Chapter 13 use vectors in any form, e.g. (4), AB, p, ai + bi Use position vectors and unit vectors find the magnitude of a vector; add and subtract vectors and multiply vectors by scalars compose and resolve velocities.CAMBRIDGE IGcSe™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK Before you start. Combridge | |Drowvectorclagrams | y.(2) ya (4) IGCSE/O Level | to represent the 3 1 Mathematics | addition and Draw vector diagrams to represent: subtraction of vectors. i a+b b Cambridge | Add and subtract (3) (3) IGCSE/O Level } column vectors 3 B= \_2) Mathematics — | without using a Write these vectors as column vectors: diagram a he BE Cambridge _ | Multiply a vector by (3) IGCSE/O Level | a scalar. ae Mathematics Write these vectors as column vectors: a 2p b -3p Cambridge | Recognise parallel Which of these vectors are parallel? IGCSE/O Level | vectors. (3) (2) (=) ( 4) (4) Mathematics 4 5 2 -8 4 ‘Cambridge Calculate the ( ay IGCSE/O Level | magnitude (modulus) a l-8 Mathematics | of a vector. Find [al Cambridge __| Use the sum and c IGCSE/O Level | difference of two Mathematics | vectors to express given vectors in 7 ‘terms of two “ coplanar vectors. a c= B ABC's a triangle. M is the midpoint of 4B. Cd = pand CB =q Find in terms of p and q: a 4B b aM © BM od GM x6)13.1 Further vector notation ‘The yeetor 4B in the diagram can be written in 4 2 component form as component formas (3) f AB can also be written as 4i + 3}, where: ‘is a vector of length 1 unin in the 4 positive x-direction i and fisa veetor of length 1 unit in the oP Pa positive y-direction. Note: | Avector of length 1 unit is called a unit veetor Memccekse Wigan a Write PQ in the form ai + bj. b Find |POL. Pa Answers a PO=4i-3 b Using Pythagoras, |PO|= (a) + 2)" = 9120 = 2/5. ‘You could be asked to find the unit vector in the direction of a given veetor. ‘The method is outlined in the following example, EF = a+ 3 FFind the unit vector in the direction of the veotor EF. Answers. ay First find the length of the veetor EF: EP=4°+3 using Pythagoras EF=5 Hence the unit vector in the direction of EF is: 5 \ sais 13 Vectors 327)> CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK WORKED EXAMPLE 3 a=-21+3), b=4i-j and Find 4 and p such that 4a + jib Answers Jat pb=e A(-2i + 3H) + Gi — = -221 + 18) Equating the #’s gives + 4p =-22 “At2=-11 (1) Equating the js gives, 3A w= 18 1-2 =36 Q) Adding equations (1) and (2) gives SA =25 ass Substituting for 4 in equation (1) gives -5+2y =-11 Exercise 13.1 1 Write each vector in the form ai + j. a b ae © 4D a aE e BE * DE a Ea: h DB i De 2. Find the magnitude of each of these vectors. a -4 b 443] c 2 Ti+ 24 f 151-8) 9 ~4i+4j h 51-10) 3. The vector 4B has a magnitude of 20 units and is parallel to the vector 4i + 3). Find 4B. 328 >13 Vectors 4 The vector PQ has a magnitude of 39 units and is parallel to the vector 121 ~ 5}. Find PO. 5 Find the unit vector in the direction of each of these vectors. a 6it8 ob SHI c ~~ od H-IS] 34H 6 -2i + 3jand r= Loi 1 1 b 2 p+q © apo ie dogt-p-a 7 p=9i+ 12), q=3i-3jandr= 745 Find a Ip+al bd iptaqtrl 8 p=7i-2jandq=it pi. Find 4 and p such that 4p + q = 368 13}. 