I am given the following question here.
Consider the plane $H$ in $\mathbb{R}^3$ consisting of all points satisfying the equation $x-2y+z=0$.\
a) Find an ordered basis $\mathcal{B}=\{\vec{b_1},\vec{b_2},\vec{b_3} \}$ for $\mathbb{R}^3$ with one vector normal to $H$ and two vectors parallel to $H$. Ensure that the basis vectors are ordered with positive orientation.
b) Using the basis from part $(a)$, form $P_{\mathcal{E}\leftarrow\mathcal{B}}$ and $P_{\mathcal{B}\leftarrow\mathcal{E}}$ where $\mathcal{E}$ is the standard basis for $\mathbb{R}^3$.
c) Let $T$ be the linear operator where vectors in $\mathbb{R}^3$ are projected orthogonally onto $H$. Form the transformation matrix $[T]_\mathcal{B}$ with respect to the basis $\mathcal{B}$.
d) Find the standard matrix $[T]$ for the linear operator $T$.
I was wondering if someone could check my reasoning.
a) Let $x=s$ and $y=t$. So this means $z=-s+2t$. So this means that $z=s(1,0,-1)+t(0,1,2)$. So a basis for the parallel vectors are $\{(1,0,-1),(0,1,2) \}$. Now I can take the cross product to get a vector orthogonal to both. Let $\vec{v_1}=(1,0,-1)$ and $\vec{v_2}=(0,1,2)$. $\vec{v_1} \times \vec{v_2} = \vec{v_3}=(1,-2,1)$. So $\mathcal{B}=\{(1,0,-1),(0,1,2),(1,-2,1)\}$
b) $P_{\mathcal{E}\leftarrow\mathcal{B}}=\begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & -2\\ -1 & 2 & 1 \end{bmatrix}$ (Pretty much just reading off...)
$P_{\mathcal{B}\leftarrow\mathcal{E}}=P^{-1}_{\mathcal{E}\leftarrow\mathcal{B}}=\begin{bmatrix} 5/6 & 1/3 & -1/6\\ 1/3 & 1/3 & 1/3\\ 1/6 & -1/3 & 1/6 \end{bmatrix}$
c) I'm not sure about this one. Would I used the gram schmidt process to get an orthogonal basis and then project $\vec{e_1}$,$\vec{e_2}$,$\vec{e_3}$ onto each of the orthogonal vectors?
d) I'm not really sure what this is even asking.