# Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length is 1. A unit vector is often written with a “hat”, thus: î.

In Euclidean space, the dot product of two unit vectors is simply the cosine of the angle between them. This follows from the formula for the dot product, since the lengths are both 1.

The normalized vector û of a non-zero vector u is the unit vector codirectional with u, i.e.,

${\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{\|\mathbf {u} \|}}.}$

where ||u|| is the norm (or length) of u. The term normalized vector is sometimes used simply as a synonym for unit vector.

As opposed to a general vector, which represents direction and magnitude, a unit vector just represents direction. The components are called direction cosines, because each is the cosine of the angle between the vector and one coordinate axis.

The elements of a basis are often chosen to be unit vectors. In the 3-Dimensional Cartesian coordinate system, these are usually i, j, and k—unit vectors along the x, y, and z axes, respectively:

 ${\displaystyle \mathbf {\hat {i}} ={\begin{bmatrix}1\\0\\0\end{bmatrix}}}$ ${\displaystyle \mathbf {\hat {j}} ={\begin{bmatrix}0\\1\\0\end{bmatrix}}}$ ${\displaystyle \mathbf {\hat {k}} ={\begin{bmatrix}0\\0\\1\end{bmatrix}}}$

These are not always written with a hat; but it can generally be assumed that i, j, and k are unit vectors in most contexts. Very often these unit vectors are written as e1, e2 and e3 respectively.

Other coordinate systems, such as polar coordinates or spherical coordinates use different unit vectors; notations vary.