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Free unit vector calculator and normalize vector calculator for 2D and 3D vectors. Enter components, get magnitude ‖v‖ and the direction vector û = v/‖v‖ with clear formulas—ideal for linear algebra, mechanics, and computer graphics.
Last updated: April 13, 2026
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Original v
(3, 4)
Magnitude ‖v‖
5
Unit vector û
(0.6, 0.8)
Formula
û = v / ‖v‖ = (x, y) / √(x² + y²)
Interpretation
The unit vector û points in the same direction as v and satisfies ‖û‖ = 1. It is found by dividing each component by the magnitude ‖v‖.
Quick rules
Magnitude
‖v‖ = √(x² + y²)
Common in kinematics, complex numbers as vectors, and screen-space directions.
Magnitude
‖v‖ = √(x² + y² + z²)
Used for cross products, normals to planes, and 3D physics.
Core formula
û = v / ‖v‖
Same as “normalizing” or finding the versor of v.
Also called
L2 norm
‖v‖ appears in distance formulas and dot product identities.
Property
‖û‖ = 1
Useful when only direction matters, not how fast or how far.
Pairs with
Projection & dot product
Combine with our vector addition and multiplication calculators below.
For v = (3, 4) in 2D:
‖v‖
5
û
(0.6, 0.8)
Check: 0.6² + 0.8² = 1 — classic 3-4-5 triangle scaled to unit hypotenuse.
The calculator reads your components, evaluates the Euclidean magnitude using the Pythagorean pattern in 2D or 3D, then divides each component by that positive length. That is exactly the definition of the unit vector in the direction of v, whenever v is not the zero vector.
û = v / ‖v‖ , ‖v‖ = √(x² + y² (+ z²))Each component of û is the corresponding component of v divided by the same scalar ‖v‖, so direction is preserved and the new length becomes 1.
v scales down to û on the unit sphere (3D) or unit circle (2D)
Unit vectors form a convenient basis for expressing directions. The set of all unit vectors in ℝ³ is the sphere of radius 1. Multiplying û by a scalar s gives a vector of length |s| in the same (or opposite) direction.
Any positive multiple of (3, 4)—such as (30, 40) or (6, 8)—has the same unit vector (0.6, 0.8).
û = (0.6, 0.8), ‖û‖ = 1
v = (1, 0, 0) → û = (1, 0, 0)
Already a unit vector
v = (−3, −4)
û = (−0.6, −0.8) — opposite direction
Share it with classmates working on vectors
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