9 a= 51-6), b= 1+ and e=-13i+ 18), Find 4 and ye such that a+ 13.2 Position vectors The position vector of a point P relative to an origin, O, ‘means the displacement of the point P from O. For this diagram, the positon vector of Pis oP = ( 3) or OF ¥ Now consider two points 4 and B with position vectors a and b, AB means the position veetor of B relative to A. AB = 40 +08 =-04 +08> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK Relative to an origin O, the position vector of P is 4i + Sj and the position vector of Dis 104-35 a Find PO. ‘The point R lies on PQ such that PR = 40. b Find the position vector of R. Answers a PO=00- OF __ = M0 = 3) - i+ 3p) PO =6i-% is b pR=1rg =] a = 151-2 OR = OP + PR = (i + 5) + (51-25) OR = 5.5143) ), the position vectors of points A, Band C are ~2i + 5), 10i~ j and 4@2 +i) respectively. Given that Clies on the line 42, find the value of J Answers AB = OB - 0a OB = \0i~ jand Oa = -24+ 5) 10i-j)-(-21+5)) collect sand js 2i~ 6) If Clies on the line AB, then AC = KAB. AC = 00-08 OC = A2i+ jp and Od = -21+ 5} = AQi+})- (2+ 5} collects und fs =Qi+ 2-56-25 KAB = K(12- 6) = 12ki - 6Kj Hence, Qi + 2)5 — (5 ~ajj = 12Ai ~ 6Aj 330)13 Vectors CONTINUED Equating the is gives: 24+2=12k (1) Equating the j’s gives: 5-4=6k — multiply both sides by 2 10-22= 12 (2) Using equation (1) and equation (2) gives 2+2=10-22 4=8 a=2 Exercise 13.2 1. Find 4B, in the form ai + dj, for each of the following, 2 AG, Nand BG,4) — b A(0,6)and BQ, -4) © AQ, ~3)and B(6,-2) d AC. and BRS,3)@ A(-4,-2)and B-3,3) FAG, -6) and BI, -7. 2 a Oisthe origin, Piste point (1,5)and PQ = (3). Find 0. b Os the origin, Zis the point (~3, 4) and EF = (3). Find the position vector of F. © Ois the origin, 1s the point (4, 2) and Wad = (_3), Find the position vector of WV. 3 The vector O4 has a magnitude of 25 units and is parallel to the vector ~3i+4j. ‘The vector OB has @ magnitude of 26 units and is parallel to the vector 12i + Sj. Find a On b OB c 4B d |ABL 4 Relative oan origin 0, the position yector of 4 is ~7i~ 7jand the position vector of Bis 91+ 5) The point Clies on AB such that AC = 3CB. a Find AB. b Find the unit vector in the direction of AB. © Find the position vector of C. 5 Relative to an origin O, the postion vector of P is ~21 ~4j and the position vector of Qis 81+ 20) 2 Find 70. b Find PO ¢ Find the unit vector in the direction of PO. __ Find the position vector of M, the midpoint of PQ, 6 Relative to an origin 0, the position vector of 4 is 4i~ 2] and the position yector of Bis Al + 2). ‘The unit vector in the direction of AB is 0.31 + 0.4, Find the value of 2> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK 10) a Find 4B, The points 4, B and Clic on a straight line such that AC = 248. b Find the position vector of the point C. 10) the pasion vor 5 (_1? 7 Relative to an origin O, the position vector of 4 is, : “a a 24 8 Relative to an origin O, the position vector of 4 is(_31) and the position vector of Bis (4) a Find: i (Oa) ii (08) 8 The points 4, Band Clie on a straight line such that AC = CB. b Find the position vector of the point C. 9 Relative to an origin 0, the position vector of is 31~ 2j and the position vector of Bis 151+ 7}, a Find AB. ‘The point C'lies on ABsuch that AC b Find the position vector of C: 10. Relative to an origin 0, the position vector of is 61 + 6 and the position vector of Bis 121 ~ 2 a Find AB ‘The point C lies on AB such that AC. b Find the position vector of C. B 3 q4 11. Relative to’an origin 0, the postion vector of 4 is (}) and the postion vetor of Bis (3). ‘The points 4, Band Care such that BC =24B. Find the posi 12 Relative to an origin O, the position vectors of points 4, Band Care ~Si~11j, 231~4j and 2(4— 3}) respectively. Given that Clies on the line 4B, find the value of 2 13 Relative to an origin O, the position vectors of A, Band Care ~2i+ 7j, 24 —jand 61+ 2j respectively 2 Find the value of when AC = 17 b Find the