Angle Between Two Vectors Calculator
Calculate the angle between two vectors in 2D or 3D space using dot product formula
Calculate Vector Angle
Vector A
Vector B
⚠️ Vector A is a zero vector. Please enter non-zero components for both vectors.
Example Calculation
3D Vector Angle Example
Given: Vector a = (3, 6, 1), Vector b from point A(1,1,2) to B(-4,-8,6)
Step 1: Calculate vector b = (-4-1, -8-1, 6-2) = (-5, -9, 4)
Step 2: Dot product a·b = (3×-5) + (6×-9) + (1×4) = -15 - 54 + 4 = -65
Step 3: |a| = √(3² + 6² + 1²) = √46 ≈ 6.782
Step 4: |b| = √((-5)² + (-9)² + 4²) = √122 ≈ 11.045
Step 5: α = arccos(-65 / (6.782 × 11.045)) = arccos(-0.867) ≈ 150.2°
Vector Angle Formulas
Vector Properties
Angle range: 0° to 180° (0 to π radians)
Perpendicular vectors: angle = 90°
Parallel vectors: angle = 0° or 180°
Acute angle: 0° < α < 90°
Obtuse angle: 90° < α < 180°
Quick Tips
Zero vectors have undefined angles
Dot product positive → acute angle
Dot product negative → obtuse angle
Used in physics, engineering, and computer graphics
Understanding Vector Angles
What is the Angle Between Vectors?
The angle between two vectors is the angle formed when the vectors are placed tail-to-tail. It's always measured as the smaller positive angle, ranging from 0° to 180°.
Dot Product Relationship
- •Geometric definition: a·b = |a| |b| cos(α)
- •Algebraic definition: a·b = Σ(aᵢ × bᵢ)
- •Angle formula: α = arccos(a·b / (|a| |b|))
- •Independent of vector positioning
Mathematical Foundation
Vector Magnitude:
• 2D: |v| = √(x² + y²)
• 3D: |v| = √(x² + y² + z²)
• Based on Pythagorean theorem
Special Cases
α = 0°: Vectors point in same direction
α = 90°: Vectors are perpendicular
α = 180°: Vectors point in opposite directions
Physics Applications
Force components, work calculation (W = F·d cos α), velocity direction analysis in mechanics.
Computer Graphics
Surface normal calculations, lighting models, 3D rotation and transformation operations.
Engineering
Structural analysis, signal processing, robotics path planning and control systems.