I am reading Kane - Reflection Groups and Invariant Theory and need help with two of the properties of reflections stated on page 7
(see attachment - Kane _ Reflection Groups and Invariant Theory - pages 6-7)
On page 6 Kane mentions he is working in dimensional Euclidean space ie where has the usual inner product (x,y).
In defining reflections with respect to vectors Kane writes:
" Given let be the hyperplane
We then define the reflection by the rules
Then Kane states that the following two properties follow:
(1) for all
(2) is orthogonal, ie for all
I would appreciate help to show (1) and (2) above.