# Definition:Vector Length

## Definition

The length of a vector $\mathbf v$ in a normed vector space $\struct {V, \norm {\, \cdot \,} }$ is defined as $\norm {\mathbf v}$, the norm of $\mathbf v$.

### Arrow Representation

Let $\mathbf v$ be a vector quantity represented as an arrow in a real vector space $\R^n$.

The length of $\mathbf v$ is the length of the line segment representing $\mathbf v$ in $\R^n$.

## Examples

### Real Number Line

Let $\mathbf v$ be a vector represented by an arrow on the real number line.

Let:

the initial point of $\mathbf v$ be $a \in \R$
the terminal point of $\mathbf v$ be $b \in \R$

The length of $\mathbf v$ is defined as:

$\norm {\mathbf v} = \size {b - a}$

the absolute value of $b - a$.

### Real Vector Space

Let $\mathbf v$ be a vector represented in the real $n$-space $\R^n$ by an ordered $n$-tuple of components $\tuple {v_1, v_2, \ldots, v_n}$.

The length of $\mathbf v$ is defined as:

$\norm {\mathbf v} = \ds \sqrt {\sum_{i \mathop = 1}^n v_i^2}$

### Complex Plane

Let $\mathbf v$ be a vector represented in the complex plane $\C$ by the complex number $z = a + b i$.

The length of $\mathbf v$ is defined as:

$\norm {\mathbf v} = \cmod z$

where $\cmod z = \sqrt {a^2 + b^2}$ is the modulus of $z$.

## Also denoted as

$\size {\mathbf v}$ is often also seen for the length of $\mathbf v$.

Some authorities state that this is not recommended since it can lead to confusion with absolute value.

$\mathsf{Pr} \infty \mathsf{fWiki}$ is less convinced that this is a problem, and both notations can be found here.

It is commonplace to use the non-bold italic form of the symbol for the length, hence using $v$ for the length of $\mathbf v$.

Some sources use $\bmod \mathbf v$ for the length of $\mathbf v$.

## Also known as

The length of a vector is also referred to as its module in some older books.