RI Vectors C2A Tut Sect A (Soln)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ____________________ Tutorial 2A: Vectors I Page 1 of 5 Tutorial 2A: Vectors I Vector Algebra, Ratio Theorem, Scalar and Vector Products Section A (Basic Questions) 1 (a) The vectors , ab and c are such that 2ab c 0  . If 4 ai j k and 2bi k , find a unit vector in the direction of c. (b) Find a vector a such that a is parallel to the vector 84ij k and is equal in magnitude to the vector 22ij k . (c) Find the position vector of P if OP is of length 5 units and is in the direction 24ij k . [(a) 2 1 1 30 5    (b) 8 1 13 4    or 8 1 13 4    (c) 2 5 1 21 4   ] Solution: (a) a 4 1 1   , b 1 0 2   Given that 2ab c 0  , we have 2cb a 14 2 20 1 1 21 5               Unit vector in direction of c 22 2 22 11ˆ 11 30(2 ) 1 (5 ) 55 cc c              [Note: The question states that the unit vector and c must be in the same direction] (b) Given a 88 8 / / 1 1 44 4                 a for some \0  . and  222 1 21 2 29 3 2 a       

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________ Tutorial 2A: Vectors I Page 2 of 5   22 2 8 4 3    2 2 81 9 1 9 1 3       Hence, a 8 1 13 4    OR : 22 2 88 8 13 1ˆ 31 1 1 381814 44 4                       aa a [Note: The question states that a and 84ij k are parallel to each other. i.e. they could be in the same direction or opposite direction. That is the reason for 2 possible answers. On the other hand, the question ask for a vector a, so you can just give any of the 2 possible answers.] (c) Given 5OP   and 2 1 4 OP      222 2 ( 2 )()( 4 ) 5 21 25 5 21          But since OP  is in the direction of 2 1 4   , 5 21   i.e. 2 5 1 21 4 OP     OR : 22 15unit vector in the direction of 5 1 1 4 1 16 21 44 OP OP OP                     

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________ Tutorial 2A: Vectors I Page 3 of 5 2 The points P and Q have position vectors 31 7 ij k and 892ijk respectively when referred from the origin O . (a) Evaluate OP OQ     and OP OQ    . (b) Find the position vector of the point R if (i) R is the mid-point of PQ ; (ii) R lies on the line