value of a when ABCis a steaight line. ¢ Find the value of A when ABC isa right-angle. vector of C 14. Relative to an origin O, the position vector of 4 is ~6i + 4j and the position vector of Bis 18i + 6). Clies on the y-axis and OC = OA + A0B Find OC. 15_ Relative to an origin O, the position vector of is 8i+ 3] and the position vector of Qis~12i~ 7}. R lies on the x-axis and OR = OP + 100. Find OR 332)13 46 CHALLENGE QUESTION Relative to an otigin O, the position vectors of points P, Qand Rare ~6i-+ 8), ~4i + 2) and 5i + Sj respectively 2 Find the magnitude of: i PO i PR OR b Show that angle POR is 90° HOP =200 + wOR, find the value of 4 and the value of 13.3 Vector geometry Od =a, 08 =b, BY = 2 BA A and OF =304. ig a Find in werms of a.and b: i Bi i BY ee si ii OF iv BY b Given that OF = 20N, find OP in terms of 2, aandb. © Given that BP = BY. find OP in terms of 1, » and b, Find the value of 2 and the value of 4. Answers a i BA=04-OB=a-b — 3—> BY = 5 BA = 3(a~b) We OX=08 +BY =b+2-v) Ww BY =80+0¥=-v+304=3a-b b OP =10K ni(fasb) 4% ayatse < OP =0R+BP = b+ BY =b+u(ja-b) 3u. yard wp> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK CONTINUED d_ Equating the coefficients of a for OF gives: mae _ BaF divide both sides by 3 4 ar multiply both sides by 20 4i=Su Equating the coefficients of b for OP gives: = multiply both sides by 5 2=5-Sn 2) ‘Adding equation (1) and equation (2) gives: ees =5 a6 Substituting 2= in equation (1) gives p= 3. = 4 22. Hence, A= Gand y= 3. i Peet = 4h and Mis the midpoint of OB a :3.and BY = ABP. a Find in terms of w and b: u AB a MA. hy b Find in terms of 2, wand b: 4 i Bx i MX. If M,Xand A are collinear, find the value of 2. Answers a 4B = 40 + 0B +4b ‘MO +04 2b +30 =3a-2b13. Vectors b BX = ABP = ABO + OP) ey son’ oe ni(-w+4at) wt =s =a = ta Aib ii MX=MB+BY use BY = 2a — ai =m +H aaah collect a’sand b's = +(2-4i)b IM, Yand A are collinear, then MX = kA. Bea + 2 ab =kGa- 20) Equating the coefficients of a gives: uk= 122 divide both sides by 3 a-# a Equating the coefficients of b gives: -2k=2- 44 divide both sides by ~2 k=2-1 Q) Using equation (1) and equation (2) gives: Exercise 13.3 1 OA =a, OB = Ris the midpoint of OA and OP = 308. AO = 1AB and RO = pRP. a Find 0 in terms of 4, aandb. = 2 bb Find OG in terms of 4, and b ‘0 © Find the value of a and the value of yu. 2 R “ 335 >> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK a > © 2 Find in terms of aand b, ‘ aD, DB. b Findin terms of 2, 4, andlor b, i aR, ii DY. z ia # © Find the value of 2 and the value of a PY =2,and 00 =30 O8 = 40¥ and OX = nOP. 2 Find OF in terms of 2, aandb. b Find O¥ in terms of u, a and b. € Find the value of 4 and the value of p. 4 O4=2,08=b > Bis the midpoint of OD and AC = 304. AR = 14D and BX = BC. r 8 Find O¥ in terms of , aad ‘ hy b Find OF in terms of ju and b, © Find the value of 2 and o > 4 c the value of p. 5 O¢=a,0B=b B Mis the midpoint of 4 and oY =304. O¥ = 10M and BY = BY. u a Find in terms of a and i i OM b Find O¥ in terms of 4, a and b. # * 4 © Find OF in terms of ya and b. Find the value of 4 and the value of 6 Of =a, OB =», BY = 3BA and OF = 308. p OP = 20 and BP = pBY. a Find OP in terms of 2, aand b. 2 b Find OP in terms of y, wand b. cS ¢ Find the value of 2 and the value of 2 elas o Y 4 336 >13. Vectors |, OB = band is the origin. OA and OF = nOB. 2 i Find BY in terms of 4,aand b. fi Find AY in terms of yz. aand b. SBP = 2BX and AY =4P¥. Find OP in terms of J,aandb. 