Let q = a + bi + cj + dk be a unit quaternion with b, c, d not all 0, and let f: R3 > R3 the corresponding rotation. That is, f(x, y, z) = (x', y', z') where q(xi+yj+zk)q_1=x'i+y'j+z'k. 1. Show that if (x, y, z) = (ill), Ac, ild) for some scalar A 6 IR (that is, (x, y, z) is in the line spanned by (b, c, d)), then f(x, y, z) = (x, y, z). 2. If (x, y, z) is perpendicular to (b, c, d), then find the angle between (x, y, z) and f (x, y, z) and show that this angle is independent of the choice of (x, y, z) (hint: this is much easier if you use the formulas for quaternion multiplication with dot and cross product, i.e. (1'1, 5002, Z32) = (1'11'2 3132, r152 + 1351+ 51>