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 $1$.
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 $1$.
user
In the case of the Euclidean norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^1$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^2$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^3$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^4$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^5$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^6$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^7$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^8$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^9$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^10$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^11$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^12$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^13$ norm, the unit vector associated to a vector $\vec{v
assistant
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 $1$.
user
## Unit vectors in $\mathbb{R}^3$

In $\mathbb{R}^3$, there are three unit vectors that are orthogonal to each other:

- $\hat{x}$
- $\hat{y}$
- $\hat{z}$

These three vectors are the basis of the unit sphere.

## Unit vectors in $\mathbb{R}^2$

In $\mathbb{R}^2$, there is only one unit vector:

- $\hat{x}$

This vector is the basis of the unit circle.

## Unit vectors in $\mathbb{R}$

In $\mathbb{R}$, there is only one unit vector:

- $\hat{x}$

This vector is the basis of the unit interval.

## Unit vectors in $\mathbb{C}$

In $\mathbb{C}$, there is only one unit vector:

- $\hat{x}$

This vector is the basis of the unit circle.

## Unit vectors in $\mathbb{R}^n$

In $\mathbb{R}^n$, there are $n$ unit vectors:

- $\hat{x}_1$
- $\hat{x}_2$
- $\cdots$
- $\hat{x}_n$

These $n$ vectors are the basis of the unit sphere.

## Unit vectors in $\mathbb{C}^n$

In $\mathbb{C}^n$, there are $n$ unit vectors:

- $\hat{x}_1$
- $\hat{x}_2$
- $\cdots$
- $\hat{x}_n$

These $n$ vectors are the basis of the unit sphere.

## Unit vectors in $\mathbb{R}^n$

In $\mathbb{R}^n$, there are $n$ unit vectors:

- $\hat{x}_1$
- $\hat{x}_2$
- $\cdots$
- $\hat{x}_n$

These $n$ vectors are the basis of the unit sphere.

## Unit vectors in $\mathbb{C}^n$

In $\mathbb{C}^n$, there are $n$ unit vectors:

- $\hat{x}_1$
- $\hat{x}_2$
- $\cdots$
- $\hat{x}_n$

These $n$ vectors are the basis of the unit sphere.

## Unit vectors in $\mathbb{R}^n$

In $\mathbb{R}^n$, there are $n$ unit vectors:

- $\hat{x}_1$
- $\hat{x}_2$
- $\cdots$
- $\hat{x}_n$

These $n$ vectors are the basis of the unit sphere.

## Unit vectors in $\mathbb{C}^n$

In $\mathbb{C}^n$, there are $n$ unit vectors
assistant
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 $1$.
user
In the case of the Euclidean norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^1$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^2$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^3$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^4$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^5$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^6$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^7$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^8$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^9$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^10$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^11$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^12$ norm, the unit vector associated to a vector $\vec{v}$ is the vector that points in the same direction as $\vec{v}$ but has length $1$.

In the case of the $L^13$ norm, the unit vector associated to a vector $\vec{v
assistant