9 a4 ji Find OP in terms of and b Ili Find the value of 4 and the value of. 8 0,A, Band Care four points such that 04 = 1a ~ 5b, OB =2a+ Shand OC = ~2a+ 13 a Find i aC if ZB, b Use your answers to part ato explain why B lies on the line AC. 9 CHALLENGE QUESTION o 04 =a.and OB =b ie 04: AE=1:3and AB: BC = J » on= BD Find, in terms of aandior b, as a i OE i OD oe. b Find, in terms of a and/or b, i i @ ii DE. © Use your answers to part b to explain why C, D and E are collinear. Find the ratio CD : DE, 13.4 Constant velocity problems If an object moves with a constant velocity, ¥, where v= (41 - 2j)ms be represented on a diagram, as shown here. Velocity is a quantity that has both magnitude and direction, ‘The magnitude of the velocity is the speed. If v= (4i—2j)ms" then, speed = ya)’ + (-2)° =v =2/S5ms" You should already know the formula for an object moving with constant speed:> CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK Similarly, the formola for an object moving with constant velocity is: t Splitting a velocity into its components The velocity ofa particle traveling north-east at 4/Zms"* can be written in the form (ai + 4j)ms"! 6 and sings? = —— 42 a= 402 x cosas? b= 4/2 x sings? a=4 Hence the velocity vector is (41 + 4j) ms 7 The velocity of a particle travelling on a bearing of 120° at 20ms" can be written in the form (xi + 9j)ms x 2 sin60° = 35, a = 20 x sin 60° y= 20 008 60° x= 10/3 y=l0 Hence the velocity vector is (10/31 ~ 10j)ms~'. (ences cun ‘An object travels at a constant velocity from point 4 to point # AB = (321 — 24})m and the time taken is 4s. Find the velocity b the speed. Answers displacement _ 32i~ 24} slosh = ‘time taken 4 = (8i- 6j)ms" speed = \(8)° + (-6) Consider a boat sailing with velocity (_3) kmh"! ‘At 1200 hous the boat is at the point 4 with position vector ( 3 km relative to an origin 0. 338 >13. Vectors ‘The diagram shows the positions of the boat at 1200 hours, 1pm, 2pm, 3pm, 4pm... Thepostionst4pm= (,3)-+4(_3)=(8) ‘Hence the position vector, r, of the boat r hours afier 1200 hours is given by the expression: (3)*(2) ‘This leads to the general rule: WORKED EXAMPLE Particle A starts moving at time ¢ = 0 from the point with position vector 524+ j with a speed of 15ms" in the direction ~3i + 4j, a Find the velocity vector, v4, of 4. b Find the position vector, r,, of 4 after / seconds. Particle B starts moving at time r= 0 from the point with position vector —I1i~ 8] with velocity v,= 121+ 15ims ¢ Show that 4 and B collide, and find the value of ¢ when they collide and the position vector of the point of collision.> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK CONTINUED va= 9+ ims 1g = S24 5+ (91+ 13) ry =—11i- 8) + (125 + 15)) Particles A and B collide if r« =r» ata particular time. SQA | + (91 + 12}) = —11i — 8) + 125 + 15)) Equating thes: 52-91=~11 + 12¢ 2 = 63 1=3 Equating thejs: 1+ 12¢=-8 + 15? 3r=9 1=3 Particles A and Beollide when 1 = 3. When (= 3,14 = 521+) +3(-91-+ 12j) = 251 + 37) Fy =—I1i~ 8) + 3(124 + 15}) = 258 + 37} Hence particles 4 and B collide when 1 = 3 at the point with position vector 251+ 37}. Exercise 13.4 1 2 Displacement = 14 + 54])m, time taken = 6 seconds. Find the velocity, ‘Velocity = (Si ~ 6j)ms”, time taken = 6 seconds, Find the displacement. Velocity = (-4i + 4j)kmh'™, displacement = (—50i + 50j) km. Find the time taken. 2 A car travels from a point A with position vector (604 ~ 40j)km to a point B with, position vector (~S0i + 18}) km. The car travels with constant velocity and takes 5 hours to complete the journey. Find the velocity vector. 