Unit Vector Calculator

A unit vector is a vector with a magnitude (length) of exactly 1. It points in the same direction as the original vector but is scaled so that its length equals 1. Unit vectors are often used to describe direction without regard to magnitude.

To find the unit vector, divide each component of the original vector by the vector's magnitude. For vector u = ⟨a, b, c⟩, the magnitude is |u| = √(a² + b² + c²), and the unit vector is ê = ⟨a/|u|, b/|u|, c/|u|⟩.

The magnitude of a vector is its length, calculated as the square root of the sum of the squares of its components. For a vector ⟨a, b⟩ in 2D, the magnitude is √(a² + b²). For ⟨a, b, c⟩ in 3D, it is √(a² + b² + c²).

Yes. Enter two comma-separated values for a 2D vector (e.g. 3, 4) or three comma-separated values for a 3D vector (e.g. 1, 2, 3). The calculator automatically detects the dimension based on the number of components you enter.

By definition, a unit vector is normalized to have a magnitude of exactly 1. This property makes unit vectors useful for representing pure direction in physics, engineering, and mathematics, without any scaling effect from the original vector's length.

No. The zero vector ⟨0, 0⟩ or ⟨0, 0, 0⟩ has a magnitude of 0, and dividing by zero is undefined. Therefore, the zero vector has no unit vector and the calculator will indicate an error if you enter it.

Unit vectors are commonly written with a hat symbol (ê or û), read as 'e-hat' or 'u-hat'. The standard basis unit vectors along the x, y, and z axes are denoted î, ĵ, and k̂ respectively.

Unit vectors are widely used in physics to express direction of forces, velocities, and fields. In computer graphics, they are used for surface normals and lighting calculations. In engineering, they simplify vector decomposition and projection problems.