Vector Calculators

Comprehensive collection of vector operations for 2D, 3D, and 4D vectors

Basic Vector Operations

Addition
Add two vectors component-wise
Subtraction
Calculate vector difference
Multiplication
Multiply two vectors component-wise
Division
Divide two vectors component-wise

Scalar Operations

Scalar Multiplication
Scale vector by a scalar value
Scalar Division
Divide vector by a scalar value

Products & Metrics

Dot Product
Calculate scalar dot product
Cross Product
Calculate perpendicular vector
Triple Product
Calculate parallelepiped volume
Magnitude
Calculate vector length
Squared Magnitude
Calculate length squared (optimized)
Distance
Calculate distance between points
Distance Squared
Calculate distance squared (optimized)

Advanced Operations

Normalization
Convert to unit vector
Reflection
Reflect vector across normal
Interpolation
Linear interpolation between vectors

About Vector Mathematics

Vector mathematics is fundamental to physics, engineering, and computer science. Vector operations form the foundation for:

  • Physics - Forces and motion
  • Computer Graphics - 3D transformations
  • Engineering - Stress and strain
  • Navigation - GPS coordinates
  • Machine Learning - Feature vectors
  • Robotics - Motion control
Important Vector Properties
Commutative Property
v₁ + v₂ = v₂ + v₁
v₁ · v₂ = v₂ · v₁
Associative Property
(v₁ + v₂) + v₃ = v₁ + (v₂ + v₃)
a(bv) = (ab)v
Distributive Property
a(v₁ + v₂) = av₁ + av₂
(a + b)v = av + bv
Identity Elements
v + 0 = v
1 · v = v
Tip: Use our calculators to verify vector operations. Start with basic operations (addition, subtraction) before exploring advanced operations (dot product, cross product, triple product).

Practical Application Examples

Physics
  • Force Vectors: Magnitude and direction
  • Velocity: Speed and direction
  • Acceleration: Change in velocity
Computer Graphics
  • Position: 3D coordinates
  • Normals: Surface orientation
  • Transformations: Rotation and scaling
Navigation
  • Displacement: Change in position
  • Direction: Vector heading
  • Distance: Between two points
Data Science
  • Features: Data dimensions
  • Similarity: Dot product
  • Distance: Euclidean metrics
Quick Reference
v₁ + v₂
Addition
v₁ - v₂
Subtraction
v₁ · v₂
Dot Product
v₁ × v₂
Cross Product
|v|
Magnitude
|v|²
Squared Magnitude