Find the orthogonal projection of a vector

**cristi92** · December 17th 2011, 11:29 AM

Find the orthogonal projection of $\vec{v}$ on the $span[\vec{v_{1}},\vec{v_{2}},\vec{v_{3}}]$

$\vec{v}=(6,3,9,6) , \vec{v_{1}}=(2,1,1,1) , \vec{v_{2}}=(1,0,1,1) , \vec{v_{3}}=(-2,-1,0,-1) ;$

$\vec{v},\vec{v_{1}},\vec{v_{2}},\vec{v_{3}} \in \mathbb{R}^{4} .$

Can you please tell me how to do this?

**FernandoRevilla** · December 17th 2011, 11:58 AM

One way: find an orthonormal basis $\{e_1,e_2,e_3\}$ of $\textrm{Span}[v_1,v_2,v_3]$ . The solution is $p=<v,e_1>e_1+<v,e_2>e_2+<v,e_3>e_3$ .

**ILikeSerena** · December 17th 2011, 12:17 PM

Another way: find a vector $\vec w$ perpendicular to each of $\vec v_1, \vec v_2, \vec v_3$ .

The solution is $\vec p = \vec v - {\langle \vec v,\vec w \rangle \vec w \over \langle \vec w,\vec w \rangle}$ .

**cristi92** · December 17th 2011, 12:46 PM

Thank you!

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