# Thread: Find the orthogonal projection of a vector

1. ## Find the orthogonal projection of a vector

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

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

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

Can you please tell me how to do this?

2. ## Re: Find the orthogonal projection of a vector

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

3. ## Re: Find the orthogonal projection of a vector

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

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

Thank you!