3 A helicopter fies from a point P with position vector (50i + 100})km to a point Q. The helicopter flies with a constant velocity of (30i ~ 40j) kmh” and takes. 2.5 hours to complete the journey. Find the position vector of the point @. 340)13. Vectors 4 a Acartravels north-east with a speed of 18/2kmh”'. Find the velocity veetor of the ea: b A boat sails on a bearing of 030° with a speed of 20kmir! Find the velocity vector of the boat. © A plane fies on a bearing of 240° with a speed of 100ms" Find the velocity vector of the plane. 5 A particle starts at a point P with position vector (~80i + 60j)m relative to an origin O. ‘The particle travels with velocity (12i ~ 16]) ms” 2 Find the speed of the particle. b Find the position vector of the particle after 1 second, i 2seeonds ¢ Find the position vector of the particle r seconds after leaving P. ji 3seconds. 6 At 1200 hours, a ship leaves a point Q with position vector (101 + 38]) km relative to an origin O. The ship travels with velocity (61 — 8)) kmh”. @ Find the speed of the shij b Find the position vector of the ship at 3pm. € Find the position vector of the ship s hours after leaving Q. Find the time when the ship is at the point with position vector (61i ~ 30) km. 7 At 1200 hours, a tanker sails from a point P with position veetor (Si-+ 12j) km relative to an origin O. The tanker sails south-east with a speed of 12/2 kmh! Find the velocity vector of the tanker. b Find the position vector of the tanker at i 1400hours fi 1245 hours ‘¢ Find the position veetor of the tanker ¢ hours after leaving P. 8 At 1200 hours, a boat sails from a point P. ‘The position vector, rkm, of the boat relative to an origin O, # hours after 1200 is, aiven by r= ('0) +¢( 3) 2 Write down the position vector of the point P. b Write down the velocity vector of the boat © Find the speed of the boat. d_ Find the distance of the boat from P after 4 hours. 9 At 1500 hours, a submarine departs from point 4 and travels a distance of 120km toa point B. ‘The position vector, rkm, of the submarine relative to an origin O, ¢ hours after (ist 20+ 61} 2 Write down the position vector of the point a. b Write down the velocity vector of the submarine, © Find the position vector of the point B. 1500 is given by SSeS ee ees 341)> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSESOOK 10 AC1200 hours, boats 4 and B have position vectors (101 + 40})km and (70i + 10})km and are moving with velocities (201 + 10])kmh™ and (-10i + 30)) kmh respectively. Find the position vectors of A and Bat 1500 hours. b Find the distance between 4 and Bat 1500 hours. 11. Attime r= 0, boat P leaves the origin and travels with velocity (3i + 4]) kmh” Also at time = 0, boat Q leaves the point with position vector (~10i + 17})km ple toa friend and travels with velocity (Si + 2)) kmh” Datweet speed anid a Write down the position vectors of boats A and B after 2 hours. velocity. b Find the distance between boats P and Q when = 2. AB means the position vector of B relative to A. AB=OB-OA or AB=b-a If an object has intial position a ancl moves with a constant velocity , the position vector r, at time f, is given by the formula: r= a + w. Past paper questions Worked example Particle A is at the point with position vector (3) at time = 0 and moves with a speed of 10ms" in these dinetn as (3). 38 a i Given that A is at the point with position vector ( ) when r= 6s, find the value of the constant a. BI Particle 2s atthe point with position vector (4) at time ¢= 0 and moves with velocity (3) ms” ji Write down, in terms of ¢, the position vector of Bat time 1s. au Verify that particles A and B collide. 