(a) Prove that an orthogonal 2x2 matrix must have the form
[a -b
b a]
OR
[a b
b -a]

where
[a
b] is a unit vector.

(b) Using part (a), show that every orthogonal 2x2 matrix is of the form
[cosx -sinx
sinx cosx]
OR
[ cosx sinx
sinx -cosx]

where 0 < x < 2 pai.

(c) Show that every orthogonal 2x2 matrix corresponds to either a rotation or a reflection in R^2.

(d) Show that an orthogonal 2x2 matrix Q corresponds to a rotation in R^2 if detQ=1 and a reflection in R^2 if det Q=-1.