Chapter 15 VECTORS EXERCISE 15A. Scale: 1 cm 10 km h 1 1 cm 10 ms 1 N. 35 ms km h-1 W S. Scale: N cm 10 m. 60 ms -1.
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1 Chate VECTORS EXERCISE a Sale: Sale: m 0 km h m 0 ms W E ms - 0 km h- S W S E W 0 0 E Sale: m 0 m 60 ms - m unway S Sale: m 0 ms a Sale: m newtons Sale: m newtons 7 newtons W S E 60 newtons W S E a Sale: m 0 km h Sale: m kg m s W Sale: m 0 km 60 km h- S W E km S E W Sale: m 0 km h kg m s- 90 km h- 0 E 0 S unway
2 Mathematis MYP+ (n en, Chate VECTORS 9 EXERCISE B a Vetos a,, an e ae eual in magnitue Vetos an ae eual in magnitue Vetos a,,, an ae aallel Vetos a an ae in the same ietion one of the vetos ae eual Vetos an ae in the same ietion e Vetos a an ae negatives of one anothe Vetos an ae negatives of one anothe a False (! RS has same length, ut oosite ietion Tue False (! QS has same length, ut iffeent ietion Tue e Tue EXERCISE C a e f + + a! QR +! RS! QS! PQ +! QR! PR! PS +! SR +! RQ! PR +! RQ! PQ! PR +! RQ +! QS! PQ +! QS! PS a i ii + + Yes, + + fo any two vetos an
3 0 Mathematis MYP+ (n en, Chate VECTORS Sale: m km C km 0 B 0 km a 0 t oint, (µ + 60 ± 0 ± 0 ± fangles in a evolutiong The two oth lines ae aallel, so (0 ± + BC+(µ + 80 ± fo-inteio anglesg 0 ± + BC +0 ± 80 ± BC 90 ± Using Pythagoas theoem: C 0 + C 0 + km To fin µ, we notie that tan µ 0 µ tan ( µ ¼ 6:9 ± So, the eaing is 0 ± +6:9 ± ¼ 7 ± Paolo is km fom his stating oint on a eaing of 7 ± C The two oth lines ae aallel, so ± + ( ± + BC 80 ± fo-inteio anglesg BC 90 ± B km km 676 km Using Pythagoas theoem: C : +: C : +: ¼ 6:76 km To fin the eaing µ, we see that tan(µ + ± : : µ + ± tan ( : : µ tan ( : : ± µ ¼ 6:60 ± Gina is 6:76 km fom he stating oint on a eaing of 006:60 ± 6 a Sale: m 00 km h Sale: m 00 km h 00 km h- 00 km h- 600 km h- 00 km h- 00 km h- 00 km h- the see of the aeolane is 600 km h ue noth the see of the aeolane is 00 km h ue noth
4 Mathematis MYP+ (n en, Chate VECTORS Sale: m 00 km h The magnitue of the esulting veto is fpythagoasg 00 km h- ¼ 0 km h 00 km h- To fin the eaing, we notie that tan µ µ tan ( µ ¼ : ± The see of the aeolane is 0 km h on a eaing of 0: ± 7 a Sale: m km h The oat s atual see will e 0 km h Sale: m km h 0 km h- 0 km h- The oat s atual see will e 0 km h, tavelling east Sale: m km h The magnitue of the esulting veto is 0 km h- 0 km h fpythagoasg ¼ : km h The oat s atual see is : km h, tavelling noth east 8 00 km h- B 60 km h- C Using Pythagoas theoem: C C ¼ 0 km h To fin the eaing, we notie that tan µ µ tan ( µ ¼ 8: ± an 90 ± +8: ± ¼ 98: ± The aiaft is tavelling at a see of 0 km h at a eaing of 098: ±
