Vector Algebra. Lecture programme. Engineering Maths 1.2
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1 Leue pogmme Engneeng Mh. Veo lge Conen of leue. Genel noduon. Sl nd veo. Cen omponen. Deon one. Geome epeenon. Modulu of veo. Un veo. Pllel veo.. ddon of veo: pllelogm ule; ngle lw; polgon lw; veo lw fo ddon. D P.. Slegh Eml: P..Slegh@leed..u Tle of onen Conen of leue... Sl nd Veo.... Cen Co-odne.... Deon Cone..... Emple...7. Geome Repeenon...8. Veo noon found n e oo...9. Mgnude (o Modulu) The Un Veo Emple:....7 Equl Veo....8 Pllel Veo... Veo ddon.... The o polem.... Geome ddon lw.... Veo Lw fo ddon nd uon.... Veo ddon Emple...6 Veo Mulplon...8. The Sl Podu...8. Rule fo he l (o do) podu.... Sl Podu Emple.... The Veo (o Co) Podu.... The Momen of foe F e lulon....7 Lw fo he Veo (o Co) podu....8 Pllel veo Cen (omponen) fom Emple...7. Tple podu of Veo... Veo Equon of Lne nd Plne.... The Veo Equon of Lne.... Equon of lne emple:.... The Veo Equon of Plne.... Emple nvolvng oh lne nd plne...6. Mulplon of veo: he l podu; wo done foe; ngle eween veo; omponen of foe n gven deon.. Veo podu: momen of foe; e of ngle; pllel veo.. Tple veo podu nd ple l podu: volume of pllelepped; o-pln veo. 6. Veo emen of lne: veo nd Cen fom; neeon of lne. 7. Veo emen of plne: equon of plne fom o-odne pon; veo nd Cen fom. 8. Fuhe emple: neeon of lne wh plne; pependul dne fom pon o plne. Engneeng Mhem.: Veo lge Engneeng Mhem.: Veo lge
2 Sl nd Veo In engneeng muh of he wo n oh nl nd degn nvolve foe. You wll e fml wh foe n uul meme n pe fme nd now h foe n gven deon wh gven mgnude. foe hee dmenonl qul nd one of he mo ommon emple of veo. Two ohe ommon veo qune e eleon nd velo. In h eon of he module we wll nodue ome foml mheml noon nd ule fo he mnpulon (mulplon ddon e.) of hee veo qune. In he el ge wll e e o eogne he geome menng of veo ddon nd mulplon u he polem eome moe omple (nol when he e hee-dmenonl) he foml ule eome moe mpon. The genel defnon of l nd veo e gven elow: Sl qun one h defned ngle nume wh ppope un. Some emple e lengh e volume m nd me. ) empeue of o C ) n eleon of 9.8 m/ owd eh ) The wegh of g m d) The um of. e) Noh eel wnd of no nwe: ) Sl ) Veo ) Veo d) Sl e) Veo Noe: To qulf veo he qun mu lo f ome ohe ule of omnon (ddon mulplon e.) whh we wll ee le n h oue. Fo emple ngul dplemen qun h h oh deon nd mgnude u doe NT oe he ddon ule - o NT veo. Cen Co-odne The heo of veo oed ve loel wh o-odne geome o we hll nodung he o-odne em. We ue engul Cen o-odne em wh e o hown elow. The poon of n pon P gven o-odne o omponen () o ( ). - Veo qun defned ompleel when we now oh mgnude (wh un) nd deon of pplon. Some emple e foe velo nd eleon. o P() - Two ve mple (nd ommon) emple demonng he dffeene eween l nd veo e peed - l nd velo - veo. peed of m/h l qun velo of m/h degee veo qun. - em Emple: Wh e hee qune veo o l? Engneeng Mhem.: Veo lge Engneeng Mhem.: Veo lge
