Let $\displaystyle \underline{\vec e}$ be an orthonormal basis in $\displaystyle E^4$ and $\displaystyle W=\{\vec u = \underline{\vec e}X: x_1+2x_2+2x_3=0\}$.

Create a new orthonormal basis $\displaystyle \underline{\vec f}$ so that $\displaystyle W=\{\vec u = \underline{\vec f}Y: y_4=0\}$.

I know how to transform coordinate matrices from one basis to another, but I don't really know where to begin on this one.

EDIT: I guess you can use $\displaystyle X=TY \implies T=XY^{-1}$ where $\displaystyle X=(1, 2, 2, 0)^t$ and $\displaystyle Y=(0, 0, 0, 1)^t$