5 Mathematis MYP+ (n en, Chate VECTORS 9 Sale: m knots knots C B knots Using Pythagoas theoem: C + C + ¼ : knots To fin the eaing, we notie that tan µ µ tan ( µ ¼ 9:87 ± the eaing is ± +9:87 ± ¼ ± The shi is tavelling at a see of : knots at a eaing of ± EXERCISE D a e f a! QR +! RS! QS! RS +! SP +! PQ! RP +! PQ! RQ e! PS! RS! PS +! SR! PR! QP! RP +! RS! QP +! PR +! RS! QR +! RS! QS f! RS +! SR! RR 0 (the zeo veto! RS! PS! QP! RS +! SP +! PQ! RP +! PQ! RQ
6 Mathematis MYP+ (n en, Chate VECTORS Let n e the veloity 00 km h noth, w the win veloity, an x the veloity the aeolane heas fo Then x + w n x n w x w n By Pythagoas theoem: jxj jxj lso, tan µ 0 00 µ tan ( 0 µ ¼ :7 ± ¼ 0:6 km h 0 km h- \ x\ 00 km h- So, in oe to tavel noth at 00 km h, the lane must aim to fly :7 ± west of noth at 0:6 km h Let e the oat s veloity in still onitions, e the uent, an x e the veto to the safe haven Then x + x a ow sin µ 0 0 µ sin ( µ ¼ 9:7 ± an ± 9:7 ± ¼ : ± The oat must hea : ± west of noth By Pythagoas theoem, jxj +0 0 jxj jxj 800 x ¼ 8:8 The oat will e tavelling at 8:8 km h 0 km h- x \ x\ 0 km h- EXERCISE E a a u units units own (- units u ight units The veto is The veto is 0
7 Mathematis MYP+ (n en, Chate VECTORS units left (- units ight units own (- units u 6 units left (-6 units ight e The veto is f 6 The veto is 0 units ight units u 6 units left (-6 units ight units own (- units u 6 The veto is The veto is a Ois (0, 0 is (,! 0 O 0! O Bis (, Cis (,! BC! BC is (, Bis (,! B! B e Cis (, is (,! C! C Cis (, Ois (0, 0! 0 CO 0! CO a Let the hoizontal omonent e x x os 0± 60 0 x 60 os 0 ± x 0 60 ms - y ¼ :0 ms Let the vetial omonent e y y sin 0± 60 y 60 sin 0 ± 0ms :0 The veto is 0 Let the hoizontal omonent e x x sin ± x sin ± x ¼ 9:8 km Let the vetial omonent e y y os ± km y y os ± y : km x 9:8 The veto is :
8 Mathematis MYP+ (n en, Chate VECTORS Let the hoizontal omonent e x x os 9± 60 x 60 os 9 ± 60 km h- y x ¼ 8 km h 9 Let the vetial omonent e y x unway y sin 9± 60 y 60 sin 9 ± y :0 km h The veto is 8 :0 EXERCISE E a a a e a f + a g a + a h + a a e f
9 6 Mathematis MYP+ (n en, Chate VECTORS a! BC! B +! C! B +! C ! C! B +! BD +! DC! B +! BD! CD a length units length ( +( + 6 units length units e length ( units length +( 9+ units f length (6 +( units a i is (, Bis (,! B i is (, Bis (,! B i is (, Bis (0, 0! 0 B 0 i is (, Bis (, 0! B 0 ii B ( +( +9 units ii B +( + 9 units ii B ( +( 9+6 units ii B ( units EXERCISE F a -s -s -s -s Qw_
10 Mathematis MYP+ (n en, Chate VECTORS 7 e f s -s -s -s + s -s -s -s -s g h Qw\s + Qw\s s + s Qw\( + s a 6 s 6 µ s e + s f s g s h ( + s a o any two vetos with the same magnitue an same ietion o any two vetos that have the same length, an ae aallel, ut have oosite ietion o any two vetos whih ae aallel, an have the same ietion, ut with veto thee times as long as o any two vetos that ae aallel, an have the same ietion, ut with veto having the length of
