Understand the subject
What is vector magnitude?
Magnitude is the Euclidean length of a vector: the straight-line size of all its components together.
Use it to turn component measurements into one distance, speed, or norm value.
See the structure
What the calculation is doing
input matrix→bounded numerical method→labelled approximation
Worked interpretation
Read the result in context
‖v‖=√Σᵢvᵢ²
Interpret with care
Important boundary
Magnitude is always non-negative and is zero only for the zero vector.
Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.
Quick guide
How to use this calculator
- Choose the number of components.
- Enter every signed component.
- Read an exact rational length when possible or a labelled approximation.
Calculation method
Square, add, and take the non-negative root
The squared magnitude is a · a. Its principal square root is exact only when the reduced numerator and denominator are both perfect squares.
Worked example
Length of a 3–4 vector
The squared magnitude of ⟨3,4⟩ is 9 + 16.
‖a‖ = √25 = 5
Supported inputs
Precision and limits
Vector size
General vectors may contain 2–8 components; three-dimensional products require exactly three.
Entry precision
Each signed decimal may contain up to 30 digits and 15 decimal places.
Square roots
Squared magnitudes remain exact. Irrational roots and normalized components are explicitly labelled approximations.
