Two cyclists start from the same intersection O. Cyclist A rides to his home along the displacement vector a=6i^+3j^−2k^ km, while Cyclist B rides to her home along the displacement vector b=−4i^+9j^+2k^ km.
(i) Determine whether the routes of the two cyclists from O are along perpendicular directions.
(ii) A third cyclist C reaches her home along the displacement vector c=9i^−4.5j^−1k^ km. Decide whether the route of cyclist C is parallel to the route of cyclist A.
(iii) Calculate the obtuse angle between the routes of cyclists B and C.
(iv) If cyclist D must choose a displacement vector of length 10km that is perpendicular to the route of cyclist B, write one possible vector for cyclist D’s route.