11 8 Mathematis MYP+ (n en, Chate VECTORS e o any two vetos that ae aallel, have oosite ietion, an whee veto is times as long as a Let a e a a has length jaj a + a a an ka k a ka ka jkaj (ka +(ka k a a k (a jkj a fsine jkaj must e > 0g jkjjaj as euie EXERCISE G a x x x x e x + x + x x ( g x +x x + x +x + x +x x + x x ( i + x e + x e x e x (e f h x x x x x x x x ( x ( m x n m x + x n + x n + x m n + x n m n x m n x (m n a y s y s y s +y y s y ( s µ y y y 6 s y s y y (s y
12 Mathematis MYP+ (n en, Chate VECTORS 9 a x x x 8 x +x x x ( x x x µ x x x x ( x x 6 x 6 x 9 6 x 8 x + x x x + x + + x + x e x + x x + x x ( f x + x x + x x ( x x x x 8 8 x µ x x x µ x EXERCISE H a is aallel to an jj jjjj jj is twie as long as, an they have the same ietion is aallel to an jj jj jj is half as long as, an they have the same ietion is aallel to an jj jjjj jj is thee times as long as, ut has the oosite ietion is aallel to an jj jj jj is as long as, ut has the oosite ietion
13 0 Mathematis MYP+ (n en, Chate VECTORS If an k ae aallel, then a k, whee a is a onstant ka a a a an so k k 0 a If an 9 t ae aallel, then k 9 t, whee k is a onstant 9k tk 9k k 9 an so t t 6 If an t then ae aallel, k t, whee k is a onstant tk k k k an so t t 0 If a an a+ 8 ae aallel, Sustituting ( into (: then k a a+ 8, whee k is a onstant 8 a + k ak a+ a 8 a + k a + ( a an ak 8 a +a k 8 a +a 0 ( a (a + (a 0 a o!! PQ SR P(-, S(-, Q(,! PQ! SR R(, - PQ an SR ae aallel, an eual in length This is suffiient eviene to onlue that PQRS is a aallelogam 6 a! B! B 7 Let C e the oint (x, y Let D e the oint (x, y! DC x y x+ y ow, sine BCD is a aallelogam,! B! DC x+ y x + an y x an y oint C is (,! CD x7 y x7 y+ ow, sine BCD is a aallelogam,! B! CD x7 y+ x 7 an y + x 9 an y 0 oint D is (9, 0
14 Mathematis MYP+ (n en, Chate VECTORS EXERCISE I a i ² + + ii jj an jj + + ² jjjjos µ 0 os µ os µ 0 µ os ( 0 µ ± i ² + 0 ii Sine ² 0, the angle etween an is 90 ± (they ae eeniula e i ² ii jj an jj ( ² jjjjos µ os µ os µ 6 µ os ( 6 µ ¼ 9: ± i ² ii jj an jj + + ² jjjjos µ 0 0 os µ 0 0 os µ os µ µ 0 ± i ² + +8 ii jj ( an jj + 9+ ² jjjjos µ 7 os µ os µ µ os ( µ ¼ 70: ± f i ² + 8 ii jj +( +6 7 an jj ( + 9+ ² jjjjos µ 7 os µ os µ µ os ( µ ¼ 7:7 ±
15 Mathematis MYP+ (n en, Chate VECTORS t a s?, t ² 0 t + 0 t 60 t 6 s t+ t?, t+ t ² 0 (t + ( + t 0 (t ++t 0 t 8+t 0 t 80 t 8 s t? 6 t, t ² 6 t 0 6+t t 0 + t 0 t t ( s t? t+, t ² t+ 0 (t + t 0 t +8 t 0 t t 80 (t + (t 0 t o If BC 90 ±, then! B ²! BC 0!! k B an BC k+ 7! B ²! BC (k ++((7 0 k ++70 k +0 k k a! B 6 j! Bj an ( +(6! BC an j! BCj +( ! ow, B ²! BC j! Bjj! BCj os µ ( + (6 6 os µ 7 70 os µ os µ 7 70 B C µ os ( 7 70 BC µ ¼ 7:9 ±!! B an C 6 j! Bj ( +( an j! Cj ( ! ow, B ²! C j! Bjj! Cj os µ C B ( + ( 7 0 os µ 80 os µ os µ 80 µ os ( 80 B C µ ¼ :0 ±
