Direction of Vector Calculator
Calculate direction angles, unit vectors, and direction cosines
Vector Direction Analysis
Vector Components
Magnitude: 5.000000
Results
Step-by-Step Calculation
Direction Angle Calculation (2D)
Given: Vector v = (3, 4)
Formula: θ = arctan(y/x) with quadrant adjustment
Steps:
1. Calculate magnitude: |v| = √(3² + 4²) = 5.000000
2. Use atan2(4, 3) for proper quadrant handling
3. Raw angle = 0.927295 radians
4. Adjust to [0, 2π) range if needed
Result: θ = 53.130102°
Quadrant: I
Example: Finding Direction Angle
Given Vector
Vector: v = (3, 4)
Goal: Find the direction angle θ
Step-by-Step Solution
1. Calculate magnitude: |v| = √(3² + 4²) = √(9 + 16) = √25 = 5
2. Use atan2 function: θ = atan2(4, 3) ≈ 0.9273 radians
3. Convert to degrees: θ ≈ 53.13°
4. Vector is in Quadrant I (both x > 0 and y > 0)
Result: θ ≈ 53.13° or 0.9273 rad
Vector Direction Concepts
2D Direction Angle
- • θ = arctan(y/x)
- • Measured from positive x-axis
- • Range: [0°, 360°) or [0, 2π)
- • Use atan2 for proper quadrants
3D Direction Angles
- • α: angle with x-axis
- • β: angle with y-axis
- • γ: angle with z-axis
- • cos²α + cos²β + cos²γ = 1
Unit Vector
- • û = v / |v|
- • Magnitude = 1
- • Same direction as original
- • Useful for scaling
2D Quadrant Guide
Quadrant I: (x > 0, y > 0)
Quadrant II: (x < 0, y > 0)
Quadrant III: (x < 0, y < 0)
Quadrant IV: (x > 0, y < 0)
Note
Understanding Vector Direction
Direction Angle (2D)
The direction angle θ is the angle a vector makes with the positive x-axis, measured counterclockwise. It uniquely determines the vector's direction in 2D space.
Direction Angles (3D)
In 3D space, we use three direction angles (α, β, γ) measured from the positive x, y, and z axes respectively. These angles help describe the vector's orientation in three-dimensional space.
Unit Vectors
A unit vector has magnitude 1 and points in the same direction as the original vector. It's calculated by dividing each component by the vector's magnitude.
Applications
Physics
Force directions, velocity components, field orientations
Engineering
Structural analysis, robotics, navigation systems
Computer Graphics
3D rotations, lighting calculations, camera orientation
Mathematics
Vector analysis, coordinate transformations, optimization
⚠️ Important Notes
- • Zero vector has no defined direction
- • Direction cosines sum of squares equals 1
- • Use atan2() for proper quadrant calculation
- • Unit vectors preserve direction but normalize magnitude