Q4. (a) (i) [8 marks] Use Hamilton's quaternions to prove that the unit 3- sphere 53 in R4 is a disjoint union of isometric circles, parametrized by S². (ii) [7 marks] Determine the rotation in Ri + Rj + Rk induced by conjugation by the unit quaternion t =+++ (b) [10 marks] State the Cayley-Salmon theorem, explaining the assump- tions of the theorem. Give an example of a smooth cubic surface containing twenty seven lines and describe all the lines.