Naj�t ortogon�ln� projekci znamen� naj�t ortogon�ln� dopln�k {$ U $}
prostoru {$ W $} v~ {$ \er_{3} $} a vektor {$ \vec v $} zapsat
(jednozna�n�)
 ve tvaru
{$ \vec v = \vec u +\vec w, \ \vec u \in  U, \vec w \in W.$} Vektory {$
 \vec u, \vec w$} jsou pak po �ad� ortogon�ln� projekce vektoru {$ \vec
 v $} do prostor� {$ U,\,W. $}
 \par
 V na�em p��pad� n�s nezaj�m� vektor {$ \vec u $} a nemus�me ho ani
 po��tat. Vyu�ijeme pouze toho, �e je kolm� k {$ W $} a tedy
  {$ (\vec u,\vec w_{1})=
     (\vec u,\vec w_{2})=0 $} a toho, �e vektor {$ \vec w $} je line�rn�
     kombinac� vektor�            {$ \vec w_{1}, \vec w_{2}. $}
Rovnici {$ \vec v = \vec u +\vec w =\vec u + \alpha \vec w_{1} + \beta
\vec w_{2}$} vyn�sob�me postupn� vektory  {$ \vec w_{1}, \vec w_{2} $}
 a z podm�nky {$ (\vec u,\vec w_{i}=0) $} dostaneme syst�m rovnic
 $$
   \align
            (\vec v,\vec w_{1})&=\alpha (\vec w_{1},\vec w_{1})+\beta (\vec  w_{2},w_{1})\\
            (\vec v,\vec w_{2})&=\alpha (\vec w_{1},\vec w_{2})+\beta (\vec  w_{2},w_{2})\\
   \endalign
$$
respektive po dosazen� za p��slu�n� skal�rn� sou�iny
$$
  \align
  17&=17\alpha +17\beta  \cr
  16&=17\alpha +19\beta  \cr
  \endalign
$$
    odkud {$ \alpha =\frac{3}{2},\beta =-\frac{1}{2}, \vec w
    =\frac{1}{2}(5,3,6). $}

Kontrola: spo�teme {$ \vec u = \vec v -\vec w $} a p�esv�d��me se, �e {$
(\vec u, \vec w_1)=
(\vec u, \vec w_2)=0$}.

�hel {$ \gamma  $} vektor� {$ \vec v, \vec w_{1} $} spo�teme ze
skal�rn�ho sou�inu: je {$ \cos \gamma =\frac{(\vec v,\vec w_{1})}{|\vec
v|\,|\vec w_{1}|} $}. Proto�e je  {$ (\vec v,\vec w_{1}) = 17 >0, $} je
�hel {$ \gamma  $} men�� ne� {$ \frac{\pi }{2} $}.



