Unit Vector Calculator
Normalize a 2D or 3D vector: get the unit vector û = v/|v|, the magnitude, the exact form with a simplified root, and the direction angles with each axis.
How it works
|v| = √(x² + y² + z²) · û = v / |v| = (x/|v|, y/|v|, z/|v|) · cos α = x/|v|, cos β = y/|v|, cos γ = z/|v|
A unit vector has length 1 and points in the same direction as the original vector. To normalize a vector, divide each component by its magnitude (Euclidean length). The components of the unit vector are the direction cosines, so the angle with each axis is the arccosine of that component. The zero vector cannot be normalized because it has no direction.
Worked example
For v = (3, 4, 0): |v| = √(9 + 16) = 5, so û = (3/5, 4/5) = (0.6, 0.8). The direction angles are arccos 0.6 = 53.1301° with the x-axis and arccos 0.8 = 36.8699° with the y-axis, and |û| = 1.
Sources: Unit vector — Wikipedia · Vectors in Three Dimensions — OpenStax Calculus Volume 3, 2.2
Frequently asked questions
How do you find a unit vector?
Divide the vector by its magnitude: û = v/|v|. For (1, 2, 2), |v| = 3 and û = (1/3, 2/3, 2/3).
What is a unit vector used for?
To describe a pure direction: normals in geometry and graphics, directions of forces in physics, and the basis vectors i, j, k.
Does a unit vector have units?
No. Dividing a vector by its own magnitude cancels the units, leaving a dimensionless direction.
How do I find a vector of a given length in the same direction?
Multiply the unit vector by the length you want. A length-10 vector along (3, 4) is 10 × (0.6, 0.8) = (6, 8).
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This calculator is for educational purposes. Double-check important results before acting on them.