RI Vectors C2A Add Prac (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ______________________________ Additional Practice C2AVectors I Page 1 of 6 Additional Practice Questions for Chapter 2A: Vectors I Vector Algebra, Ratio Theorem, Scalar and Vector Products 1 Referred to an origin O, points A and B have position vectors given respectively by and22 236 .OA OB    ij k ij k    The point P on AB is such that :: 1 .AP PB  Show that ( 1 ) ( 25 ) ( 28 ) .OP      ij k  (i) Find the value of  for which OP is perpendicular to AB. (ii) Find the value of  for which angles AOP and POB are equal. [(i) 5 18 (ii) 3 10 ] 2 Referring to an origin O, the position vectors of points A, B and C are given by 7OA i  , 4OB ik   and 4OC ij   respectively. A parallelepiped has OA, OB, OC as three edges, and the remaining vertices are X, Y, Z and D as shown in the diagram. (i) Write down the position vector of Z in terms of i, j and k. [1] (ii) The point P divides CZ such that CP PZ  . Given that OP  is perpendicular to CZ  , find the value of  and evaluate OP  . [6] [(i) 84ik (ii) 4 ] A O B Z Y D X C

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ___________________________ Additional Practice C2AVectors I Page 2 of 6 3 The position vectors of A and B with respect to the origin O are a and b respectively. M is the mid-point of OA and C is the point on MB such that 2CB MC      . Given that 6a , 5b and the angle between a and b is 45 , evaluate the exact value of the scalar product of b and c . [   5 53 23 ] 4 Referred to the origin O, the points A and B have position vectors a and b such that aij k and 22 bi j k . (i) Find the size of angle OAB. [2] The point C has position vector c given by  cab , where λ and µ are positive constants. Given that the area of triangle OAC is twice that of triangle OBC, (ii) find µ in terms of , [3] (iii) hence, if OC = 118, find the position vector c. [4] [(i) 144.7 (ii) 2  (iii) 32 52 52     ] 5 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] [(ii) 2 (iii) 2 3 ba ]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ___________________________ Additional Practice C2AVectors I Page 3 of 6 6 Relative to the origin O, the points A, B, and C, have non-zero position vectors a, b and 3 a respectively. D lies on AB such that ADA B , where 01  . (i) Write down the position vector of the point D. [1] (ii) The point E is the midpoint of BC. Find the value of  if E lies on the line OD, and write down the ratio OE : ED. [4] [(i)  1,OD    ba    (ii) 2 : 1 ] 7 Referred to origin O, the points P, Q, R and S have position vectors p, q, r and s respectively. (i) Given that   pq r 0 and 10   , where  and  are non-zero constants. Show that P, Q and R are collinear. [3] It is given that the point R lies on PQ produced. The point T lies on line RS produced such that :3 : 2RT ST  . (ii) Find the position vector of the point T in terms of r and s. [2] (iii) Give a geometrical interpretation of    pr sr sr . [1] (iv) Find the area of triangle PRT in the form  pspr rs , where γ is a constant to be determined. [3] [(ii) 32OT sr   (iv) 3 2 ps pr rs ]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ___________________________ Additional Practice C2AVectors I Page 4 of 6 8 The diagram shows a semi-circle with diameter OB. Point A is on the circumference of the semi- circle and point C lies on chord AB such that 3AC CB . Referred to the origin O, the points A and B are such that OA a  and OB b  where a and b are non-zero and non-parallel vectors. (i) Show that 2 . = ab a . [2] (ii) Find the position vector of C in terms of a and b. [1] (iii) Hence find the length of projection of OC   onto OB  if 3a and 2b . [4] [(ii) 3 4 ab (iii) 15 8 ] 9 Referred to the origin O, points A and B have position vectors a and b respectively. Point P is on the line AB such that ::AP PB m n where m and n are positive integers. Point C is on OP extended such that :1 : 2OP PC  . (i) Show that 23nm mAC mn mn        ab   . [3] (ii) If ab and the angle between vectors a and b is 3  , find the area of the triangle ABC in terms of a . [4] (iii) Find the ratio :AP PB such that AC is parallel to OB. [2] [(ii) 23 2 a (iv) 2:1 ] O B A x y

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ___________________________ Additional Practice C2AVectors I Page 5 of 6 10 Referred to the origin O, three distinct and non-collinear points A, B and C have position vectors , and ab c respectively. Point L is the mid-point of BC. The position vector of a point P is given by  1 2 kkab c , where k is a non-zero constant and 1k  . (i) Show that A, L and P are collinear. [3] (ii) Show that 11 22 kCP CB    abb ac c    . [3] For the rest of the question, let 1 2k  . Let point Q be a point on the line passing through A and L. P and Q are distinct points and the areas of triangle CPB and triangle CQB are equal. (iii) By considering part (ii), find the position vector of Q in terms of , a n d ab c . [4] (iv) Given that 1BC    , interpret geometrically LP BC     . [1] [(iii)  0.5 0.75 ab c ] 11 With reference to the origin O, the points A and B have position vectors a and b respectively, where a and b are perpendicular. A point P lies on AB between A and B such that :: 1AP PB   , 01  . (i) Show that   (1 )cos 1AOP      a ab . [4] (ii) Prove that    222 211 1         ab ab a b . Hence, given also that OP bisects AOB , find the ratio of a b , leaving your answer in terms of . [6] [(ii) 1   ]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ___________________________ Additional Practice C2AVectors I Page 6 of 6 12 Relative to the origin O, the position vectors of the points P, Q and R are i, 2 j  tk and tk respectively, where t is a fixed constant. The points A and B divides both line segments PQ and QR respectively in the same ratio of µ : 1 – µ, where µ is a parameter such that 0 1. (i) Find the vector AB  in terms of t and µ. [3] (ii) Determine whether the points O, A, B are collinear. [1] (iii) Find the values of µ such that the length of projection of AB  onto 3 4 0    is 1 5 unit. [2] (iv) Given that angle AOB is a right angle, find the set of possible values of t, justifying your answer c