Vector Magnitude Calculator
Calculate the magnitude (length) of vectors in 2D, 3D, 4D, and 5D space
Calculate Vector Magnitude
Input Method
Vector Magnitude
Magnitude
Vector Information
Dimension: 2D
Components: (0, 0)
Sum of squares: 0.000000
Square root: √0.000000 = 0.000000
Step-by-step Calculation
Example Calculations
2D Vector Example
Vector: v = (3, 4)
Calculation: |v| = √(3² + 4²) = √(9 + 16) = √25 = 5
Result: The magnitude is 5 units
3D Vector Example
Vector: v = (1, 2, 2)
Calculation: |v| = √(1² + 2² + 2²) = √(1 + 4 + 4) = √9 = 3
Result: The magnitude is 3 units
4D Vector Example
Vector: v = (3, -1, 2, -3)
Calculation: |v| = √(3² + (-1)² + 2² + (-3)²) = √(9 + 1 + 4 + 9) = √23 ≈ 4.796
Result: The magnitude is approximately 4.796 units
Magnitude Formulas
2D Vector
3D Vector
n-D Vector
Dot Product Form
Vector Properties
Non-negative: Magnitude is always ≥ 0
Zero vector: |0| = 0
Unit vector: |û| = 1
Scalar multiple: |kv| = |k| × |v|
Real-World Applications
Physics
Speed (magnitude of velocity), force strength
Computer Graphics
Distance calculations, normalization
Machine Learning
Feature vector lengths, similarity measures
Navigation
Distance between coordinates
Understanding Vector Magnitude
What is Vector Magnitude?
Vector magnitude, also called the length or norm of a vector, represents the distance from the origin to the point defined by the vector's components. It's always a non-negative scalar value that tells us "how much" of the vector quantity we have.
Mathematical Foundation
The magnitude formula is derived from the Pythagorean theorem extended to multiple dimensions. In any n-dimensional space, the magnitude is the square root of the sum of squares of all components.
|v| = √(v₁² + v₂² + ... + vₙ²)
Physical Interpretation
In physics, vector magnitude often represents measurable quantities:
- •Velocity vector: Speed is the magnitude
- •Force vector: Magnitude shows force strength
- •Displacement vector: Magnitude is distance
- •Electric field vector: Magnitude is field strength
Unit Vectors
A unit vector has magnitude 1 and represents pure direction. Any vector can be normalized to a unit vector by dividing each component by the vector's magnitude: û = v / |v|.