Similarly it would not be sensible to travel 60 km due north in order to go from London to Brighton when Brighton is 60 km due south of London.

Transcription

1 HPTER HPTER 1 nd vector nottion When ll is thrown, the direction of the throw is s importnt s the strength of the throw So if netll plyer wnts to pss the ll to nother plyer to her right it would e no use throwing the ll to her left Similrly it would not e sensile to trvel 60 km due north in order to go from London to righton when righton is 60 km due south of London N London righton In mthemtics there re mny quntities tht need direction s well s size in order to descrie them completely The netll exmple illustrtes the fct tht to descrie force the direction is importnt How ody moves when it is pushed or pulled will depend on the direction of the push or pull s well s the size or mgnitude of the push or pull The second exmple illustrtes the fct tht to descrie chnge in position or displcement,it is necessry to give the direction of the movement s well s the distnce moved To tell someone how to get from London to righton it is not enough to tell them tht righton is 60 kilometres from London It would e necessry to tell them tht righton is lso due south of London Forces nd displcements tht need size nd direction to descrie them re exmples of vectors vector needs mgnitude nd direction to descrie it completely Velocity is nother exmple of vector To descrie velocity it is necessry to give its mgnitude (speed) nd direction, for exmple 0 km/h north In this chpter, only displcement vectors will e considered ut the results pply to other vectors The displcement from to is cm on ering of 00 This displcement is written to show tht it is vector nd it hs direction from to In the digrm the line from to is drwn cm long in direction of 00 nd it is mrked with n rrow to show tht the direction is from to cn lso e lelled with single old letters such s, nd c N 0 cm 6

2 1 nd vector nottion HPTER Exmple 1 vector hs mgnitude cm nd direction 080 Drw the vector Solution 1 N Drw line cm long (mgnitude) on ering of 080 (direction mrked with n rrow) 80 cm The vector hs een drwn on centimetre grid The displcement represented y cn e descried s to the right nd up s with trnsltions this cn e written s the column vector So we cn write Exmple Point hs coordintes (1, 6) nd point hs coordintes (, 1) Write s column vector The point is such tht Find the coordintes of Solution y Mrk the points nd on grid 6 x The coordintes of re (, ) To move from to go to the right nd down For, from go to the left nd up to find Exercise 1 Drw ccurtely nd lel the following vectors i Vector with mgnitude cm nd direction est ii Vector with mgnitude cm nd direction with ering 00 iii Vector c with mgnitude cm nd direction with ering 0 iv Vector with mgnitude 6 cm nd direction v Vector PQ with mgnitude cm nd direction 10 6

3 HPTER n squred pper drw nd lel the following vectors i 1 ii iii c iv v D 0 The point is (1, ), the point is (6, 9) nd the point is (, ) Write s column vectors i ii iii Wht do you notice out your nswers in? The points,, nd D re the vertices of qudrilterl where hs coordintes (,1),, nd D 1 n squred pper drw qudrilterl D Write s column vector D c Wht type of qudrilterl is D? d Wht do you notice out nd D? The points,, nd D re the vertices of prllelogrm hs coordintes (0, 1), nd D 0 n squred pper drw the prllelogrm D Write s column vector i D ii c Wht do you notice out i nd D ii D nd? Equl vectors s vectors need mgnitude (size) nd direction to descrie them, vectors re equl only when they hve equl mgnitudes nd the sme direction c d e The vectors nd re equl, tht is They hve the sme mgnitude nd direction The vectors nd c re not equl They do not hve the sme mgnitude lthough they hve the sme direction The vectors nd d re not equl They hve the sme mgnitude nd re prllel ut they re in opposite directions nd so do not hve the sme direction The vectors nd e re not equl They hve the sme mgnitude ut they do not hve the sme direction 6

4 The mgnitude of vector HPTER The mgnitude of vector The mgnitude of the vector is written or The mgnitude of the vector is, tht is, the length of the line segment Exmple 6 Find the mgnitude of the vector Give your nswer i s surd ii correct to significnt figures Solution 6 mens to the right nd 6 down Drw right-ngled tringle to show this i 1 ii 71 (to sf) Use Pythgors theorem to find the length,, of the hypotenuse Notice tht in this exmple 6 the mgnitude of the vector is () ( 6) 16 6 In generl the mgnitude of the vector is x y x y Exmple Find the mgnitude of the vector Solution ( ) ) ( 9 16 Sustitute x nd y into x y 6