3 Z- Z- o P() Y- o β P() Y- X- X- em The ode nd deon of e e umed o e 'gh hnded'. Th defnon ome fom mgnng gh hnded ew - f ou un he ew n he deon fom o he ew dvne long. (nd o long fom o long ). Th onvenon nd MUST e dheed o. Z- The dne fom pon o pon P n e luled ung he Phgo heoem o gve P() ( ) / o γ Y-. Deon Cone We wll ee le h ueful o now he ngle he lne P me wh eh. Founel h n e el luled. The hded e of he fgue elow eh fom gh-ngled ngle wh he e. o α Z- P() Y- X- Engneeng Mhem.: Veo lge X- So nowng he lengh nd he ppope o-odne eh ngle n e luled. The ngle The ngle The ngle P gven P gven P gven l o α m o β n o γ Engneeng Mhem.: Veo lge 6
4 The one of hee ngle α β γ e (ofen wen) l m n nd hee e nown he deon one of he lne P. I n el e hown h l m n.. Emple ) If P h o-odne ( -9). Wh he lengh P nd deon one. Deon one: P 6 8 P l m n 9 Clule he deon one: We need he hd deon one. Ung l m n l oα o6. m o β o.68 m oγ.8 γ. o l m The lengh T () n e luled we now he o-odne.e. fom l Fom he ohe deon one l Co-odne of he one e: ( ) ( ) ) The poon of he op one of he leue oom w equed. To fnd h heodole w e up n he fgue elow. The dne o he de wll long he - 7m nd he followng ngle wee meued: T 7 o nd T o. Clule he o-odne of he one elve o he poon of he heodole. T. Geome Repeenon We epeen veo geomell lne egmen n pe. The lengh of he lne epeenng he mgnude nd he deon of he lne he deon of he veo. Fom h defnon he ng pon elevn. Wh he dffeene eween hee wo veo? 7m Engneeng Mhem.: Veo lge 7 Engneeng Mhem.: Veo lge 8
5 The lengh he me. The deon he me. (The ow nde he deon long he lne) The un veo The un veo n he o-odne deon e denoed nd. () () nd () Thee wo veo e equvlen (equl). he me gumen he wo veo e lo equvlen o veo of he me lengh nd deon whh he ogn. We n heefoe ue o-odne noon of hee nume - ued pevoul fo P - o epeen n veo (nwhee n hee dmenonl pe). n veo n e epeed n em of omponen wh epe o he un veo () The noon () nepeed ( ). We hve hown veo deed n wo w: n em of o-odne ( ) nd lo epeened geomell lne fom n ogn o h pon. We dul defnon n e ued ehe geomel o o-odne (omponen) one.. Veo noon found n e oo veo n e wen down n mn w. Some of he moe ommon (nd eple) w ou wll ome o e:.6. Emple: veo ( ) The modulu of ( ) / 6 So he un veo n he deon of ˆ ( ).7 Equl Veo Two veo nd e equl f he hve he me mgnude nd deon.e.. Mgnude (o Modulu) The mgnude modulu o lengh of he veo wen o nd gven n Ung he omponen defnon omponen fom ( ) / hen ( ) ( ) Geomell h he lengh of he lne.6 The Un Veo veo whoe modulu lled un veo omeme wen â nd ˆ Engneeng Mhem.: Veo lge 9 Engneeng Mhem.: Veo lge
6 .8 Pllel Veo If λ l nd λ hen f λ> he veo n he me deon nd h mgnude λ f λ< he veo n he oppoe deon o nd h mgnude λ In omponen fom Veo ddon. The o polem o em no due E fo one hou. The de unnng Noh-Noh-E no. Whee wll he o e fe one hou? dgm of he veo nvolved loo le h elow whee epeen he velo of he o nd epeen he velo of he de. λ λ λ Th n e ummed he veo nd e Noh NNE C Pllel f λ > n-pllel f λ <. E The ne velo of he o epeened he lne C whh he um of nd.. Geome ddon lw Th led o he pllelogm ule fo veo ddon whh m e wen: The um o euln of wo veo nd found fomng pllelogm wh nd wo den de. The um he veo epeened he dgonl of he pllelogm. C In omponen fom ( ) Engneeng Mhem.: Veo lge Engneeng Mhem.: Veo lge
