The formula this page uses
û = v / |v|, where |v| = √(v₁² + v₂² + v₃²)
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Unit Vector Calculator
Solve unit vector problems with clear steps, notation, and a final check.
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Calculate without using AI.
Evaluate the governing formula locally in your browser. Define each known quantity once, then change values to test another case instantly.
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Vectors
Unit Vector Calculator explained
The short version
- A unit vector points the same way as the original but has a length of exactly 1.
- You get it by dividing every component by the vector's own length.
- Because its length is 1, a unit vector carries direction and nothing else.
What each part means
SymbolWhat it means
v — The original vectorIts components can be any units: metres, newtons, or plain numbers.
|v| — MagnitudeThe length of v, always positive, in the same unit as the components.
û — Unit vectorSame direction, length 1, and no unit at all because the units cancel in the division.
cos α, cos β, cos γ — Direction cosinesThe three components of û. Each is the cosine of the angle between v and one axis.
Show your work: a full example
- The vectorv = (3, −6, 6)
- Square each component3² = 9, (−6)² = 36, 6² = 36
- Add and take the square root|v| = √(9 + 36 + 36) = √81 = 9
- Divide every component by 9û = (3/9, −6/9, 6/9)
- Simplifyû = (1/3, −2/3, 2/3) ≈ (0.3333, −0.6667, 0.6667)
- Check the length is 1√(1/9 + 4/9 + 4/9) = √(9/9) = √1 = 1
- Read the direction anglescos α = 1/3 gives α = 70.53° from the x-axis, and cos γ = 2/3 gives γ = 48.19° from the z-axis
A second, different case
- A different case: a 2D vector and its compass anglev = (−5, 12)
- Magnitude√(25 + 144) = √169 = 13
- Unit vectorû = (−5/13, 12/13) ≈ (−0.3846, 0.9231)
- Find the angle from the positive x-axisθ = 112.62°, in the second quadrant because x is negative and y is positive
- Confirm from the angle(cos 112.62°, sin 112.62°) = (−0.3846, 0.9231), matching û exactly
- Now try the zero vectorv = (0, 0, 0) has |v| = 0
- The division fails0/0 is undefined, so the zero vector has no unit vector and no direction
Copy-ready example
v = (3, −6, 6)
The vector
Questions about the Unit Vector Calculator
What is a unit vector actually for?
It separates direction from size. A force of 90 N along (3, −6, 6) is just 90 × û, and a directional derivative needs the direction alone with the length stripped out, which is exactly what û supplies.
What are direction cosines?
They are the three components of û, and each one is the cosine of the angle between v and an axis. For (3, −6, 6) they are 1/3, −2/3 and 2/3, giving axis angles of 70.53°, 131.81° and 48.19°. Their squares always add to 1.
Does the zero vector have a unit vector?
No. Its length is 0, and dividing by 0 has no answer. That matches the geometry: the zero vector has no direction to preserve, so any code doing this needs a guard before it divides.
Is a unit vector the same thing as a normalized vector?
Yes, they are two words for the identical object. 'Unit vector' names the result and 'normalize' names the action of producing it, which is why i, j and k are the standard unit vectors along the three axes.