RI 5 Vectors Qn

RAFFLES INSTITUTION H2 Mathematics (9758) 2024 Year 6 ____________________________________ Y6 H2 Math Term 3 Revision Session 5: Vectors Page 1 of 2 Term 3 Revision Session 5: Vectors Questions 1 With respect to the origin O, the position vectors of the points A , B and C are a, b and c respectively. Point C lies on AB such that : 1:2AC CB = . It is given that a is a unit vector and the length of OB is 2 units. (i) Give a geometrical interpretation of ac . [1] (ii) It is given that the angle AOB is 60°. By considering ( ) ( )22−−ab ab , find 2 −ab . [3] (iii) Find c in terms of a and b. [1] (iv) Hence by considering cosine of angle AOC and cosine of angle COB, determine if the line segment OC bisects the angle AOB. [3] 2 The points A, B and R have position vectors ,a b and r respectively. (a) The point C has position vector 2 77 3−ab and the point D is such that the origin O is the midpoint of the line segment CD. The point R lies on BD extended such that the ratio of BD to BR is 4 : 7. Show that the points A , O and R are collinear and state the ratio of OA to OR. [4] (b) It is given that the point R has position vector , x y z  = r and that 1 3, 2  = a and 1 5. 3 − = b (i) Determine the exact area of the triangle AOB. [2] (ii) Give the geometrical interpretation of the point ,R given that ( ) 0.⋅×=rab [2] (iii) Find the shortest distance between the point ( 8, 2, 9)−− and the collection of all points R satisfying ( ) 0.⋅×=rab [2]

Raffles Institution H2 Mathematics 2024 Year 6 _________________________________________________________________________________________ ____________________________________ Y6 H2 Math Term 3 Revision Session 5: Vectors Page 2 of 2 3 The plane p passes through the points with coordinates ( ) , 2, 5k− , ( )0, 2, 1− and 1, 3, 12 −− , and the line l has equation 24 23 xz y k +− =−=− , where k is a constant. (i) Show that the cartesian equation of the plane is 63 6x y kz k++= − . [2] (ii) Show that line l cannot be perpendicular to p. [2] For the rest of this question, let 2k =− . (iii) Given that l meets p at point N, find the coordinates of N. [3] (iv) Another plane π is parallel to the plane p. Given that the distance between p and π is 11 units, find the possible points of intersection between l and π . [3] 4 One day, Eddie came home from a birthday party and brought back a helium filled balloon. After playing with it, he accidentally released the balloon at the point (1, 2, 3) and it floated vertically upwards at a speed of 1 unit per second. t seconds later, a sudden gust of wind caused the balloon to move in the direction of i +4j+ 6k. You may assume that z = 0 refers to the horizontal ground. (i) Find the angle in which the balloon has changed in direction after the gust of wind blew it away. [3] (ii) Find the Cartesian equation of the pla