NJC 1A Vectors Tutorial

National Junior College Mathematics Department 2025 – 2026 / H2 Maths / Vectors I / Tutorial Page 1 of 6 National Junior College 2025 – 2026 H2 Mathematics Topic 1A Vectors I Tutorial Check Your Understanding 1 The diagram below shows a rectangular cuboid OADBFCEG with dimensions OA = 6 units, OB = 2 units and OC = 3 units. (a) Find the following vectors, giving your answers in column vector notation. (i) ,OE ⎯⎯ → (ii) ,OG ⎯⎯ → (iii) ,CB ⎯⎯ → (iv) .CD ⎯⎯ → State another two displacement vectors that are equal to OE ⎯⎯ → and CB ⎯⎯ → respectively. (b) Find the magnitude of .CD ⎯⎯ → Hence, or otherwise, find two distinct vectors that are parallel to CD ⎯⎯ → and has length 21 units each. (c) The lines OG and CD intersect at point P. Find the position vector of P. (d) The point Q is such that E lies on the line segment BQ and BQ : EQ = 3 : 2. Find the position vector of Q.

National Junior College Mathematics Department 2025 – 2026 / H2 Maths / Vectors I / Tutorial Page 2 of 6 2 For each of the following pairs of vectors, find their scalar product. (a) 3+−i j k and 2,+ik (b) 1 4 3  − and 1 3 1 − − . 3 Relative to an origin O, the position vectors of the points A and B are 2−+i j k and 2−+ij respectively. Find (a) ,AOB (b) .OBA 4 Relative to an origin O, A and B have position vectors 2c+−i j k and 2c− + +i j k respectively, where c is a real constant. Find the value of c (a) if OA ⎯⎯ → is perpendicular to ,OB ⎯⎯ → (b) if instead 120 .AOB =  5 For each of the following pairs of vectors, find their vector product. (a) 3+−i j k and 2,+ik (b) 1 4 3  − and 1 3 1 − − , (c) 26−ik and 3 0 9 −   . 6 Relative to an origin O, the position vectors of the vertices A, B and C of a triangle are ,+−i j k 23++i j k and 54+−i j k respectively. (i) Find the exact area of triangle ABC. (ii) Find the length of projection of AB ⎯⎯ → onto .AC ⎯⎯ → (iii) Find the projection vector of AB ⎯⎯ → onto AC ⎯⎯ → in the form ( )1 5 abc++i j k for some integers a, b and c to be determined. (iv) Find the length of the vector component of AB ⎯⎯ → perpendicular to AC ⎯⎯ → in the form n for some integer n to be determined. (v) Find the vector component of AB ⎯⎯ → perpendicular to AC ⎯⎯ → in the form ( )1 5 p q r++i j k for some integers p, q and r to be determined.

National Junior College Mathematics Department 2025 – 2026 / H2 Maths / Vectors I / Tutorial Page 3 of 6 Practice Questions 1 (a) Relative to the origin O, the position vectors of two points A and B are t=+a i k and 2= − +b i j k respectively, where t . Given that the angle between a and b is 60 , evaluate the possible exact values of t. (b) Relative to the origin O, the position vectors of the points P and Q are p and q respectively. The point R lies on PQ produced such that : 3: 5QR PR = and OR is perpendicular to OP. Show that 23 5=p q p . [2015 / NJC / SH2 / T1LT / Q1] 2 (i) Find a unit vector n such that ( )2 3 6 . − + =n i j k 0 (ii) Given that the vector u has a magnitude of 5 units and is parallel to the vector 2 3 6 ,−+i j k find the possible vectors of u. (iii) Find the cosine of the acute angle between u and the z-axis. [MI Promo 9758/2018/PU2/01/Q6 modified] 3 (a) Given that u and v are non-zero vectors such that ( ) ( ), = −  +u v u v u v what can be deduced about u and v? (b) Referred to an origin O, the points A and B have position vectors a and b respectively, where a and b are not parallel to each other. Point C is the mid-point of AB. Point D lies on OA produced such that : 1:OA OD p= , where 1p . Point E lies on BD, between B and D, such that : : 2BE ED p= . Given that points O, C and E are collinear, find the value of p. [EJC Promo 9758/2018/Q5] 4 Relative to the origin O, the position vectors of the points A, B and P are 3 2 ,− − +i j k 52+ik and ( ) ( )1 2 2 2+ + − +i j k respectively, where ,1 −R . (i) Show that A, B and P are collinear. (ii) Find the value of  such that P is on the line BA produced and area of triangle OAP is 162 5 square units. Give a reason for your choice. [2012 / NJC / Prelims / P1 / Q2] 5 Referred to the origin O, the points A and B have position vectors a and b such that = − +a i j k and 23=+b i k . The point C has position vector c given by =+c a b , where  and  are constants. (i) Find the exact area of triangle OAB. (ii) Given that OABC forms a parallelogram, write down the values of  and  . (iii) Given instead that 2 = and that 59OC= , find the possible coordinates of C, leaving your answers in exact form. [2016 / MJC / Promo / Q4]

