# Definition:Vector Length

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## Contents

## Definition

The **length** of a vector $V$ in a vector space $\left({G, +_G, \circ}\right)_K$ is defined as $\left\Vert{V}\right\Vert$, the norm of $V$.

## Examples

### Real Numbers

Given a vector $V$ in the real numbers, its length is defined as:

- $\left\Vert{V}\right\Vert = \left\vert{V}\right\vert$

the absolute value of $V$.

### Euclidean Space

A vector $V$ in Euclidean $n$-space has components $v_i$, $1 \le i \le n$, $i \in \N$.

Then its length is defined as:

- $\displaystyle \left\Vert{V}\right\Vert = \sqrt{\sum_{i \mathop = 1}^n v_i^2}$

### Complex Numbers

Given a vector $V$ in the complex plane where $V = a + b i$, its length is defined as:

- $\left\Vert{V}\right\Vert = \left\vert{V}\right\vert$

where $\left\vert{V}\right\vert = \sqrt{a^2 + b^2}$ is the modulus of $V$.

## Note

$\left\vert{V}\right\vert$ is sometimes also seen for the length of $V$, although this is not recommended since it can lead to confusion with absolute value.