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 of a nonzero vector v is the unit vector codirectional with v, i.e.,
Normalized vector is thus sometimes used as simply a synonym for unit vector.
The elements of a basis are often chosen to be unit vectors. In the 3dimensional Cartesian coordinate system, these are usually i, j, k, unit vectors along the x, y , z axes respectively. Thus:
\begin{bmatrix} 1 \\ 0 \\ 0\end{bmatrix}\quad \hat \mathbf{j} =
\begin{bmatrix} 0 \\ 1 \\ 0\end{bmatrix}\quad \hat \mathbf{k} =
\begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix}</math>
These are not always written with a hat; but it can generally be assumed that i, j, k are unit vectors in most contexts.
Other coordinate systems, such as polar coordinates or spherical coordinates use different unit vectors; notations vary.
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