Can u help me to finish this question about group and symmetry ?

3. Recall rom Example 9 in the lecture notes for Chapter 3 that the centralizer in 0(3, IR) of 001 A2010 100 consists of the group of matrices g = {iG1[a), :tG2(o:) | or E R} with 1 1+coso \\/sinoz 1|cosor G1(a]= E sino: 2cosoz sina , 1+cosoc \\/sinoz 1+coso: 1 1|coso: sino: l+coscr G202] = E sinoz 2 cosa sino: 1+cosa x/iisino,r 1+cosor (a) Show that each matrix Gl (a) represents a rotation of 0: around the line passing through the origin parallel to the vector 17": (1,0, 1). Conclude that {91(0) | a 6 IR} :s 30(2, R). (b) Show that each matrix G201) represents a rotation of order 2. What is the axis of rotation of G203]? [Your answer will depend on or.) (e) Let R = G200). Show that: R represents a rotation of order 2 around the line passing through the origin parallel to the vector 13 = [1,0, 1); Rh?) 2 13'; and G202] = G1(or) R. (d) Show that RG1[CE)R = Ga] and that 1. G101) Gli) = 0'10?! -- l3) 2- G201] G203) = G1 (a l3) 3. G1(Ct] G2U3) = G202 )3) 4- 0203) (31(0) = G203 - 0') (e) Explain why {G E g I det(G) = 1} 2 {63(0), G201) |a E IR} % O(2,R). (f) Find a shape in 1&3 for which 9 is the symmetry group, and illustrate all the symmetries