Angle Between Two Vectors Calculator
Calculate the angle between two vectors using the dot product formula with comprehensive analysis. Our calculator supports both 2D and 3D vectors, provides vector projections, cross products, and detailed step-by-step solutions for complete linear algebra understanding and engineering applications.
Last updated: December 15, 2024
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Vector Angle Analysis
Input Vectors:
Angle Between Vectors:
Vector Properties:
Unit Vectors:
Vector Projections:
Calculation Steps:
- Step 1: Given vectors u = (3, 4) and v = (1, 2)
- Step 2: Calculate dot product u · v = 3 × 1 + 4 × 2 = 11
- Step 3: Calculate |u| = √(3² + 4²) = 5.000000
- Step 4: Calculate |v| = √(1² + 2²) = 2.236068
- Step 5: Calculate cos θ = (u · v) / (|u| × |v|) = 11 / (5.000000 × 2.236068) = 0.983870
- Step 6: Calculate θ = arccos(0.983870) = 0.179853 radians
- Step 7: Convert to degrees: θ = 0.179853 × (180/π) = 10.304846°
Quick Example Result
For vectors u = (3, 4) and v = (1, 2):
θ = 0.1799 rad = 10.30°
How This Calculator Works
Our vector angle calculator uses the fundamental dot product formula to determine the angle between two vectors in 2D or 3D space. The calculation involves computing the dot product, finding vector magnitudes, and applying the inverse cosine function. Additional features include vector projections, cross products, and comprehensive vector analysis for engineering and mathematical applications.
Vector Angle Calculation Process
Step 1 - Calculate Dot Product:
u · v = u₁v₁ + u₂v₂ + u₃v₃Step 2 - Find Magnitudes:
|u| = √(u₁² + u₂² + u₃²)Step 3 - Calculate Angle:
θ = arccos((u · v) / (|u| × |v|))Shows two vectors with their angle θ and projections
Mathematical Foundation
The angle between two vectors is derived from the geometric interpretation of the dot product. For vectors u and v, the dot product u · v = |u||v|cos(θ), where θ is the angle between them. Rearranging gives us cos(θ) = (u · v)/(|u||v|), and taking the inverse cosine yields the angle. This fundamental relationship connects algebraic vector operations with geometric properties.
- Dot product: Measures how much vectors point in the same direction
- Vector magnitude: The length or norm of a vector in n-dimensional space
- Angle range: Always between 0° and 180° (0 and π radians)
- Special cases: 0° (parallel), 90° (perpendicular), 180° (anti-parallel)
Sources & References
- Strang Linear Algebra - Vector Geometry and Dot ProductsComprehensive treatment of vector operations and geometric interpretations
- MIT OpenCourseWare - Linear Algebra: Vector OperationsDetailed examples and applications of vector angle calculations
- Khan Academy - Vector Dot Product and AngleInteractive tutorials on vector operations and geometric applications
Need help with other vector operations? Check out our dot product calculator and cross product calculator.
Get Custom Calculator for Your BusinessExample Calculation
Given Vectors:
- Vector u: (3, 4)
- Vector v: (1, 2)
- Find: Angle θ between them
Calculation Steps:
- Dot product: u · v = 3×1 + 4×2 = 11
- |u| = √(3² + 4²) = √25 = 5
- |v| = √(1² + 2²) = √5 ≈ 2.236
- cos θ = 11/(5 × 2.236) = 0.9839
- θ = arccos(0.9839) ≈ 0.1798 rad ≈ 10.3°
Result: θ = 0.1799 radians = 10.30° (Acute angle)
This acute angle indicates that the vectors point in generally the same direction with a small angular separation.
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