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Unit vector

A unit vector in a normed vector space is a vector whose length is 1. A unit vector is often written with a "hat" thus: .

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 non-zero vector v is the unit vector codirectional with v, i.e.,

<math>\hat\mathbf{v} = \frac{1}{||\mathbf{v}||} \mathbf{v}</math>

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 3-dimensional Cartesian coordinate system, these are usually i, j, k, unit vectors along the x, y , z axes respectively. Thus:

<math>\hat \mathbf{i} =
 \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}

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 co-ordinate systems, such as polar coordinates or spherical coordinates use different unit vectors; notations vary.

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