Vectors. a Write down the vector AB as a column vector ( x y ). A (3, 2) x point C such that BC = 3. . Go to a OA = a

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1 Streth lesson: Vetors Streth ojetives efore you strt this hpter, mrk how onfident you feel out eh of the sttements elow: I n lulte using olumn vetors nd represent the sum nd differene of two vetors grphilly. I n identify olumn vetors tht re prllel. hek-in questions omplete these questions to ssess how muh you rememer out eh topi. Then mrk your work using the nswers t the end of the lesson. If you sore well on ll setions, you n go stright to the Revision heklist nd Exm-style Questions t the end of the lesson. If you don t sore well, go to the lesson setion indited nd work through the exmples nd prtie questions there. 1 In the digrm, is the point (, ) y nd is the point (7, 7). (7, 7) Write down the vetor s olumn vetor ( x y ). (, ) Write down the oordintes of 0 x point suh tht =. Go to.1 O is prllelogrm. is prllel to O. O is prllel to. O = O = O Express eh vetor in terms of nd. i ii O N is the midpoint of. Express ON in terms of nd. Go to.1 Edexel GSE (9-1) Mths for Post-16 HrperollinsPulishers 017

2 .1 Vetors slr quntity n e fully desried y its mgnitude or size, for exmple, temperture. vetor quntity is fully desried y its mgnitude nd diretion, for exmple, veloity hs mgnitude (lled speed) nd diretion. Four types of nottion n e used to represent vetors. The vetor indited y n rrow in the shpe elow n e written s: or or or The diretion of the vetor is usully shown y n rrow. 5 If DE = k, then nd DE re prllel nd the length of DE is k times the length of. = 10 DE = = 5 D E DE = If two vetors re equl, they re prllel nd equl in length. is equl to. If the vetor =, then the vetor is in the opposite diretion to, so = 4. The olumn vetor 4 mens units to the left nd 4 units down. dding vetors The resultnt of two vetors is found y dding them. Vetors must lwys e dded end to end so tht the rrows follow on from eh other. resultnt is usully lelled with doule rrow. Edexel GSE (9-1) Mths for Post-16 HrperollinsPulishers 017

3 = ( 4) ( ) = 5 + To tke the route diretly from to is equivlent to trvelling from to nd then from to. The strting points nd finishing points re the sme. This mens = +. + = 4 ( ) ( + 5 ) ( = 9 0 ) Sutrting vetors n e interpreted s + ( ). + ( ) = 4 ( ) ( + 5) ( 6 1) Therefore = Position vetors The position vetor of point is the vetor O, where O is the origin. In the digrm, the position vetors of nd re nd respetively. O Using this nottion, = + = Exm tips On n exm pper, vetors represented y lower se letters will e shown in old. In your working, you re expeted to underline lower se letters representing vetors, for exmple,. lwys put rrows to show diretion on lines tht represent vetors. Edexel GSE (9-1) Mths for Post-16 HrperollinsPulishers 017

4 Exmple 1 Q O is tringle. N splits in the rtio 1 :. O = nd O =. Express, in terms of nd, the vetors i ii ON. N O i = O + O Go from to vi O. = + ii ON = O + N N = 1 = + 1 ( + ) = = + 1 = 1 ( + ) Ftorise the expression. Prtie questions 1 Express eh of these s olumn vetor. d e f g h i j Drw digrm on squred pper to represent eh vetor. 0 k = l = 0 m = 1 n = 0 p = q = r = Edexel GSE (9-1) Mths for Post-16 HrperollinsPulishers 017

5 dd the olumn vetors nd drw tringle on grid to show the resultnt. The first one is done s n exmple = d p =, q = 1 1 nd r = Drw vetor digrm on grid to represent eh of these. i p ii q iii r iv p + q v r vi r + p Write eh resultnt s single olumn vetor. i p + r ii q + r iii p q iv q r v p vi q vii r viii q r 1 5 G K P F J N E I M O D H L O = nd OD= Write down the resultnt of eh of these. The first one is done s n exmple. i + = OE ii + iii + iv + v + Write eh of these vetors in terms of nd/or. i OH ii O iii OI iv OJ v ON REVISION HEKLIST The diretion of vetor is shown with n rrow. The resultnt of two vetors is found y vetor ddition or sutrtion. Vetors must lwys e omined end to end. Edexel GSE (9-1) Mths for Post-16 HrperollinsPulishers 017

6 Exm-style questions 1 1 = = Work out eh of these: i ii iii + 4 = = 1 Work out: i + ii iii The digrm shows regulr hexgon, EFG with entre D. Express these vetors in terms of nd. F FE GD d F F G E D Edexel GSE (9-1) Mths for Post-16 HrperollinsPulishers 017

7 hpter Streth lesson: nswers hek-in questions (5, 4) i ii 1 ( + ) 4 p q r p + q r r + p.1 Vetors 1 = = 1 = d = 4 0 e = 0 i 0 ii 1 iii 5 iv 0 5 f = 4 0 g = = = 6 h = i = j = 4 1 v ii OF iv OG vi 9 iii OI v OM i ii iii + or + iv + or + v + or + Exm-style questions 1 i i ii 0 ii vii iii iii 1 viii d d + = Edexel GSE (9-1) Mths for Post-16 HrperollinsPulishers 017

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