number of vectors in a basis
  • Linear Independence: No vector in the set can be written as a linear combination of the others
  • Span: The set of all linear combinations of a set of vectors
  • Subspace: A subset of a vector space that is itself a vector space

Vector Operations

Common vector operations and their formulas:

Vector Addition
A + B = (a₁+b₁, a₂+b₂, ..., aₙ+bₙ)
Scalar Multiplication
cA = (c·a₁, c·a₂, ..., c·aₙ)
Dot Product
A·B = a₁b₁ + a₂b₂ + ... + aₙbₙ
Cross Product (3D)
A×B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)
Magnitude
|A| = √(a₁² + a₂² + ... + aₙ²)
Angle Between Vectors
cosθ = (A·B) / (|A||B|)
Projection of A onto B
proj_B A = [(A·B) / (B·B)] B

Vector Space Q&A

A vector space is a fundamental concept in linear algebra. It is a set of objects called vectors, which may be added together and multiplied by scalars (numbers).

Formally, a vector space over a field F (often the real numbers ℝ) is a set V equipped with two operations:

  • Vector addition: V × V → V
  • Scalar multiplication: F × V → V

These operations must satisfy the eight axioms listed in the knowledge base section.

Scalars are single numbers that represent magnitude but not direction. They can be real or complex numbers.

Vectors are mathematical objects that have both magnitude and direction. They are typically represented as arrays of numbers.

Key differences:

  • Vectors have direction, scalars do not
  • Vectors can be added geometrically (tip-to-tail), scalars cannot
  • Vectors have components, scalars are single values
  • Scalars scale vectors through multiplication

The dot product (or scalar product) is a fundamental operation in vector algebra with many applications:

  • Angle calculation: cosθ = (A·B) / (|A||B|)
  • Projection: Finding the component of one vector along another
  • Orthogonality test: A·B = 0 indicates perpendicular vectors
  • Work calculation: In physics, work = F·d (force dot displacement)
  • Vector similarity: In machine learning and data science
  • Geometric transformations: In computer graphics

A basis of a vector space is a set of linearly independent vectors that span the full space. This means:

  • Every vector in the space can be expressed as a linear combination of basis vectors
  • No basis vector can be written as a linear combination of the others

Key properties:

  • All bases of a vector space have the same number of vectors (dimension)
  • The standard basis for ℝⁿ is {(1,0,...,0), (0,1,0,...,0), ..., (0,0,...,1)}
  • Basis vectors are not unique - a vector space has many possible bases
  • Changing basis is fundamental in linear transformations

Vector spaces have numerous applications across various fields:

  • Physics: Representing forces, velocities, and fields
  • Computer Graphics: 3D modeling, transformations, rendering
  • Engineering: Structural analysis, electrical circuits
  • Economics: Input-output models, optimization
  • Data Science: Vector space models for text analysis
  • Quantum Mechanics: State spaces as complex vector spaces
  • Machine Learning: Feature spaces, support vector machines
  • Navigation: GPS systems, flight paths