7 . Veo Lw fo ddon nd uon eue he veo he me n ohe veo whh pllel nd n he me Commuve lw deon o ould e "moved" n he fgue ove o fom ngle. Th led o he ngle lw. If wo veo nd e epeened n mgnude nd deon he wo de of ngle de NT mpon en n ode hen he um epeened n mgnude nd deon he long hd de. The ngle ule n e mde moe genel o ppl o n geomel hpe - o polgon. Th hen eome he polgon lw. If fom pon n he fgue elow lne e dwn o epeen he veo d nd e. Then he long de epeen he veo nd he um of he veo d nd e. In omponen fom ove lw ( ) ( ) ( ) ( ) e E D d C E e D d C Th follow fom he omponen defnon nd geomell - fom he ngle o polgon lw. In omponen fom () () d e Engneeng Mhem.: Veo lge Engneeng Mhem.: Veo lge
8 ( ) ( ) ( ) ( ) ( ) ( ) - Duve lw ( ) λ λ λ - Th follow fom he omponen defnon o geomell fom ml ngle In omponen fom ( λ ( ) λ( ) λ( )) ( λ λ λ λ λ λ ) Noe Th n mpon eul how h he veo epeened he lne onng wo pon gven ng he veo gvng he poon of he f pon fom h fo he eond λ( ) λ λ Suon λ λ. Veo ddon Emple () ( ) ( ) ( ) ( ) ( ) ( ) ( ) - (-) Modulu of In omponen fom: ( ) The un veo n he deon of ˆ ( ) / No h he modulu of ˆ ˆ Geomell: Engneeng Mhem.: Veo lge Engneeng Mhem.: Veo lge 6
9 ) Fom nl of he foe n new dge luled h he followng foe e mpoed one of he dge uppo (n Newon). F () F 6N nd n he deon (-) F N nd n he deon (-) In ode o degn he dge uppo he euln foe equed. Wh he foe he uppo mu mpoe on he dge o edue he euln foe o eo. The f foe gven n he uul veo fom. The eond wo e n vld fom u need o e onveed o h we n pefom he uul veo ddon. F lule he un veo n he deon of F nd F ˆ F F F ˆ F F F So he n veo fo he wo foe e ( ) ( ) 9 6 F 6 F The euln foe F found veo ddon F F F F ( ) ( ) ( ) ( 6 8) ( ) ( ) ( 6 8) ( 9 ) The foe he uppo mu mpoe on he dge equl nd n he oppoe deon he euln.e. Reon foe ( 9) Veo Mulplon We hve een how veo n e dded nohe nul opeon n mh o mulpl qune ogehe. The de of mulplon of veo eome ve ueful n engneeng. In pul we wll how how o ue hem o lule momen nd wo-done wh ge ee. Two pe of mulplon e. The e lled he veo nd l podu.. The Sl Podu (lo nown he do podu o he nne podu) P N The omponen of he veo n he deon of P el luled equvlen o he lengh N N o Condeng he onn foe F n he fgue elow whh hough he pon. If h foe moved long he lne long he veo hen we n lule he wo done he foe. (Rememe: wo done he omponen of he foe n he deon of movemen mulpled dne moved he pon of pplon.) F F The omponen of F long F o. Th foe moved he dne gvng wo done F o Engneeng Mhem.: Veo lge 7 Engneeng Mhem.: Veo lge 8
10 Th he l podu of he wo veo F nd nd n e een geome defnon. () Ue of he "do" eenl o nde h he lulon l podu. n equvlen omponen defnon n e wen. Sl podu defnon: The l podu of wo veo ( ) nd ( ) wen wh do ( ) eween he o veo. I defned n omponen fom nd n geomel fom o Whee he ngle eween he wo veo nd π.. Rule fo he l (o do) podu. Commuve Lw o o ove Lw You nno hve l podu of hee veo 'dong' he f wo gve l. Duve Lw (fo l mulple) e e mulpled ou n he uul w. ( λ ) ( λ) λ( ) The wo defnon n e poved o e equvlen he one ule fo ngle Whh n e epnded o how h ( )( ) o Thee mpon pon ou l podu: () The l podu of wo veo gve nume ( l) () The l podu podu of veo ( nno e of wo l no veo nd l) Engneeng Mhem.: Veo lge 9 o Duve Lw (fo veo ddon) Powe of veo ( ) o No ohe powe of veo e pole ohe hn. Noe h fo un veo Pependul Veo π π If nd e pependul hen nd o o Hene UT doe no mpl h nd e pependul eue ould e eo o ould e eo. Noe: Suppoe [ ( ) ] Th doe no men h ne ould e eo o pependul o -. Engneeng Mhem.: Veo lge