5 HPTER Exercise 1 Here re 8 vectors c e d f g h There re pirs of equl vectors Nme the equl vectors Work out the mgnitude of ech of these vectors Where necessry, nswers my e left s surds i ii 1 iii c 1 1 iv d v 8 vi PQ In tringle, 0 nd 1 7 Work out the length of the side of the tringle Show tht the tringle is n isosceles tringle In qudrilterl D,, 0, D Wht type of qudrilterl is D?, D 0 ddition of vectors The two-stge journey from to nd then from to hs the sme strting point nd the sme finishing point s the single journey from to Tht is, or This is written s to followed y to is equivlent to to followed y is equivlent to Notice the pttern here gives This leds to the tringle lw of vector ddition This does not men tht The sum of the lengths of nd is not equl to the length of 66

6 ddition of vectors HPTER Tringle lw of vector ddition Let represent the vector nd represent the vector Then if represents the vector c c c Prllelogrm lw of vector ddition PQRS is prllelogrm In prllelogrm, opposite sides re equl in length nd re prllel S R So since PQ nd SR re lso in the sme direction PQ SR ( ) P Q Similrly PS QR ( ) From the tringle lw PQ QR PR so tht PR Hence PR PQ PS s PQ nd PS So if in prllelogrm PQRS, PQ represents the vector nd PS represents the vector, the digonl PR of the prllelogrm represents the vector oth of these lws llow vectors to e dded ut the tringle lw is the esier to use When c the vector c is sid to e the resultnt of the two vectors nd Exmple Find, y drwing, the sum of the vectors nd Solution Use the tringle lw of vector ddition Move vector to the end of vector so tht the rrows follow on Drw nd lel the vector to complete the tringle could lso hve een found y moving the vector to the eginning of vector The nswer is the sme s the two tringles re congruent 67

7 HPTER Exmple 6 In the qudrilterl D,, D nd D c Find the vectors i ii D c Solution 6 i so ii D D so D ( ) c D c Exmple 7 nd 8 Find Use the tringle lw of vector ddition Mke sure tht the s follow ech other Use Vector expressions like this cn e treted s in ordinry lger The rckets cn e removed Solution Use the tringle lw of vector ddition Drw sketch From to is to the right From to is 8 to the right So from to is 8 11 to the right From to is up From to is down So from to is 1 up Exmple 7 shows tht Exmple 8 6 nd Find Solution 8 6 c d 6 c d dd cross 68

8 Prllel vectors HPTER Exercise 1 vector hs mgnitude cm nd direction 00 vector hs mgnitude 7 cm nd direction 10 Drw the vector i ii iii Work out 6 c d e PQ QR 1 Work out PR p q 1 r Work out i p q ii q p Wht do you notice? c Work out i (p q) r ii p + (q + r) d Wht do you notice? DEF is regulr hexgon n Explin why ED n m D p Find i ii D c Wht is FD? F E D Prllel vectors Using the ordinry rules of lger ut wht does this men? Here is the vector Here re nd is vector in the sme direction s nd with twice the mgnitude For, 10 tht is, 69

9 HPTER Similrly is vector in the sme direction s nd with mgnitude times the mgnitude of 6 1 nd The vector is the displcement from to nd is the displcement from to These displcements hve the sme mgnitudes ut re in opposite directions so followed y is the zero displcement (0) s there is no overll chnge in position This is written 0 Using the usul rules of lger it follows tht nd hve the sme mgnitude ut opposite directions negtive sign in front of vector, reverses the direction of the vector so showing tht the reverse of to the right nd down is to the left nd up The vector hs the sme mgnitude s ut is in the opposite direction The vector hs the sme mgnitude s ut is in the opposite direction So the vector hs times the mgnitude s ut is in the opposite direction tht re prllel either hve the sme direction or hve opposite directions For ny non-zero vlue of k, the vectors nd k re prllel The numer k is clled sclr; it hs mgnitude only If p then k k p kp q q kq Exmple 9 Here is the vector Drw the vectors i ii 70

10 Prllel vectors HPTER Solution 9 i is in the sme direction s ut with times the mgnitude Drw line in the sme direction s ut times longer ii is in the opposite direction to ut with times the mgnitude Drw line in the opposite direction to nd twice s long Exmple 10 Drw the vector Solution 10 mens is the vector reversed in direction Use the tringle lw of vector ddition to dd nd Move vector to the end of vector so tht the rrows follow on Drw nd lel the vector to complete the tringle 71

11 HPTER With origin, the points,, nd D hve coordintes (1, ), (, 7), ( 6, 10) nd ( 1, 10) respectively Write down s column vector i ii Work out i s column vector ii D s column vector c Wht do these results show out nd D? Solution 11 i 1 ii 7 Exmple 11 i Method 1 1 Method ii D c D D The lines D nd re prllel nd the length of the line D is times the length of the line From to is 1 cross nd up From to is cross nd 7 up to, tht is, (1, ) to (, 7) is 1 cross nd up nother wy to otin is to use the tringle lw of vector ddition to D is to the right nd 0 up nd k re prllel vectors With the origin, the vectors nd re clled the position vectors of the points nd p In generl the point (p, q) hs position vector q 7

