Unit vector

In mathematics, a unit vector in a normed vector space is a vector (most commonly 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.,

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

The term normalized vector is sometimes used simply as a synonym for unit vector.

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:

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

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

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

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.

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

Categories: Linear algebra