National Junior College Mathematics Department 2025 – 2026 / H2 Maths / Vectors I / Tutorial Page 4 of 6 6 Referred to the origin O, the points A and B are such that OA ⎯⎯ → = a and OB ⎯⎯ → = b where a and b are non-parallel vectors. It is also given that C, D and E are the midpoints of OA, OB and AB respectively, and M is the point of intersection of the lines AD and BC. (i) Find OM ⎯⎯ → in terms of a and b. (ii) Show that M lies on the line OE. (iii) Find the area of the quadrilateral OCMD, expressing your answer in the form k ab , where k is a constant to be determined. 7 (i) Given that =u v 0 , what can be deduced about the vectors u and v? Vectors a, b and c are such that a0 and 32 = a b a c . (ii) Show that 32 −=b c a , where  is a constant. (iii) It is now given that a and c are unit vectors, the modulus of b is 4 and that the angle between b and c is 60 . Using a suitable scalar product, find exactly the two possible values of  . [2018 / A-Level / P1 / Q6(modified)] 8 Relative to the origin O, two points A and B have position vectors given by a and b respectively such that they are non-zero, non-parallel and =ab . (i) Show that ( ) ( ) 0+  − =a b a b . (ii) Give a geometrical interpretation of the result shown in part (i). (iii) Suppose 1==ab , show that 22 1 +  =a b a b . [2016 / NJC / SH2 / T2 CT / Q6] 9 Relative to the origin O, the position vectors of A and B are 11 7 433 ++i j k and 24ik+ respectively. It is given that R lies on BA produced such that 23BR AR= . (i) Show that OR ⎯⎯ → is given by 7 12i j k+− and hence find the exact length of projection of OR ⎯⎯ → on .OB ⎯⎯ → (ii) Find the vector ,NR ⎯⎯ → where N is the foot of perpendicular from R to OB. (iii) Find the position vector of the point R , which is a reflection of the point R in the line passing through OB. [2016 / SRJC / J1 CT / Q7]

National Junior College Mathematics Department 2025 – 2026 / H2 Maths / Vectors I / Tutorial Page 5 of 6 10 During a camp, a group of scouts builds a structure using wooden poles as shown in the diagram. The structure has a rectangular base ABCD on the horizontal ground, with AB = 6 metres and AD = 4 metres. The top of the structure, T, is 2 metres vertically above O, the centre of ABCD. T is connected to P, Q, R and S, the mid-points of AB, BC, CD and DA respectively. (i) If i, j and k are the unit vectors in the directions of OQ, OR and OT respectively, write down the position vectors of Q, R and T. Hence find angle QTR to the nearest degree. (ii) The scouts tie a rope from P to a point E on ST such that 4.SE ST ⎯⎯ → ⎯⎯ → = Find the length of the rope between E and P, assuming that the rope is taut. (iii) Another rope is tied from P to a point F on TQ. Find the position vector of F such that the length of the rope between F and P is minimised. (iv) A flagpole of 5 metres is erected vertically at the point ( ), , 0G  . Find the values of  and  if the top of the flagpole is collinear with A and T. [2016 / PJC / J1 CT / Q10