Vector Norms

You can compute vector norms in Scientific Notebook for every positive integer n, where

$\displaystyle \Vert$v$\displaystyle \Vert_{{n}}^{}$ = $\displaystyle \left(\vphantom{ \sum \left\vert v_{i}\right\vert ^{n}}\right.$$\displaystyle \sum$$\displaystyle \left\vert\vphantom{ v_{i}}\right.$vi$\displaystyle \left.\vphantom{ v_{i}}\right\vert^{{n}}_{}$$\displaystyle \left.\vphantom{ \sum \left\vert v_{i}\right\vert ^{n}}\right)^{{% \frac{1}{n}}}_{}$

and for ∞, with

$\displaystyle \left\Vert\vphantom{ v}\right.$v$\displaystyle \left.\vphantom{ v}\right\Vert _{{\infty }}^{}$ = max$\displaystyle \left(\vphantom{ \left\vert v_{i}\right\vert }\right.$$\displaystyle \left\vert\vphantom{ v_{i}}\right.$vi$\displaystyle \left.\vphantom{ v_{i}}\right\vert$$\displaystyle \left.\vphantom{ \left\vert v_{i}\right\vert }\right)$

as illustrated by the following examples.


$\blacktriangleright$ Evaluate

$\Vert$$\left(\vphantom{ a,b,c}\right.$a, b, c$\left.\vphantom{ a,b,c}\right)$$\Vert_{{1}}^{}$ = $\left\vert\vphantom{ a}\right.$a$\left.\vphantom{ a}\right\vert$ + $\left\vert\vphantom{ b}\right.$b$\left.\vphantom{ b}\right\vert$ + $\left\vert\vphantom{ c}\right.$c$\left.\vphantom{ c}\right\vert$ 6pt

$\Vert$$\left(\vphantom{ a,b,c}\right.$a, b, c$\left.\vphantom{ a,b,c}\right)$$\Vert_{{3}}^{}$ = $\sqrt[3]{{\left( \left\vert a\right\vert ^{3}+\left\vert b\right\vert ^{3}+\left\vert c\right\vert ^{3}\right) }}$ 6pt

$\Vert$$\left(\vphantom{ -1,2,1}\right.$ -1, 2, 1$\left.\vphantom{ -1,2,1}\right)$$\Vert_{{5}}^{}$ = $\sqrt[5]{{34}}$ 6pt

$\Vert$$\left[\vphantom{ a,b,c}\right.$a, b, c$\left.\vphantom{ a,b,c}\right]$$\Vert_{{\infty }}^{}$ =  max$\left(\vphantom{ \left\vert a\right\vert ,\left\vert b\right\vert ,\left\vert c\right\vert }\right.$$\left\vert\vphantom{ a}\right.$a$\left.\vphantom{ a}\right\vert$,$\left\vert\vphantom{ b}\right.$b$\left.\vphantom{ b}\right\vert$,$\left\vert\vphantom{ c}\right.$c$\left.\vphantom{ c}\right\vert$$\left.\vphantom{ \left\vert a\right\vert ,\left\vert b\right\vert ,\left\vert c\right\vert }\right)$ 6pt

$\Vert$$\left[\vphantom{ 8,-10,2}\right.$8, - 10, 2$\left.\vphantom{ 8,-10,2}\right]$$\Vert_{{\infty }}^{}$ = 10 6pt

$\left\Vert\vphantom{ \left( \begin{array}{c} a \\ b \end{array} \right) }\right.$$\left(\vphantom{ \begin{array}{c} a \\ b \end{array} }\right.$$\begin{array}{c} a \\ b \end{array}$$\left.\vphantom{ \begin{array}{c} a \\ b \end{array} }\right)$$\left.\vphantom{ \left( \begin{array}{c} a \\ b \end{array} \right) }\right\Vert _{{2}}^{}$ 6pt = $\sqrt{{\left( \left\vert a\right\vert ^{2}+\left\vert b\right\vert ^{2}\right) }}$

$\left\Vert\vphantom{ \left( \begin{array}{ccc} a & b & c \end{array} \right) }\right.$$\left(\vphantom{ \begin{array}{ccc} a & b & c \end{array} }\right.$$\begin{array}{ccc} a & b & c \end{array}$$\left.\vphantom{ \begin{array}{ccc} a & b & c \end{array} }\right)$$\left.\vphantom{ \left( \begin{array}{ccc} a & b & c \end{array} \right) }\right\Vert _{{4}}^{}$ = $\sqrt[4]{{\left( \left\vert a\right\vert ^{4}+\left\vert b\right\vert ^{4}+\left\vert c\right\vert ^{4}\right) }}$


$\blacktriangleright$ To enter the two pairs of vertical lines used in the norm symbol

Before doing the next set of examples, make the following definition.


$\blacktriangleright$ Define + New Definition

v = [3, 2, 1]


$\blacktriangleright$ Evaluate, Evaluate Numerically

$\left\Vert\vphantom{ v}\right.$v$\left.\vphantom{ v}\right\Vert _{{1}}^{}$ =  6 6pt

$\left\Vert\vphantom{ v}\right.$v$\left.\vphantom{ v}\right\Vert _{{2}}^{}$ =  $\sqrt{{14}}$ =  3.7417 6pt

$\left\Vert\vphantom{ v}\right.$v$\left.\vphantom{ v}\right\Vert _{{6}}^{}$ =  $\sqrt[6]{{794}}$ =  3.043 6pt

$\left\Vert\vphantom{ v}\right.$v$\left.\vphantom{ v}\right\Vert _{{20}}^{}$ =  $\sqrt[20]{{34878\,32978}}$ =  3.000045103 6pt

$\left\Vert\vphantom{ v}\right.$v$\left.\vphantom{ v}\right\Vert _{{\infty }}^{}$ =  3


This series of examples suggests that for a vector v,

$\displaystyle \lim_{{n\rightarrow \infty }}^{}$$\displaystyle \left\Vert\vphantom{ v}\right.$v$\displaystyle \left.\vphantom{ v}\right\Vert _{n}^{}$ = $\displaystyle \left\Vert\vphantom{ v}\right.$v$\displaystyle \left.\vphantom{ v}\right\Vert _{\infty}^{}$

The default $\left\Vert\vphantom{ v}\right.$v$\left.\vphantom{ v}\right\Vert$ is the 2-norm, which is also known as the Euclidean norm.




\begin{example} The area of the parallelogram in the plane with vertices $\left... ...) \times \left( 2,1,0\right) \right\Vert =\,3 \end{displaymath} \end{example}

Since

A⋅B = $\displaystyle \left\Vert\vphantom{ A}\right.$A$\displaystyle \left.\vphantom{ A}\right\Vert$$\displaystyle \left\Vert\vphantom{ B}\right.$B$\displaystyle \left.\vphantom{ B}\right\Vert$cosθ

where θ is the angle between the vectors A and B, you can use the dot product to find the angle between two vectors.



\begin{example} Define $A=\left( 1,2,-3\right) $\ and $B=\left( -2,1,2\right) $... ...\textsf{Evaluate Numerically} to get $\theta =\,2.1347.\medskip $ \end{example}