RI Vectors C2A Tut (Qn)

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   ] 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 segment PQ such that :2 : 3PR RQ  ; (iii) R lies on PQ produced so that :3 : 2PR QR  . [(a) 155 1, 142 19    (b)(i) 5 1 82 19    (ii) 75 3 11    (iii) 30 29 28    ] 3 (a) Find the value of p so that the vectors 2ij k and ij kpp are perpendicular. (b) If 2 ai j k and bj k , find a unit vector perpendicular to both a and b . [(a) 1 3p (b) 3 1 1 11 1   or 3 1 1 11 1    ]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________ Tutorial 2A: Vectors I Page 2 of 5 B Q M P A O Section B (Discussion Questions) 1 The position vectors a, b and c are given by 236 ,p ppai j k 22 bi j k and 32 02p pp ci j k where p > 0. It is given that ab . (i) Find the exact value of p. [2] (ii) Show that () . () 0 . abab [3] (iii) The three points A, B and C have position vectors a, b and c respectively relative to the origin O . Show that A, B and C are collinear. [3] [(i) 3 7p ] 2 Referred to the origin O, the points A and B are such that OA a  and OB  b  . The point P on OA is such that :1 : 2OP PA  , and the point Q on OB is such that :3 : 2OQ QB  . The mid- point of PQ is M (see diagram). (i) Find OM  in terms of a and b and show that the area of triangle OMP can be written as k ab , where k is a constant to be found. [6] (ii) The vectors a and b are now given by 263a n d 2 ,ppp   ai j k b i j k where p is a positive constant. Given that a is a unit vector, (a) find the exact value of p , [2] (b) give a geometrical interpretation of ab , [1] (c) evaluate ab . [2] [(i) 1 20k  (ii)(a) 1 7p (ii)(c) 9 1 77 8    ]

Raffles Institution H2 Mathematics