University of Florida/Egm4313/s12.team11.gooding.k/R7

Scalar Product

$<f,g>=\underset{a}{\overset{b}{\int }}f(x)g(x)dx$

$<f,g>=\underset{-2}{\overset{10}{\int }}x\cos (x)dx$

Using integration by parts;

$<f,g>=[x\sin (x)+\cos (x){]}_{-2}^{10}$

                      $<f,g>=-7.68$

Magnitude

$\Vert f\Vert =<f,f{>}^{1/2}=\underset{a}{\overset{b}{\int }}{f}^2(x)dx$

$=\underset{-2}{\overset{10}{\int }}[\cos (x){]}^{2}dx$

$=[.5(x+\sin (x)\cos (x){|}_{-2}^{10}{]}^{1/2}$

                       $\Vert f\Vert =2.457$

$\Vert g\Vert =\underset{a}{\overset{b}{\int }}{g}^2(x)dx$

$=\underset{-2}{\overset{10}{\int }}{x}^{2}dx$
$=[x^3/3{]}_{-2}^{10}$

                       $\Vert g\Vert =\frac{1008}{3}$

Angle Between Functions

$cos(\theta )=\frac{<f,g>}{\Vert f\Vert \Vert g\Vert }$

$cos(\theta )=\frac{-7.68}{\frac{1008}{3}(2.457)}$

                      $\theta =89.47$

                      The two functions are nearly orthogonal.