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 $$\|V\|=\sqrt{\sum_{i=1}^n V_i^2}$$.

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.