I am given with the inner product, $$\phi(a,b) = a_1b_1+a_2b_3 + a_3b_2$$ where $a=(a_1,a_2,a_3)\text{ and } b= (b_1,b_2,b_3)\in \mathbb{R}^{3}.$ Consider the vector space $F = \text{span}(1,1,1).$ Then I want to find the $P_{F}$ or the projection along $F$ explicitly. Let $u=(1,1,1)$ then $\phi(u,u)=3.$ Then $\hat{u}=(1,1,1)/\sqrt{3}.$ And so $$P_F(x) = \phi(x,u)\hat{u} = (x_1+x_2+x_3)\hat{u}.$$ Similarily, $$ \begin{align*} P_{F^{\perp}}(x) &= x-P_{F}(x) \\ &= (x_1,x_2,x_3)-(x_1+x_2+x_3)\hat{u}\\ &= \frac{1}{\sqrt{3}}\langle(\sqrt{3}-1)x_1+x_2+x_3,x_1+(\sqrt{3}-1)x_2+x_3,x_1+x_2+(\sqrt{3}-1)x_3\rangle \end{align*} .$$
Is this calculation correct?