12 Prllel vectors HPTER Exmple 1 Simplify i ii 1 ( ) Solution 1 i ii 1 ( ) The ordinry rules of lger cn e pplied to vector expressions like this 1 ( ) Exmple 1 is stright line where Express in terms of nd Solution 1 Use the tringle lw of vector ddition s, Exercise D 1 The vector hs mgnitude cm nd direction 10 The vector hs mgnitude cm nd direction 0 Drw the vector i ii iii iv Here is the vector p Drw the vector i p ii 1 p p m n 6 p 6 Find s column vector i m ii n iii m p iv m n p Find i the mgnitude of the vector m ii the mgnitude of the vector m p The points P, Q, R nd S hve coordintes (, ), (, 1), ( 6, 9) nd (1, ) respectively Write down the position vector, P, of the point P Write down s column vector i PQ ii RS c Wht do these results show out the lines PQ nd RS? 7

13 HPTER The point hs coordintes (1, ), the point hs coordintes (, ), the point hs coordintes (, ) Find the coordintes of the point D where D 6 6 i Express in terms of nd ii Where is the point such tht 1? 7 Here re vectors m n, D 6m 1n, EF m 8n, GH m n, IJ 6m 16n Three of these vectors re prllel Which re the prllel vectors? Simplify i 8p q p 8q ii (m n) (m 6n) 8 Here is regulr hexgon DEF In the hexgon F is prllel to nd twice s long m Express F in terms of m D n Express FD in terms of m nd n x c Express in terms of m nd x The lines nd FD re prllel nd equl in length d Find n expression for x in terms of m nd n F E D 6 Solving geometric prolems in two dimensions To solve geometric prolems the following results re useful i Tringle lw of vector ddition so tht PQ QR PR ii When PQ, QP iii When PQ krs, k sclr (numer), the lines PQ nd RS re prllel nd the length of PQ is k times the length of RS iv When PQ kpr then the lines PQ nd PR re prllel ut these lines hve the point P in common so tht PQ nd PR re prt of the sme stright line Tht is, the points P, Q nd R lie on the sme stright line Exmple 1 7 In tringle the point M is the midpoint of nd the point N is the midpoint of i Express in terms of nd N ii Express MN in terms of nd iii Explin wht the nswers in i nd ii show out nd MN M

14 6 Solving geometric prolems in two dimensions HPTER Solution 1 i Use the tringle lw of vector ddition so ii M 1 M is the midpoint of so M 1 Similrly N MN M N MN iii MN Use the tringle lw of vector ddition M so tht M nd MN This mens tht nd MN re prllel nd tht the length of is twice the length of MN Exmple 1 is qudrilterl in which, nd i Find in terms of nd nd explin wht this nswer mens ii Find in terms of nd D is the point such tht D nd X is the midpoint of Find in terms of nd iii D iv X nd v explin wht these results men Solution 1 i Express in terms of known vectors using the tringle lw of vector ddition nd re prllel nd the length of is twice the length of ii so Express in terms of known vectors using the tringle lw of vector ddition so could lso hve een used 7

15 HPTER X D D mens tht the point D is on extended so tht D nd hve the sme length Redrw the digrm with D in nd X the midpoint of iii D D Use the tringle lw of vector ddition for D, D D 6 D D 6 iv s X is the midpoint of X 1 1 ( ) X 1 X X 1 X 1 v D X Use the tringle lw of vector ddition for X D 6 X 1 So the lines D nd X re prllel with the point in common This mens tht X nd D re prt of the sme stright line Tht is, XD is stright line such tht the length of D is times the length of X In other words X is the midpoint of D Exercise E 1 The points, nd hve coordintes (, 1), (, ) nd (11, 0) respectively Find s column vectors i ii Wht do these results show out the points, nd? D is qudrilterl is the point (1, ), is the point (, 6) nd P is the midpoint of Write down the coordintes of P is the point (7, ) nd Q is the midpoint of Write down the coordintes of Q D is the point (7, ) nd R is the midpoint of D c Write down the coordintes of R S is the midpoint of D d Write down the coordintes of S e Find s column vectors i PQ ii SR f Explin with resons wht the nswers to e show out the qudrilterl PQRS 76

