Vector Calculator
Vector Calculator
Perform vector operations including addition, dot product, cross product, and more
Vector Calculator: Complete Linear Algebra Guide
Vectors are mathematical objects with both magnitude and direction, fundamental to physics, engineering, and computer graphics.Vector operations allow us to manipulate and analyze these quantities in 2D and 3D space, enabling solutions to complex problems in mechanics, electronics, and geometric modeling.
Quick Answer
To perform vector operations: Enter your vector components separated by commas (e.g., "3, 4" for 2D or "3, 4, 5" for 3D), select the operation, and get instant results with detailed explanations and analysis.
Vector Mathematics Foundation
Vector Notation
Vector representation in component form and unit vector notation
Vector Addition
v⃗₁ + v⃗₂ = (x₁+x₂, y₁+y₂, z₁+z₂)Add corresponding components of each vector
Vector Magnitude
|v⃗| = √(x² + y² + z²)Length of vector using Pythagorean theorem
Dot Product
v⃗₁ · v⃗₂ = x₁x₂ + y₁y₂ + z₁z₂Scalar result measuring vector alignment
Cross Product
v⃗₁ × v⃗₂ = (y₁z₂-z₁y₂, z₁x₂-x₁z₂, x₁y₂-y₁x₂)Vector perpendicular to both input vectors
Essential Vector Operations
Basic Operations
Addition & Subtraction
Component-wise operations for combining vectors. Used in physics for force addition, velocity composition, and displacement calculations.
Scalar Multiplication
Scaling vector magnitude while preserving direction. Essential for unit vector creation and vector normalization.
Magnitude Calculation
Determining vector length using the distance formula. Critical for normalization and comparing vector sizes.
Advanced Operations
Dot Product
Measures vector alignment and calculates work in physics. Zero result indicates perpendicular vectors, positive/negative shows alignment.
Cross Product
Creates perpendicular vector in 3D space. Used for torque calculations, surface normals, and determining rotation direction.
Vector Projection
Projects one vector onto another, showing component in specific direction. Essential for force analysis and component decomposition.
Applications of Vector Mathematics
Physics & Engineering
Mechanics
Force analysis, velocity calculations, acceleration vectors, and equilibrium problems
Electromagnetics
Electric and magnetic field calculations, Lorentz force, and electromagnetic induction
Structural Engineering
Stress analysis, load distribution, and structural stability calculations
Aerospace
Flight dynamics, orbital mechanics, and attitude control systems
Computer Science & Graphics
3D Graphics
3D transformations, lighting calculations, surface normals, and camera positioning
Game Development
Character movement, collision detection, physics simulation, and AI pathfinding
Machine Learning
Feature vectors, gradient calculations, dimensionality reduction, and optimization
Robotics
Robot kinematics, path planning, sensor data processing, and control systems
Example Problems with Solutions
Example 1: Force Addition in Physics
Two forces act on an object: F₁ = (10, 5) N and F₂ = (3, -8) N. Find the resultant force.
Answer: Resultant force = (13, -3) N with magnitude 13.3 N
Example 2: Dot Product for Work Calculation
A force F = (4, 3) N acts through displacement d = (2, 1) m. Calculate work done.
Answer: Work done = 11 Joules
Example 3: Cross Product for Torque
Position vector r = (2, 0, 0) m and force F = (0, 5, 0) N. Find torque τ = r × F.
Answer: Torque = (0, 0, 10) N·m (along z-axis)
Vector Input Guide
Input Formats
Operation Types
Important Notes
- • Cross product requires 3D vectors (z-component will be added if missing)
- • Angle calculations return results in degrees
- • Unit vectors have magnitude = 1
- • Zero vectors cannot be normalized
- • Dot product of perpendicular vectors equals zero
Related Mathematical Tools
Vector Calculator
Perform vector operations including addition, dot product, cross product, and more