11 .e. veo nno e nelled n he me w l. Fo un veo whh e pependul Pependul mpon n engneeng fo emple peue noml o ufe o foe pe un e p nˆ whee p he peue nd nˆ he un noml veo. We ofen hve o fnd veo h noml o nohe veo. Componen of veo The omponen of veo F n gven deon gven nd we n we o gve he omponen of F n he deon on.. Sl Podu Emple F ˆ F ˆ o F o F o. () Fnd he ngle eween he wo veo ( ) nd ( ) Ung o nd 6 8 / ( ) ( 9) ( ) ( 9 6) 9 () Fnd he wo done he foe F ( ) n movng ple fom P o Q whee poon veo of P nd Q e ( ) nd ( ) epevel. Wo done he foe F F PQ Q P ( ) ( ) ( ) un of wo. () Fnd he omponen of he veo F ( - ) n () he deon () he deon ( -) () The deon he veo () o he omponen n he deon () F ( ) ( ) ( ) I he veo ( -) un veo? / ( 9 ) Un veo ( ) Theefoe no un veo. Componen of F n deon ( -) ( ) F ( ) ( ) / 8 9 o 8 o () Fo ( -) ( ) ( ) fnd ) ) ( ) ) ( ) ) ( ) ) ( ) ( ) ( ) ( 8 ) ) ( ) ( )( ) ( ) ( ) Noe h nd e pependul ne nehe o e eo. Rememe h u nume (no veo). Engneeng Mhem.: Veo lge Engneeng Mhem.: Veo lge
12 . The Veo (o Co) Podu The mn pl ue of he veo podu o lule he momen of foe n hee. The Momen of foe F. dmenon. I of ve lmed ue n wo dmenon. n Fo wo veo nd he veo podu defned : F ( ) nnˆ whee he ngle eween he veo nd ( π ) nd nˆ he un veo noml o P oh nd. nd nˆ e hee veo whh fom gh-hnded e. (Rememe how gh hnded e em w defned ele.) If he foe F pe hough he pon P nd P hen he momen of he foe ou I ve mpon o ele h he eul of veo podu elf veo. defned M F M veo n he deon of he noml nˆ. n So wh ll veo momen dd he pllelogm ule. The ue of lulng momen he veo (o) podu ome no elf when ued n uon (pull n hee dmenon) whh e ve dfful o vule. n.6 e lulon Noe how fom h defnon ode of mulplon me: D C The veo gven ( ) n he oppoe deon o. h (Th follow fom he gh-hnd ew ule) o ( ) ( ) The e of he pllelogm CD gven e h D n D Engneeng Mhem.: Veo lge Engneeng Mhem.: Veo lge
13 The e of he ngle PQR gven Q Duve lw wh ddon ( ) ( ) ( ) h P e h PR PQ n PR PQ PR.7 Lw fo he Veo (o Co) podu n-commuve ( ) ( ) Th follow fom he defnon h nlude he 'gh-hnded e' nˆ hnge deon when he mulplon eveed. Non-ove (The ple veo podu) ( ) ( ) The veo pependul o oh nd nd he plne onnng nd. Smll ( ) n he plne of nd o ( ) nd ( ) e mu lw e ued fo moe hn wo veo n veo podu. Duve lw wh mulplon l ( λ ) λ R e dffeen veo..8 Pllel veo. Fom he defnon of he veo podu f nd e pllel hen So we n h f hen nd o o nd e pllel. If nno e dedued h. You mu f how h nd h no pllel o -..9 Cen (omponen) fom. Fo un veo eue of pependul Componen fom of he veo podu: ( n ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) So how do ou ememe h!!? Engneeng Mhem.: Veo lge Engneeng Mhem.: Veo lge 6