4 iv Write down the position vector of the point of collision. 0) Cambridge IGCSE Additional Mathematies 0606 Paper 11 QL0 Nov 201813. Vectors Answers 1 Magnitude of (3) Velocity vector of A is: v4 = 2(3) =($) Position vector of 4 at ime ris:r4 = (_2} + (8) When 1= 6.1, (38) (a= (-3)+6(3) ()-(3) Hence a= 48 Position vector of B at time tis: Particles A and # collide ifr, = (2)*s)=(5)"«G) and -5+8r=3742 (37)*#(3) ata particular time. 6r=a2 1=7 Hence particles A and 2 collide when ¢ = 7. Wy Whens ras (_3}+7(8) and ep=(39)+7(3) w=(8) mt welt Position vector of the point of collision is ($4 1. The position vectors of the points 4 and B relative o an origin O are ~24 + 17) and 61 +2 respectively i Find the veotor 4B. a fi Find the unit vector in the direction of AB. _ 2 {il ‘The position vector of the point C relative to the origin O is such that OC = O4 + mB, where m isa constant, Given that Ces on che x-axis, find the vector OC. Bl Cambridge IGCSE Additional Mathemeaies 0606 Paper 22 Q5 Mar 2015 2 a Thefourpoints O, A, Band C are such that OA = Se, OB = 15b, OC = 24h ~ 3a, Show that 2 ies on the fine AC. BI 'b Relative to an origin O, the position vector of the point P is i ~ 4j and the position vector of the point Qis 3i+ 7j. Find + POL s el the unit vector in the direction PO, uy iii the position vector of M, the mid-point of PQ. 2 Cambridge IGCSE Additional Mathematies 0606 Paper 21. Q? Jun 2015 30)> CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK 3 Relative to an origin O, points 4, Band C have positon vectors (3), (~19) ana ( _ig) respectively. All distances are measured in kilometres. A man drives at a constant speed directly from A to B in 20 minutes i Calculate the speed in kmh” at which the man drives from A to B. He now drives directly from B to Cat the same speed ii Find how long it takes him to drive from B to C. Gi 8 Cambridge 1GCSE Additional Mathematics 0606 Paper 21 Q3 Nov 2013 o 2 # ‘The position vectors of points A and B relative to an origin O are a and b respectively. The point Pis such that OF = 04. The point Qs such that OG = 108. The lines AQ and BP intersect at the point R. Express 4 in terms of 4, a and b. Express BP in terms of ,a and b Itis given that 34 = AQ and 8BR = 7BP. Express OR in terms of 4, aand b. iv Express OR in terms of ye, aand b, Vv Hence find the value of und of 2. oO m fe} Py) G8 Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q12 Nov 2014 5 a The veetors pand q are such that p = Ili ~ 24j and q =2i + oj i Find the value of each of the constants a and f such that p + 2q = (a + fi — 26} {i_Using the values of a and f found in part i, find the unit vector in the direction p + 24 b 4 ® B The points 4 and B have position vectors a and b with respect to an origin O. ‘The point C'lics on AB and is such that 4B: AC is 1:2, Find an expression for OC in terms of a, band A. © The points S and 7 have position vectors sand t with respect to an origin O. The points O, $ and T'do not lie in a straight line. Given that the vector 2s + jt is parallel to the vector (a+ 3)s + 9t, where isa positive constant, find the value of a a 8) G8 Cambridge IGCSE Additional Mathematics 0606 Paper 22 010 Mar 2016 aa)13. Vectors & @ Avedtory has magnitude of 102 unis and bas the same rection as(_,$). Find v in the form (51), where a and b are integers, el (4) pma z (e be vectorse=($)anda=(sp 5 4) aresuch thao +26 (2° Find the possible values of the constants p and q. 161 Cambridge IGCSE Additional