16 Mathematis MYP+ (n en, Chate VECTORS!! QP an QR 9 6 j! QPj ( +9 an j! QRj ( ! ow, QP ²! QR j! QPjj! QRj os µ P R ( + ( os µ 8 89 os µ os µ 8 89 Q µ os ( 8 89 P QR µ ¼ : ±!! MK an ML 7 8 j! MKj ( + an j! MLj ( ! ow, MK ²! ML j! MKjj! MLj os µ L K ( + ( os µ os µ os µ µ os ( M K ML µ ¼ :0 ± a P(-, R(, 0 Q(, -! PQ! PR 0! RQ 0 an j! PQj +( + 0 an j! PRj +( 9+6 an j! RQj +( + Fo angle Q PR:!!!! PQ ² PR j PQjj PRj os Q PR ² 0 os Q PR ( + ( 0 os Q PR 0 os Q PR os Q PR 0 Q PR os ( 7 0 ¼ 8: ±
17 Mathematis MYP+ (n en, Chate VECTORS Fo angle P QR:!!!! QP ² QR j QPjj QRj os PQR ² 0 os PQR fsine! QP! PQ an! QR! RQg ( + ( 0 os P QR 0 os P QR os P QR 0 Fo angle P RQ: Q(-, Fo angle Q PR: Fo angle P QR: P QR os ( 0 ¼ 8: ±!!!! RP ² RQ j RPjj RQj os PRQ ²!! os PRQ fsine RP PRg ( + ( os P RQ 0 os P RQ os P RQ 0 P RQ os ( ¼ : ±! PQ an j! PQj ( + + 9! PR an j! PRj ( +( P(, + 8! QR an j! QRj +( 9+6 R(, -!!!! PQ ² PR j PQjj PRj os Q PR ² 9 8 os Q PR ( + ( os Q PR 6 os Q PR os Q PR 6 Q PR os 6 ( ¼ 66:8 ±!!!! QP ² QR j QPjj QRj os PQR ²!! 9 os PQR fsine QP PQg ( + ( 9 os P QR 9 os P QR os P QR 9 P QR os ( 9 ¼ :±
18 Mathematis MYP+ (n en, Chate VECTORS Fo angle P RQ:!!!! RP ² RQ j RPjj RQj os PRQ ² 8 os PRQ fsine! RP! PR an! RQ! QRg ( + ( 8 os P RQ 8 os P RQ os P RQ 8 P RQ os ( 8 ¼ 8:9± EXERCISE I a y x has gaient y x + has gaient The line theefoe has ietion veto The line theefoe has ietion veto x y 7 has gaient x +y has gaient The line theefoe has ietion veto The line theefoe has ietion veto a x + y has gaient it has ietion veto a x +y 6 has gaient it has ietion veto If the angle etween the lines is µ, then os µ ja ² j jajjj j( + ( j + + jj 0 µ os ( 0 ¼ 8: ± the aute angle is aout 8: ± x +y has gaient it has ietion veto a x + y 8 has gaient it has ietion veto If the angle etween the lines is µ, then os µ ja ² j jajjj j( + ( j + +9 jj 0 0 µ os ( 0 ± the aute angle is ± y x + has gaient it has ietion veto a y x has gaient it has ietion veto If the angle etween the lines is µ, then os µ ja ² j jajjj j( + ( j + +6 jj 7 µ os ( ¼ 9:0 ± the aute angle is aout 9:0 ± y x + has gaient it has ietion veto a x y has gaient it has ietion veto If the angle etween the lines is µ, then os µ ja ² j jajjj j( + ( j j6j µ os ( 6 8 ¼ 9: ± the aute angle is aout 9: ±
