Vector norm
5
Euclidean length of the result vector.
Mathematics · Vectors & matrices
Vector calculator — norm, sum and unit vector
Calculate norms, vector sums, differences, scalar products kA and unit vectors in 2D or 3D.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
03
Results
Results show up to four significant digits for readability; internal calculations retain full precision.
See calculation details
01Vector norm
Formula
‖A‖ = √(∑ᵢ Aᵢ²)Substitution
√(3² + 4²)Result
5
01Formulas and symbols
Formulas used
v⃗
‖A‖ = √(∑ Aᵢ²)
A+B
A−B
kA
 = A/‖A‖
A and B: vectors; k: scalar; ‖A‖: norm; Â: unit vector.
| Symbol | Meaning |
|---|---|
| d | Dimension |
| Aₓ | Vector A — ₓ |
| Aᵧ | Vector A — ᵧ |
| A_z | Vector A — 𝓏 |
| Bₓ | Vector B — ₓ |
| Bᵧ | Vector B — ᵧ |
| B_z | Vector B — 𝓏 |
| k | Scalar k |
| ‖v‖ | Vector norm |
| v⃗ | Result vector |
02Validation example
Reference numerical case
- A = (3, 4) has norm 5 and unit vector (0.6, 0.8). A + (1, 0) = (4, 4); multiplying A by 2 gives (6, 8).
03How it works
Addition, subtraction and scalar multiplication act component by component. A unit vector divides each component by the Euclidean norm; it preserves direction and has norm one.
04Limits and conventions
Vectors must have the same dimension, either 2 or 3. The zero vector has no unit direction. 3D drawings use an explicitly labelled projection; a vector along the viewing direction may project to a point.
05Common mistakes
Multiplying by a negative scalar reverses direction. The norm of A+B generally differs from ‖A‖+‖B‖. For A·B or A×B, use the related dedicated tool.
06Practical questions
What is the difference between a vector’s magnitude and its components?
The components describe the vector along the chosen coordinate axes, while the magnitude is its length. In Cartesian coordinates, ‖v‖ = √(v₁² + v₂² + …).
How do you add or subtract vectors using components?
Add or subtract corresponding components independently: A ± B = (Aₓ ± Bₓ, Aᵧ ± Bᵧ, …). Both vectors must use the same coordinate system and dimension.