Mathematics 0606 Paper 12 @? Mar 2017 7 The diagram shows the quadrilateral OABC 4 such that Of = a, OB = band OC Wis given that AMEMC= 2:8 and OMA = i Find AC in terms of and. ul Find OM in terms of @ and e. ° 3 (2 Find OM7in terms of b. a Find 5a + 10ein terms of b. af el v Find AB in terms of a and ¢, giving your answer in its simplest form. pl Cambridge IGCSE Additional Mathematies 0606 Paper 12 Q6 Mar 2018 8 Relative to an origin O, the position vectors of the points A and B ate 2i + 12} and 64 ~ 4j respectively: i Write down and simplify an expression foraB. 2 ‘The point C lies on AB such that AC:CBis 1:3. Find the upit vector in the direction of OC. 4) ‘The point D lies on OA such that OD:DA is 1:8 Find an expression for AD in terms of 4, iand j a Cambridge IGCSE Additional Mathematics 0606 Paper 22 Q8 May 2019 9 Theposton eto of thee pins 4, Band C ate oan vin 0, are (8), (!8) and (3) expe Given that 4C = 4BC, find che unit vector in the direction of OC. fo Cambridge IGCSE Additional Mathematics 0606 Paper 22 Q4 Mar 2020 10 In this question all distances are in km. A ship P sails from a point 4, which has position vector (3). with a speed of s2kmin the direction of (73 ind the velocity veetor of the ship. b Write down the position vector of Pata time s hours after leaving. At the same time that ship P sails from A, ship Q sails from a point B, which has position vector ( 25) ate! with velocity weotor (“35 ) km Write down the position vector of Qat atime ehours after leaving B. u Using your answers to parts band e, find the displacement veetor PO al time ¢ hours. iD) Hence stiow that PQ = \'341" ~ 1681 + 208. a Find the value of ¢ when Pand Q are first 2km apart. Pe Cambridge IGCSE Additional Mathemaries 0606 Paper 12 Q8 Mar 2020 345 > Sheena> CAMBRIDGE IGCSE™ AND © LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK 11 A particle Pis moving with a velocity of 20ms” in the same direction as (3) Find the velocity vector of P. B ‘At time ¢= 0s, P has postion vector (3 ) relative to a fixed point 6. li Write down the position yeetor of P alter ts. a A parce Qs postion sor (17) lave to Oat tine r= Os and sa voc vector ($s Given that P and Q collide, ind the value of t when they collide and the position vector of the point of collision. a Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q5 Jun 2019 12a 2B o = ¢ ‘The diagram shows a figure OABC, where Od = a, OB =b and OC =e. ‘The lines ACand OB intersect atthe point M where M isthe midpoint of the line AC. i Find, in terms of a and ¢, the vector OM. 2 Wi Given that OM:MB = 23, find bin terms of a and ral b Vectors j and j are unit vectors parallel to the x-axis and y-axis respectively. ‘The vector p has a magnitude of 39 units and has the same direction as ~10i + 249. i Find p in terms of i and j. 2 fi Hence find the vector q such that 2p + q is parallel to the positive y-axis and has a magnitude of 12units, BI iil Hence show that |q] = kv/3, where k is an integer to be found. Rl Cambridge IGCSE Addtional Mauhematies 0606’ Paper 11 QS Jin 2017 19 -A particle Pi intially at the point with position vector (7) ) and moves with a constant speed of 10s"! in the same direction as (~$). 2 Find the position vector of P after 1s. Bl ‘As P starts moving, a particle Q starts to move such that its position vector after (sis given by (~89) + (3 b Write down the speed of 1 © Find the exact distance between P and Q when f= 10, giving your answer in its simplest surd form, 81 Cambridge IGCSE Additional Mathematics 0606 Paper 11 6 Nov 2020