Suppose that $\displaystyle \{u,v,w\}$ is an orthogonal set. Under what conditions will the set $\displaystyle S_1=\{u+v,u-v,w\}$ be an orthogonal set?

Repeat for $\displaystyle S_2=\{u+v-w,u-v+w,v+w\}$ and $\displaystyle S_3=\{u-v,u+v,w-u,u+v+w\}$.