Vector Magnitude Calculator
Calculate the magnitude (length or norm) of vectors in any dimension with detailed step-by-step solutions. Visualize vectors in 2D and 3D space and understand the geometric interpretation of vector magnitude with our comprehensive vector magnitude calculator.
Vector Input
Vector Magnitude Theory
What is Vector Magnitude?
Vector magnitude (also called vector length, norm, or modulus) is a scalar value that represents the "size" or "length" of a vector. It's the distance from the origin to the point represented by the vector in n-dimensional space. The magnitude is always non-negative and equals zero only for the zero vector.
Euclidean Norm (L2 norm):
||v|| = √(v₁² + v₂² + v₃² + ... + vₙ²)
For 2D vectors: ||v|| = √(x² + y²)
For 3D vectors: ||v|| = √(x² + y² + z²)
General n-dimensional: ||v|| = √(Σᵢ₌₁ⁿ vᵢ²)
Calculation Steps
Step 1: Square each component
Calculate v₁², v₂², v₃², ..., vₙ² for each vector component
Step 2: Sum the squares
Add all squared components: v₁² + v₂² + v₃² + ... + vₙ²
Step 3: Take square root
Calculate √(sum) to get the final magnitude
Step 4: Verify result
Check that the result is non-negative and makes geometric sense
Types of Vector Norms
L1 Norm (Manhattan)
||v||₁ = |v₁| + |v₂| + ... + |vₙ|
L2 Norm (Euclidean)
||v||₂ = √(v₁² + v₂² + ... + vₙ²)
L∞ Norm (Maximum)
||v||∞ = max(|v₁|, |v₂|, ..., |vₙ|)
Unit Vector
û = v / ||v|| (magnitude = 1)
Applications
Physics
Velocity magnitude, force magnitude, displacement distance
Computer Graphics
Distance calculations, normalization, lighting computations
Machine Learning
Feature normalization, distance metrics, regularization
Engineering
Signal processing, control systems, optimization
Navigation
GPS distance calculations, path planning
Data Science
Clustering algorithms, similarity measures
Worked Examples
Example 1: 2D Vector Magnitude
Problem:
Calculate the magnitude of vector v = [3, 4]
Solution:
Step 1: Square each component: 3² = 9, 4² = 16
Step 2: Sum the squares: 9 + 16 = 25
Step 3: Take square root: √25 = 5
Result: ||v|| = 5
Verification: This forms a 3-4-5 right triangle
Example 2: 3D Vector Magnitude
Problem:
Calculate the magnitude of vector v = [1, 2, 2]
Solution:
Step 1: Square each component: 1² = 1, 2² = 4, 2² = 4
Step 2: Sum the squares: 1 + 4 + 4 = 9
Step 3: Take square root: √9 = 3
Result: ||v|| = 3
Unit vector: û = [1/3, 2/3, 2/3]
Example 3: Higher Dimension
Problem:
Calculate the magnitude of vector v = [1, -2, 3, -4, 5]
Solution:
Step 1: Square each component: 1² + (-2)² + 3² + (-4)² + 5²
Step 2: Calculate: 1 + 4 + 9 + 16 + 25 = 55
Step 3: Take square root: √55 ≈ 7.416
Result: ||v|| ≈ 7.416
Note: Signs don't matter when squaring
Frequently Asked Questions
Magnitude and length are essentially the same concept for vectors. Both refer to the Euclidean norm (L2 norm) of the vector, which represents the distance from the origin to the point represented by the vector in n-dimensional space.
Squaring eliminates negative signs and ensures all contributions are positive. This gives us the true geometric distance. The square root then "undoes" the squaring to give us the actual magnitude in the original units.
A unit vector has magnitude 1 and points in the same direction as the original vector. Calculate it by dividing each component by the vector's magnitude: û = v / ||v||. Unit vectors are useful for representing direction without magnitude.
These are different ways to measure vector "size": L1 (Manhattan) sums absolute values, L2 (Euclidean) is the standard magnitude we usually mean, and L∞ (Maximum) takes the largest absolute component. Each has different geometric interpretations and applications.
No, vector magnitude is always non-negative (≥ 0). It represents a distance, which cannot be negative. The magnitude is zero only for the zero vector [0, 0, ..., 0].
Vector magnitude is crucial in ML for feature normalization, calculating distances between data points, regularization techniques (L1/L2 regularization), and in algorithms like k-means clustering and nearest neighbor classification.