I am seeking to understand Finite Reflection Groups and am reading Grove and Benson (G&B) : Finite Reflection Groups

Grove and Benson, in Section 21 Orthogonal Transformations in Two Dimensions define T as a linear transformation belonging to the groups of all orthogonal transformations $\displaystyle O( {\mathbb{R}}^2 )$.

G&B point out that the vector $\displaystyle x_1 = (cos \ \theta /2 , sin \ \theta /2) $ is an eigenvector having eigenvalue 1 for T and that similarly $\displaystyle x_2 = ( - sin \ \theta /2 , cos \ \theta /2 ) $ is an eigenvector with eigenvalue -1 and $\displaystyle x_1 \ \bot \ x_2 $[see attachment]G&B then state that if $\displaystyle x = {\lambda}_1 x_1 + {\lambda}_2 x_2 $ , then $\displaystyle Tx = {\lambda}_1 x_1 - {\lambda}_2 x_2 $ and T sends x to its mirror image with respect to the line L (see Figure 2.2(b) in attachement)

The transformation T is called the refection through L or the reflection along $\displaystyle x_2 $.

G&B then say "observe that $\displaystyle Tx = x - 2( x , x_2) x_2 $ for all $\displaystyle x \in {\mathbb{R}}^2 $"

Can someone help me show (explicitly and formally) that $\displaystyle Tx = x - 2( x , x_2) x_2 $ for all $\displaystyle x \in {\mathbb{R}}^2 $ ?

And further (and possibly more important) can someone help me get a geometric sense of what the formula above means?ie why do G&B highlight this particular relationship?

Would very much appreciate such help

Peter