Math
Unit Vector Calculator
Enter a vector and mathcheck divides it by its magnitude to produce the unit vector pointing the same way.
Components are shown exactly with rationalized denominators (such as 3√13/13) and as decimals, with the magnitude, the direction angle, the opposite unit vector, and a check that the result has length 1.
Horizontal component of Vector.
Vertical component of Vector.
Try an example
Result
Unit vector (exact)
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- Unit vector (decimal)
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- Magnitude of the original vector
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- Direction
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- Opposite unit vector
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- Length of the unit vector
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What a unit vector is
A unit A quantity with both a size and a direction, often drawn as an arrow. has The length or size of a vector, ignoring which way it points. exactly 1. To find the unit vector in the direction of v, divide every component of v by |v|. The result keeps the direction of v and throws away its length, which is why unit vectors are used to describe pure direction in physics, graphics, and geometry.
v = ⟨2, 3⟩
|v| = sqrt(4 + 9) = √13. u = ⟨2/√13, 3/√13⟩. Rationalize: ⟨2√13/13, 3√13/13⟩ ≈ ⟨0.5547, 0.8321⟩. Check: (2/√13)^2 + (3/√13)^2 = 4/13 + 9/13 = 1.
Why rationalize?
Textbooks write 2/√13 as 2√13/13 so there is no A root expression written with the √ symbol, such as a square root. in the The bottom number of a fraction; it says how many equal parts make one whole.. The calculator shows the rationalized form and reduces it, so ⟨6, 8⟩ becomes ⟨3/5, 4/5⟩, the same unit vector as ⟨3, 4⟩.
How to use this calculator
- Choose 2D or 3D.
- Enter the components of the vector.
- Read the exact unit vector, then the decimal version beneath it.
- Use the opposite unit vector when you need the direction reversed.
- Open Show the work to see the magnitude and the division, plus the length check.
How to read the answer
The unit vector points the same way as the original but has length 1; multiplying it by the original magnitude reproduces the original vector. Its components are the cosines of the angles the vector makes with each axis, so in 2D the unit vector is ⟨cos θ, sin θ⟩ where θ is the direction angle.
Common mistakes and edge cases
- Dividing by the sum of the components instead of the magnitude.
- Forgetting to square the components before adding when finding the magnitude.
- Leaving a radical in the denominator when the answer must be rationalized.
- Trying to normalize the zero vector; it has no direction.
- Assuming the unit vector has integer components. Only vectors along an axis do.
Worked examples
3-4-5 vector
v = ⟨3, 4⟩
Unit vector (exact)
⟨3/5, 4/5⟩
Radical components
v = ⟨2, 3⟩
Unit vector (exact)
⟨2√13/13, 3√13/13⟩
Diagonal
v = ⟨1, 1⟩
Unit vector (exact)
⟨√2/2, √2/2⟩
3D vector
v = ⟨1, 2, 2⟩
Unit vector (exact)
⟨1/3, 2/3, 2/3⟩
Negative components
v = ⟨-6, -8⟩
Unit vector (exact)
⟨-3/5, -4/5⟩
Decimal components
v = ⟨1.5, 2⟩
Unit vector (exact)
⟨0.6, 0.8⟩
Zero vector
Unit vector (exact)
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Frequently asked questions
How do you find a unit vector?+
Divide the vector by its magnitude: u = v / |v|. For ⟨3, 4⟩, |v| = 5 and u = ⟨3/5, 4/5⟩.
What is a unit vector used for?+
To describe a direction without a length. Physics uses unit vectors for directions of forces and fields, graphics uses them for surface normals, and geometry uses them for direction cosines.
Is the unit vector always a fraction?+
Usually. Unless the vector points along an axis, its components divided by the magnitude produce fractions or radicals such as √2/2.
How do I write the answer without a radical in the denominator?+
Multiply the numerator and denominator by the radical: 2/√13 = 2√13/13. The calculator does this automatically.
What is the unit vector of the zero vector?+
It does not exist. Dividing by a magnitude of 0 is undefined, and the zero vector has no direction.
How do I get a vector of length 5 in the direction of v?+
Multiply the unit vector by 5. In general, k × (v / |v|) has length |k| and the direction of v (reversed if k is negative).
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026