Definition:Vector Length/Examples

Examples of Vector Length

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 $z$.