RI 5 Vectors Soln

RAFFLES INSTITUTION H2 Mathematics (9758) 2024 Year 6 ____________________________________ Y6 H2 Math Term 3 Revision Session 5: Vectors Page 1 of 9 Term 3 Revision Session 5: Vectors Questions 1 MI Prelim 9758/2021/01/Q4 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] Suggested Solution (i) Since a is a unit vector, ac is the length of projection of c onto a. (ii) ( ) ( ) 22 2 o2 2 2 4( ) 2( ) 2( ) ( ) 4 4( ) ( ) 4(1) 4 cos 60 (2) 14 4(1)(2) 4 4.2 −− =−−+ = −+ = = −+ =− +=  a b a b aa ab ba bb a ab b ba ab ab        ( ) ( )22− −=ab ab 2 2 4 2 2.− =⇒ −=ab ab (iii) By Ratio Theorem, 2 3 += bac . (iv) 2 2 cos 2 3 [from ] 12 3 12 [since 1 ]3 AOC∠= +=  +=   += = = a c ac baa ( iii)ac ab aa ac ab aa ac     

Raffles Institution H2 Mathematics 2024 Year 6 _________________________________________________________________________________________ ____________________________________ Y6 H2 Math Term 3 Revision Session 5: Vectors Page 2 of 9 2 2 2 cos 2 3 [from ] 1 2( )= 3 1 2 2( ) [since 2 ]32 12 3 COB∠= +=  +  += = =  +=  bc bc bab (iii)bc bb ab bc ab bb bc ab c       Since cos COB∠ cos AOC= ∠ , line OC bisects the angle AOB. 2 DHS Prelim 9758/2021/02/Q5 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] Suggested Solution (a) ( ) 23 77 1 327 OD OC  →  → =− = −− = − d ab ba

Raffles Institution H2 Mathematics 2024 Year 6 _________________________________________________________________________________________ ____________________________________ Y6 H2 Math Term 3 Revision Session 5: Vectors Page 3 of 9 By ratio theorem, Since is parallel