RI Vectors C2A Add Prac (Soln)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ______________________________ Additional Practice C2AVectors I Page 1 of 21 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. 9205/1984/02/Q5 Solution:       By Ratio Theorem, 1 1 121 2 12 3 2 13 262 1 6 1 2 5 , that is, (1 ) (2 5 ) ( 2 8 ) (shown) 28 OA OBOP OP                                      ij k      B P A O

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ___________________________ Additional Practice C2AVectors I Page 2 of 21 (i)   21 1 32 5 62 8 0 11 2 5 5 0 28 8 1 10 16 1 25 64 0 AB OB OA OP AB OP AB                                                 25 5 90 18  (ii)       cos cos 11 12 11 25 2 25 337 28 2 28 6 11 1 4 4 1 10 16 2 6 12 2 15 4837 AOP POB AOP POB OP OA OP OB OP OA OP OB                                                                      79 2 5 3 1 6 6 5 3 10        OR : Use the fact that in a rhombus, the diagonal is an angle bisector. We can form a rhombus with the vectors andab ab which has diagonal ab ab . So OP // ab ab ie. OP k   ab ab  for some .k  By Ratio Theorem, since P on AB is such that :: 1 .AP PB       1 1 Since and are non-zero and non parallel vectors, 1( 1 ) a n d ( 2 ) 33(2) (1) 17 1 0 OP k kk                      ab abab ab ab ab a b 

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ ___________________________ Additional Practice C2AVectors I Page 3 of 21 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 a