Definition:Vector Length

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

Real Numbers
Given a vector $V$ in the real numbers, its length is defined as $\| V \| = |V|$, the absolute value of $V$.

Euclidean Space
A vector $V$ in Euclidean $n$-space has components $V_i$, $1\leq i \leq n$, $i\in\N$.

Then its length is defined as $\displaystyle \|V\|=\sqrt{\sum_{i \mathop = 1}^n V_i^2}$.

Complex Numbers
Given a vector $V$ in the complex numbers where $V = a + b i$, its length is defined as $\| V \| = |V| = \sqrt{a^2+b^2}$, where $|V|$ is the modulus of $V$.

Note
$|V|$ is sometimes also seen for the length of $V$, although this is not recommended since it can lead to confusion with absolute value.

Also see

 * Distance Formula