19 6 Mathematis MYP+ (n en, Chate VECTORS EXERCISE I Let a a a an So a ² a + a an ² a a + a a + a a ² a ² ² a fo any two vetos a an a Let a a a So a ² a ² a a a a a + a a a + a an jaj a jaj a + a a ² a a ² a jaj fo any veto a Let a a a So, a +, a a a + a + So (a + ² ( +, + a + ² a + an + an (a + ( + +(a + ( + a + a a + a + + (a + a +(a + a +( + +( + a ² + a ² + ² + ² (a + ² ( + a ² + a ² + ² + ² fo any vetos a,,, EXERCISE J! Let B! DC, an! D s! ow, BC! B +! D +! DC! B +! D +! DC + s + s! B! DC, an! BC! D s the oosite sies ae aallel an eual in length uailateal BCD is a aallelogam B s D C If OBC is a aallelogam, B then! B! OC an! CB! O a M! a So, OB! O +! B a +! C a + a O C! M ( a
20 Mathematis MYP+ (n en, Chate VECTORS 7! OM! O +! M a + ( a a + a a + (a +! OB This means that the mioint M of C is also the mioint of OB the iagonals of a aallelogam iset eah othe a i B C s P D P B a! D! B +! BC +! CD! P +! BQ +! CR + + S Q! S! D ( + + D R C + +! PQ! PB +! BQ +! SR! SD +! DR fsine! P! PBg! S! RD fs is the mioint of Dg + + fr is the mioint of CDg + Pats an show that SR is aallel to PQ, an eual in length This means that the uailateal PQRS is a aallelogam a O ow, T M! T! TM! OT! O +! T + + ( +! OB C B! PC! DP s ii! B! P +! PB + s! DC! DP +! PC s + + s (! B It means that B an DC ae aallel an eual in length, making BCD a aallelogam If the iagonals of a uailateal iset eah othe, then the uailateal is a aallelogam! Let B! OC an! O! CB! OM! OC an! M! O +! OM +! So, T! M f! T! TMg ( So,! OT is aallel to! OB Howeve, they have a ommon oint O O, T an B ae ollinea
21 8 Mathematis MYP+ (n en, Chate VECTORS 6! Fom at a, OT! OB So, OT is of the length of OB T ivies OB in the atio : O! B a + a! M! B ( a a an! OM! O +! M a + ( a a + a B M a +! B ²! OM ( a ² ( a + ² a + ² a ² a a ² fusing (a + ² ( + a ² + a ² + ² + ² g ² a ² a fsine ² a a ² g (jj jaj fusing a ² a jaj g But jaj jj fisoseles tiangleg jaj jj jj jaj 0! B ²! OM 0! B an! OM ae eeniula The line joining the aex to the mioint of the ase of an isoseles tiangle is eeniula to the ase 7 OBC is a homus, so! B! B OC an! CB! O a The iagonals ae C an OB a ow,! C a an! OB a + So,! C ²! OB ( a ² (a + ² a + ² a ² a a ² O C ² a ² a fsine ² a a ² g jj jaj But jj jaj fobc is a homusg jj jaj 0! C ²! OB 0! C an! OB ae eeniula So, the iagonals of a homus ae eeniula 8 B P a Q s O! B s! OQ! OB +! BP +! PQ! OB +! O +! B! OB +! O! B s + (s s + s + s + (s +
22 Mathematis MYP+ (n en, Chate VECTORS 9! B ²! OQ (s ² (s + [(s ² (s + ] ffom ag (s ² s + s ² ² s ² fusing (a + ² ( + a ² + a ² + ² + ² g (s ² s ² fsine s ² ² sg (jsj jj fusing a ² a jaj g But jsj jj fsine OB is euilatealg (jsj jj 0! B ²! OQ 0! B an! OQ ae eeniula [B] is eeniula to [OQ] 9 a j + j ( + ² ( + ² + ² + ² + ² an j j ( ² ( ² ² ² + ² ow, j + j j j j + j j j ² + ² + ² + ² ² ² ² + ² ( ² ( ² fsine ² ² g ( ² 0 ² 0 an ae eeniula ( + an ( ae the iagonals of a aallelogam If j + j j j, then the aallelogam is in + - fat a etangle This means that an ae eeniula REVIEW SET a Sale: m 0 ms Sale: m 0 m m unway 60 ms E a
