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

if

"

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.

Peter