So the question is:
Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} \frac{1}{\sqrt{2}}\\\frac{-1}{\sqrt{2}}\\0\end{pmatrix}$. Let $x=(1,2,2)^T$. Find the projection p of x onto S. Show that $(p-x)\perp u_2$ and $(p-x)\perp u_2$.
I understand how to show that they are perpendicular and I actually found the answer for the projection. It's:
$\begin{pmatrix} \frac{23}{18}\\\frac{41}{18}\\\frac{8}{9} \end{pmatrix}$
The problem is, i have no idea why i am doing what I am doing, I just followed my notes. Can someone explain why I was supposed to do:
$(xu_2)u_2 + (xu_3)u_3$
To find p.