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 2025 Year 5 _____________________________________________________________________________________________ __________________ Tutorial 2A: Vectors I Page 3 of 5 3 (i) Given that ab 0 , what can be deduced about the vectors a and b? [2] (ii) Find a unit vector n such that (22 ) ni j k 0 . [2] (iii) Find the cosine of the acute angle between 22i j k and the z-axis. [1] [(ii) 11 11 2o r 233 22         nn (iii) 2 3 ] 4 Find the cosine of the angle between the vectors i j k and 2i j k . Hence, find the length of projection of 2i j k on i j k . [ 22 , 18 3  ] 5 The angle between the unit vectors a and b is . By expanding the scalar product, show that (3 ) ( 3 ) 8cos  ab a b . Given that 60  , show that the length of projection of (3 )ab onto (3 )ab is 4 13 . 6 It is given that a and b are non-zero vectors. (a) Given that ab ab , by considering suitable scal ar product, comment on the relationship between a and b . [2] (b) Given that ab ab , what can you say about the relationship between a and b ? [1] (c) Given that 25 ab jk , 22 ci j k , .5ac and b is a unit vector, (i) find the value of b.c . [2] (ii) find the sine of the angle between b and c . [2] (iii) find bc , and state the geometrical meaning of this result. [2] [(c)(i) 2 (c)(ii) 5 3 (c)(iii) 5 ]

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________ Tutorial 2A: Vectors I Page 4 of 5 7 With respect to the origin O, the points A and B have position vectors a and b respectively, where a and b are non-zero and not parallel. (i) It is given that B lies on the line segment AC, such that 3BC kba   , where k is a constant. State, with a reason, the value of k. Hence find OC   in terms of a and b. [3] The point N divides the line OC in the ratio 1:  . (ii) Given that OA  is perpendicular to OB  , Show that BN OC.    can be written as 22 ,pq ab where p and q are constants to be found in terms of  . [4] (iii) Given that 0BN OC .     and 2 3ab , find the value of  . [2] [(i) 3k  , 43OC ba   (ii)   91 , 43 4pq   (iii) 4 5 ] 8 The diagram below shows a straight line l passing through the points A and B . With reference to the origin O , the position vectors of A and B are a and b respectively. It is further given that a is a unit vector, 2b and AOB 60. l P B R A b Q a (i) State the values of ab and ab . [2] (ii) The point P lies on the line l and is such that :2 : 1AB BP  . The point Q on the line OP is such that OQ OP     where 01  . Determine the value of  such that the area of triangle OBQ is 1 23 of the area of triangle OAB . [3] (iii) It is further given that the point R on the line l is such that AOR ROB  . Show that R has position v ector ab a for some   and hence find this value of . [3] [(i) 1, 3 ab a b (ii) 1 3 (iii) 1 3 ] O

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________________ __________________ Tutorial 2A: Vectors I Page 5 of 5 9 The points P and Q have position vectors ab and 32ab respectively relative to the origin .O Given that OPQR is a parallelogram, express the vectors PQ  and PR  in terms of a and b . By evaluating two scalar products, show that if OPQR is a square, then 22 2ab . [ 23PQ ab  , 4PRab  ] THE END