Quick answer
A unit vector has length 1 and points in the direction of a vector: û = a ÷ |a|. For (3, 4), the magnitude is 5 and the unit vector is (0.6, 0.8).
Enter the vector components separated by commas. The calculator returns the unit vector and the magnitude.
Related tools: Vector Magnitude Calculator: Length (Euclidean norm) of a vector. Vector Projection Calculator: Projection of vector a onto vector b. Vector Subtraction Calculator: Subtract one vector from another.
How to use the Unit Vector Calculator
- Type vector, for example 1, 2, 3.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Vector | Text, such as 1, 2, 3 | — |
Formula and method
Formula
û = a ÷ |a|
IEEE-754 double precision; shown to at most 6 decimal places.
The zero vector has no direction.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Vector = 3, 4.
| Result | Value |
|---|---|
| Unit vector | (0.6, 0.8) |
| Magnitude | 5 |
The first line is the headline result (unit vector). Below it the unit Vector Calculator lists magnitude, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Normalising
Divide each component by the magnitude: 3 ÷ 5 = 0.6 and 4 ÷ 5 = 0.8. Check: 0.6² + 0.8² = 1. The result points the same way as the original.
Why it is useful
Unit vectors represent pure direction in physics and graphics, such as surface normals and light directions. The zero vector has no direction and can't be normalised.
Direction without size
The unit vector of (3, 4) is (0.6, 0.8), since the magnitude is 5 and 0.6² + 0.8² = 1. For (1, 1) it is about (0.7071, 0.7071).
The zero vector has no direction, so it cannot be turned into a unit vector: that would mean dividing by a magnitude of 0.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Vector: entering "0, 0" shows "The zero vector has no direction." beside that field and no result.
Choosing between the Unit Vector Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Vector Magnitude Calculator](/math/vector-magnitude/) | components | Magnitude |
| [Vector Projection Calculator](/math/vector-projection-calculator/) | vector A and vector B | Projection |
| [Vector Subtraction Calculator](/math/vector-subtraction-calculator/) | vector A and vector B | a − b |
| [Vector Calculator](/math/vector/) | vector a and vector b | Sum (a + b) |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Non-zero vector
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for vector to confirm this.
Frequently asked questions
What is the unit vector of (0, 5)?
(0, 5) has magnitude 5, so its unit vector is (0, 1).
What is î?
The unit vector along the x-axis, (1, 0).
Does it work in 3D?
Yes, with three components.
Can the zero vector be normalised?
No. Its magnitude is 0, so dividing by it is undefined.
What is the unit vector for (1, 1)?
Its magnitude is √2, so the unit vector is (1/√2, 1/√2), about (0.7071, 0.7071).
What does the Unit Vector Calculator calculate?
Unit vector in the direction of a vector. It returns Unit vector and Magnitude.
What formula does the Unit Vector Calculator use?
û = a ÷ |a|. The zero vector has no direction.
What do I need to enter in the unit Vector Calculator?
You enter Vector.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Non-zero vector.
What is an example of a unit Vector result?
With vector 3, 4, the unit vector is (0.6, 0.8).
Related calculators
- Vector Magnitude Calculator — Length (Euclidean norm) of a vector.
- Vector Projection Calculator — Projection of vector a onto vector b.
- Vector Subtraction Calculator — Subtract one vector from another.
- Vector Calculator — Sum, difference, dot product, cross product and angle of two vectors.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
Sources
- OpenStax, Precalculus 2e (Rice University): vectors, polar coordinates and trigonometry
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.