Since a vector space $V$ over a field $F$ is defined with scalar multiplication, given a norm, one can associate to any non-null vector a unit vector that may be computed by multiplying the original vector by the inverse of its norm.

A typical example would be the 3-dimensional vector field $\mathbb{R}^3$ with its canonical basis $(\hat{x}, \hat{y}, \hat{z})$, for which we defined the Euclidean norm. A vector $\vec{v} = x\hat{x}+y\hat{y}+z\hat{z}\in\mathbb{R}^3$ would have length
$$
    ||\vec{v}|| = \sqrt{x^2+y^2+z^2}.
$$

The unit vector associated to $\vec{v}$ would be
$$
    \hat{v} = \frac{\vec{v}}{||\vec{v}||}.
$$

One can easily check that the norm of $\hat{v}$ is $1$.
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