Vector Calculator
Perform vector operations with detailed calculations and visualizations
Vector Operations
Vector Calculations
Vector v₁
Vector v₂
Understanding Vectors
What is a Vector?
A vector is a mathematical object that has both magnitude (length) and direction. Vectors are fundamental in physics, engineering, and computer science for representing quantities like force, velocity, and position.
Vector Notation:
v = ⟨x, y⟩ for 2D vectors
v = ⟨x, y, z⟩ for 3D vectors
Example: v = ⟨3, 4⟩ represents a vector with x=3, y=4
Vector Operations
Vector operations follow specific geometric and algebraic rules that make them useful for solving problems in physics, engineering, and computer graphics.
- •Addition/Subtraction: Component-wise operations
- •Dot Product: Scalar result representing projection
- •Cross Product: Vector perpendicular to both inputs
- •Magnitude: Length of the vector
Vector Operation Formulas
Addition: v₁ + v₂ = ⟨x₁+x₂, y₁+y₂⟩
Component-wise addition of corresponding elements
Dot Product: v₁ · v₂ = x₁x₂ + y₁y₂
Scalar result: |v₁||v₂|cos(θ)
Cross Product: v₁ × v₂ = ⟨y₁z₂-z₁y₂, z₁x₂-x₁z₂, x₁y₂-y₁x₂⟩
Result perpendicular to both input vectors
Magnitude: |v| = √(x² + y² + z²)
Distance from origin to point (x, y, z)
Real-World Applications
Physics
Force, velocity, and motion
- • Force vectors
- • Velocity and acceleration
- • Momentum and impulse
- • Electric and magnetic fields
Computer Graphics
3D modeling and animation
- • 3D transformations
- • Lighting calculations
- • Collision detection
- • Particle systems
Engineering
Structural analysis
- • Stress and strain
- • Force analysis
- • Torque and rotation
- • Fluid dynamics
Navigation
Position and movement
- • GPS coordinates
- • Course and bearing
- • Velocity vectors
- • Wind and current
Machine Learning
Data representation
- • Feature vectors
- • Weight vectors
- • Gradient vectors
- • Embedding spaces
Robotics
Motion control
- • Joint angles
- • Position vectors
- • Force sensors
- • Path planning
Special Types of Vectors
Common Special Vectors
Unit Vector
Magnitude = 1, indicates direction only
Formula: û = v/|v|
Zero Vector
All components = 0
Properties: v + 0 = v, v · 0 = 0
Position Vector
From origin to a point
Used for location representation
Normal Vector
Perpendicular to a surface
Used in lighting and physics
Vector Properties
Orthogonal Vectors
Dot product = 0, perpendicular
Used in coordinate systems
Parallel Vectors
Cross product = 0, same direction
One is scalar multiple of other
Collinear Vectors
Lie on the same line
Linearly dependent
Coplanar Vectors
Lie in the same plane
Important in 3D geometry