Definition:Vector Length

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

Real Numbers
Given a vector $V$ in the real numbers, its length is defined as:
 * $\norm V = \size V$

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:
 * $\norm V = \displaystyle \sqrt {\sum_{i \mathop = 1}^n v_i^2}$

Complex Numbers
Given a vector $V$ in the complex plane where $V = a + b i$, its length is defined as:
 * $\norm V = \cmod V$

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

Note
$\size V$ 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