Module 1: Magnitude of a Vector

The last useful computation included in this module is magnitude. The magnitude (or length or norm) of a vector $\vec{v} \in \mathbb{R} ^n$ is

$\|\vec{v} \| = \left\| \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix} \right\|:= \left( \sum_{i=1}^n v_i^2 \right)^{\frac{1}{2}}.$

For example: 

$\left\| \begin{pmatrix} 3 \\ 4 \end{pmatrix} \right\| = \left( 3^2 + 4^2 \right)^{\frac{1}{2}} = \sqrt{25} = 5.$

The magnitude computation is important because it allows us to define the distance between two vectors: Let $\vec{u}$ and $\vec{v}$ be vectors. Then the distance between them is given by

$\|\vec{u} - \vec{v} \|.$

Using this computation we can argue that the smaller the value of $\|\vec{u} - \vec{v} \|$ then the "closer'' the vectors $\vec{u}$ and $\vec{v}$ are to each other (or alternatively, how well the vector $\vec{u}$ approximates $\vec{v}$). For example, let's find a vector of the form

$\begin{pmatrix} 1 \\ t \\ 2 \end{pmatrix}$

which is as close as possible to

$\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}.$

We compute

$\left\| \begin{pmatrix} 1 \\ t \\ 2 \end{pmatrix} - \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} \right\| = \left\| \begin{pmatrix} 1 \\ t-1 \\ 1 \end{pmatrix} \right\| = \left( (t-1)^2 + 1+1 \right)^{\frac{1}{2}}.$

This is them minimized when $t=1$.