Given a plane $\pi=span(v,w)$ where $v=(3,-2,6)$ and $w=(3,5,-8)$. We have constructed orthonormal basis $B=(u_1,u_2,u_3)$ such that $\pi=span(u_1,u_2)$ and $u_3$ is normal to the plane. Using the basis $B=(u_1,u_2,u_3)$ determine the matrices $P$ and $R$ which represents the projection onto $\pi$ and reflection across $\pi$. Then also using the standard basis $E=(e_1,e_2,e_3)$ determine the matrices $Q$ and $S$ which represents the projection onto $\pi$ and reflection across $\pi$.
What I do know is
$u_1=\frac{1}{7}(3,-2,6),u_2=\frac{1}{7}(6,3,-2),u_3=\frac{1}{7}(-2,6,3)$
Let $O=\dfrac{1}{7}\begin{bmatrix}3&6\\-2&3\\6&-2\end{bmatrix}$ and the projection onto $\pi$ is given by $O(O^TO)^{-1}O^T=\dfrac{1}{49}\begin{bmatrix}45&12&6\\12&13&-18\\6&-18&40\end{bmatrix}$ and reflection across $\pi$ is given by $I-2u_3u_3^T=\dfrac{1}{49}\left[ \begin{array}{ccc}{41} & {24} & {12} \\ {24} & -{23} & -{36} \\ {12} & -{36} & {31} \\ \end{array} \right]$
I think these are in standard basis, right ?
How do I determine the matrices in both bases without using the transition matrix ?