16 6 Solving geometric prolems in two dimensions HPTER In tringle, nd i Find in terms of nd, the vector P is the midpoint of ii Find in terms of nd, the vector P iii Find in terms of nd, the vector P P is prllelogrm with nd P is the midpoint of i Use the result of question to write down P in terms of nd ii Express in terms of nd Q is the midpoint of iii Express Q in terms of nd iv Wht do your nswers to i nd iii show out the points P nd Q? v Wht property of prllelogrm hs een proved in this question? KLMN is qudrilterl where KL k, LM m, MN n nd KN m Wht type of qudrilterl is KLMN? Express n in terms of k nd m 6 is prllelogrm with nd E E is the point on such tht E 1 F is the point on such tht F 1 Find in terms of nd i ii E iii E iv F v EF Write down two geometric properties connecting EF nd F 7 M m P Q n N In tringle MN, M m nd N nthe point P is the midpoint of MN nd Q is the point such tht Q P Find in terms of m nd n i P ii Q iii MQ The point R is such tht R N Find in terms of m nd n, the vector MR MR c Explin why MQR is stright line nd give the vlue of M Q 77

17 HPTER 8 In the digrm R 6, P nd PQ The point M is on PQ such tht PM The point N is on R such tht N 1 R The midpoint of MN is the point S i Find in terms of nd/or the vector NM ii Find in terms of nd/or the vector S P 6 Q R T is the point such tht QT iii Find in terms of nd,the vector T iv Give geometric fct out the point S nd the line T v When 8 nd find the length of QR 1 hpter summry You should now: understnd nd e le to use the vector nottion, p, nd q know tht vector hs mgnitude nd direction know tht ( ) You should lso e le to: recognise equl vectors find the mgnitude of vector y using p q p q dd vectors using the tringle nd prllelogrm lws of vector ddition simplify vector expressions including those involving sclr multiples recognise prllel vectors solve geometricl prolems using vector methods including recognising when three points lie on stright line hpter review questions 1 The digrm shows two vectors nd PQ Use the resource sheet to drw the vector PQ on the grid (188 Mrch 00) P 78

18 hpter review questions HPTER is the point (0, ) Find the coordintes of is the point (, ) D is digonl of the prllelogrm D Express D s column vector E 1 c lculte the length of E (18 June 1999) PQR is trpezium PQ is prllel to R P PQ R 6 M is the midpoint of PQ N is the midpoint of R i Find in terms of nd the vector M ii Find in terms of nd the vector MN X is the midpoint of MN iii Find in terms of nd, the vector X 6 N R The lines X nd PQ re extended to meet t the point Y iv Find in terms of nd, the vector NY (18 June 000) P M Q Digrm NT ccurtely drwn PQRS is prllelogrm T is the midpoint of QR U is the point on SR for which SU : UR 1: PQ PS Write down in terms of nd, expressions for i PT P S ii TU (18 Novemer 000) U Q T R Digrm NT ccurtely drwn is the point (, ) nd is the point (, 0) i Write s column vector ii Find the length of the vector D is the point such tht D is prllel to 0 nd the length of D the length of 1 is the point (0, 0) Find D s column vector is the point such tht D is rhomus is digonl of the rhomus c Find the coordintes of (18 June 001) 79

19 HPTER 6 The digrm shows regulr hexgon, DEF, with centre 6 Digrm NT X ccurtely drwn Express in terms of nd/or i ii EF X is the midpoint of F Express EX in terms of nd/or E D Y is the point on extended such tht : Y : c Prove tht E, X nd Y lie on the sme stright line (187 June 00) 7 is prllelogrm P is the point on such tht P 6 6c i Find the vector P Give your nswer in terms of nd c 6c The midpoint of is M ii Prove tht PM is stright line (187 June 00) 6 P Digrm NT ccurtely drwn 8 DEF is qudrilterl with D, DE nd F D i Express E in terms of nd ii Prove tht FE is prllel to D M is the midpoint of DE iii Express FM in terms of nd X is the point on FM such tht FX : XM :1 iv Prove tht, X nd E lie on the sme stright line F X M E 9 is trpezium M is the midpoint of N is the midpoint of M N i Find in terms of M N P Digrm NT ccurtely drwn ii Find in terms of nd iii Find in terms of nd P is the midpoint of iv Prove tht MP is prllel to 80

20 hpter review questions HPTER 10 PQ is tringle R is the midpoint of P S is the midpoint of PQ P p nd Q q i Find S in terms of p nd q ii Show tht RS is prllel to Q R p P S Digrm NT ccurtely drwn q Q (187 Novemer 00) 11 PQR is trpezium with PQ prllel to R P PQ R 6 P M Q Digrm NT ccurtely drwn M is the midpoint of PQ nd N is the midpoint of R i Find the vector MN in terms of nd N R X is the midpoint of MN nd Y is the midpoint of QR ii Prove tht XY is prllel to R (187 June 00) 81