<s>[INST] How to find orthogonal projection of vector on a subspace? Well, I have this subspace: $V = \operatorname{span}\left\{ \begin{pmatrix}\frac{1}{3} \\\frac{2}{3} \\\frac{2}{3}\end{pmatrix},\begin{pmatrix}1 \\3 \\4\end{pmatrix}\right\}$
and the vector $v = \begin{pmatrix}9 \\0 \\0\end{pmatrix}$  
How can I find the orthogonal projection of $v$ on $V$?
This is what I did so far:  

\begin{align}&P_v(v)=\langle v,v_1\rangle v_1+\langle v,v_2\rangle v_2 =\\=& \left\langle\begin{pmatrix}9 \\0 \\0\end{pmatrix},\begin{pmatrix}\frac{1}{3} \\\frac{2}{3} \\\frac{2}{3}\end{pmatrix}\right\rangle\begin{pmatrix}\frac{1}{3} \\\frac{2}{3} \\\frac{2}{3}\end{pmatrix}+\left<\begin{pmatrix}9 \\0 \\0\end{pmatrix},\begin{pmatrix}1 \\3 \\4\end{pmatrix}\right>\begin{pmatrix}1 \\3 \\4\end{pmatrix} = \begin{pmatrix}10 \\29 \\38\end{pmatrix}\end{align}  

Is this the right method to compute this?
 [/INST] Hint
You have to construct by the Gram Schmidt procedure an orthonormal basis $(e_1,e_2)$ from the given basis of $V$ and then
$$P_v(V)=\langle v,e_1\rangle e_1+\langle v,e_2\rangle e_2$$
</s>