Normalizing a vector
The unit vector û of vector v is v/|v|, where |v| = √(v₁² + v₂² + v₃²).
The zero vector cannot be normalized because it has no direction.
û = v / |v|
Unit vector (normalization) formula.
Unit Vector Calculator
Normalize a 2D or 3D vector to find its unit vector (magnitude 1) with step-by-step work.
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Normalize
Unit Vector Calculator
Enter a vector to find its unit vector (direction with magnitude 1).
Dimension
What is a unit vector?
A unit vector points in the same direction as the original vector but has a magnitude of exactly 1. It is computed by dividing each component by the vector's magnitude.
When is it undefined?
The zero vector (all components equal to zero) has no direction and cannot be normalized to a unit vector.
Unit vector
(0.6, 0.8)
- X component
- 0.6
- Y component
- 0.8
- Original magnitude
- 5
Step-by-step
Sum of squares
3² + 4²
= 9 + 16 = 25
Magnitude |v|
√(25)
= 5
Unit vector
3 / 5, 4 / 5
= (0.6, 0.8)
Verification
The magnitude of the resulting unit vector is 1, confirming it is a valid unit vector.
Also in Linear Algebra
Linear Algebra
Unit vector calculator: normalize a vector to length 1
A unit vector calculator divides each component of a vector by its magnitude to produce a vector of length 1 pointing in the same direction.
Frequently asked questions
Unit vectors isolate direction from magnitude, used in physics to describe directions of forces and velocities.
By definition, a unit vector always has a magnitude of exactly 1.
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