Let G = R, the additive group of real numbers, and let X = R 2 be the real plane. For G and x = (x1, x2) R 2 , define the counterclockwise rotation by : (x) := cos sin sin cos x1 x2 = x1 cos x2 sin x1 sin + x2 cos . Use matrix multiplication, the addition laws for sine and cosine, and the identities cos() = cos , sin() = sin , and sin2 + cos2 = 1, to do the following. (a) Show that x = (x) makes X into a G-set. (b) Describe geometrically the orbit Ox for each x X for this action. [Hint: show that if (x) = y with x = (x1, x2) and y = (y1, y2) then x 2 1 + x 2 2 = y 2 1 + y 2 2 .] (c) Find the stabilizer subgroup Gx for each x X. (d) Find the fixed-point sets Xg for g G and the set XG of fixed points for G