# Thread: Finite Reflection Groups in Two Dimensions - Grove & Benson Section 2.1

1. ## Finite Reflection Groups in Two Dimensions - Grove & Benson Section 2.1

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

2. ## Re: Finite Reflection Groups in Two Dimensions - Grove & Benson Section 2.1

Tx = T(λ1x1 + λ2x2) = λ1T(x1) + λ2T(x2)

= λ1x1 - λ2x2 = λ1x1 + λ2x2 - 2λ2x2

= x - 2λ2x2 (this is just simple algebra, to this point).

now (x,x2) = (λ1x1 + λ2x2,x2) = λ1(x1,x2) + λ2(x2,x2)

and since x1,x2 are orthogonal, (x1,x2) = 0, therefore:

(x,x2) = λ2(x2,x2) = λ2|x2|2.

but x2 lies on the unit circle, therefore it has length (and thus length squared) of 1.

so (x,x2) = λ2, and we have:

Tx = x - 2λ2x2 = x - 2(x,x2)x2.

so why?

the quantity (x,x2)x2 is a little misleading (because we are dealing with UNIT vectors x1,x2).

normally, this is written as: [(x,x2)/(x2,x2)]x2 which is known as:

the projection of the vector x in the direction of the vector x2. geometrically this is: "the part of x that lies on the line generated by x2".

normally, we project a vector onto the unit vectors e1 = (1,0) and e2 = (0,1).

that is: if x = (x1,x2) = x1e1 + x2e2, then:

the projection of x in the direction of e1 is [(x.e1)/(e1.e1)]e1

= [(x1*1 + x2*0)/(1*1 + 0*0)]e1 = x1e1 = (x1,0).

3. ## Re: Finite Reflection Groups in Two Dimensions - Grove & Benson Section 2.1

Thanks ... that post was most helpful, particularly the bit about the misleading nature of the formula as stated ... that clarified a few issues for me.