23 0 Mathematis MYP+ (n en, Chate VECTORS a! C! C O! B! O +! OB B! O +! OB a a +! OC! O +! C a + C! BC! BO +! O +! C! OB +! O +! C + a + a! B +! BD! D! BC! DC! BC +! CD! BD! B! CB +! CD! D! B +! BC +! CD +! D! 0 (the zeo veto Let the atual see of the yaht e x km h x os ± fosine uleg x os ± x km h- 0 km h- km h- x os ± ¼ : km h sin µ sin ± an ¼ fsine uleg : sin ± sin µ ¼ : sin µ ¼ sin ± : ¼ 9:9 o 80 ± 9:9 ± ¼ 70 ± But 70 ± + ± > 80 ±, so µ ¼ 9:9 ± The atual eaing is ± +9:9 ± ¼ ± the yaht is atually tavelling at : km h, at a eaing of ± 6 If a then a is aallel to, an jaj jj jj So, a is the length of, an they have the same ietion 7 a jj +( 9+ 8 a x a a x x a a x +x +x a +x a +x x a x (a
24 Mathematis MYP+ (n en, Chate VECTORS 9 a If an k 6 ae aallel, then a k 6 fo some sala k So, 6a a an so ak k k If an k 6 ae eeniula, then ² k 6 0 ( k+( 6 0 k 8 0 k 8 k 9 0 BCD is a aallelogam! B! DC Let e the oint (x, y! x! B y an DC! So, if B! x DC then y x an y x an y x an y oint is (, a ² ( + ( Let the angle etween an e µ ² jjjjos µ os µ os µ µ 6! 0! BC an B 7 j! BCj +(7 an j! Bj +( ! ow, BC ²! B j! BCjj! Bj os C B ( + (7 0 os C B 0 os C B os C B 0 C B os ( 0 ¼ :6 ± ¼ 97: ± REVIEW SET B a They have the same length is twie the length of, an in the same ietion! B! B
25 Mathematis MYP+ (n en, Chate VECTORS Let the euie veloity e, the win veloity e w, an the still onitions veloity e x x + w x w x 00 km h- \ x\ km h- w 0 km h- a tan µ 0 00 µ tan ( 0 00 ¼ : ± The atual eaing shoul e 090 ± +: ± ¼ 0 ± jxj fpythagoasg jxj 600 jxj ¼0 km h The lane s see in still onitions woul e 0 km h - a - a a m n m + n jm + nj units 6 a If the vetos ae eeniula, then t ² 0 +t 0 6+t 0 t 6 t 6 If the vetos ae aallel, then t k fo some onstant k t k k k k an t k t t 8 7 a n x x n 8 a! B a x +x a x + x +x + x a +x a +x x a x (a istane B j! Bj +( 9+6 units
26 9 Let the angle etween a an e µ a ² jajjjos µ ( + ( + + os µ os µ os µ 0 0! RQ 7 µ os ( 0 ¼ 08: ± an! RP 6! ow, RQ ²! RP j! RQjj! RPj os Q RP ( + ( os Q RP os Q RP os Q RP Q RP os ( ¼ 6:6 ± a! DC! B B! C! B +! BC C + D x + y 8 has gaient an has ietion veto! D! B +! BC +! CD! B +! BC! DC + x y 0 has gaient an has ietion veto Let the measue of the aute angle etween the lines e µ os µ ja ² j jajjj j( + ( j + 9+ jj 6 µ os ( 6 ¼ 78:7 ± Mathematis MYP+ (n en, Chate VECTORS
SOLUTIONS TO TOPIC 6: MENSURATION
! i PR =! P +! +! R = 1 a + 1 a = ) j! PRj = jj = jaj fd rhomus ) jaj = jjg! ii SQ =! SD +! D +! Q = 1 + a 1 = a ) j! SQj = jaj From a we have! PS =! QR an! PQ =! SR ) we eue PQRS is a arallelogram. lso,
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