14 Engneeng Mhem.: Veo lge 7 Thee e evel w he deemnn fom hown hee. Emple () Gven he hee veo ( ) ( - ) ( ) Th veo pependul o he plne onnng nd. Th le n he plne onnng nd. 6 Th he me he oluon fo ove. Engneeng Mhem.: Veo lge 8 I n e hown h n genel I n lo e hown h () Fnd he e of he ngle hvng vee P ( ) Q ( - ) R ( ) e PQR PR PQ PR PQ PR PQ e 9 PR PQ P Q R
15 () foe F of 6 un hough he pon P() n he deon of he veo (). Fnd he momen of he foe ou he pon Q().. Tple podu of Veo Defnon of he wo ple podu e F P Q ( ) he Tple Veo podu. ( ) he Tple Sl podu. The un veo n he deon of he foe 6 So he foe F h omponen F 6 The poon veo of P elve o Q.e. QP ( ) The ple l podu h n neeng geomel menng: We now h Thu ( ) nnˆ ( e of he pllelogm defned nd ) ( ) ( e of he pllelogm) nˆ ( e of he pllelogm) nˆ oφ So he momen of he foe ou Q M QP F M ( ) 6 8 u o h hegh of he pllelepped noml o he plne onnng nd. (φ he ngle eween nd nˆ ). So h ( ) φ e of he pllelepped defned nd. C I follow hen h: () If n wo veo e pllel ( ) (eo volume) () If he hee veo o-pln hen ( ) (eo volume) Engneeng Mhem.: Veo lge 9 Engneeng Mhem.: Veo lge
16 () If ( ) hen ehe ) o ) o ) o v) wo of he veo e pllel o v) he hee veo e o-pln (d) ( ) ( ) ( ) he me volume. Veo Equon of Lne nd Plne. The Veo Equon of Lne Conde he lne whh pe hough he pon nd wh poon veo nd epevel nd pon P whh h poon veo nd le on h lne hown P P Fom h dgm we n ee h If ome mulple of uh h P P P hen So he equon of he lne : fo < <. ( ) ( ) In omponen fom ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) [ ( ) ( )] If we eque omponen e.g. hoe of gve Engneeng Mhem.: Veo lge Engneeng Mhem.: Veo lge
17 Engneeng Mhem.: Veo lge Smll fo nd Hene he Cen equon of lne png hough he pon ( ) nd ( ). Noe: Fo n equon n Cen fom we n edl epe he equon n veo fom. e.g. 9 In veo fom 9. Equon of lne emple: () Fnd he equon of he lne hough he pon (-) nd ()..e. In Cen fom () Fnd he equon of he lne hough he pon () nd (---7) 7 7 Engneeng Mhem.: Veo lge In Cen fom 7 () Do nd nee? Fo he lne o nee hen nd hould e uh h: 7 e ll fed. Solvng he f wo mulneoul nd -. Suung hee vlue no he hd gve -7-7.e. he hee equon e fed o he wo lne nee. Suung he vlue fo o no he epeve Cen equon gve he pon of neeon (---7) () Fnd he ngle eween he wo lne whh e n he me deon he wo veo () nd d (---7) 6 9 o d d d / / d o
18 . The Veo Equon of Plne We n ue he f h lne onng n wo pon n he plne pependul o he noml o he plne..e. n nd P e pependul. n. Emple nvolvng oh lne nd plne () Fnd he equon of he plne hough he pon - le n he plne - le n he plne () (-) (-) - ( - ) - ( -) The veo n noml o he plne he poon veo of he poon veo of P ( ( )). The veo Now - pependul o n f o o P P ( ) n n n n The ove wo e he genel fom of he veo equon of plne. If we n e n (α β γ) he equon n Cen fom α β γ p ne ( ) p P noml n o he plne ( ) ( ) ( ) ( ) n n Equon of he plne n n (( ) ( 6 ) ( 8 ) ) ( 9 ) ( 9) ( ) ( 9) 8 In omponen fom h ( ) ( 9) 9 Che : 8 : 7 : 8 ll e onen. Engneeng Mhem.: Veo lge Engneeng Mhem.: Veo lge 6
19 () () Fnd he pon whee he plne ( ) (o ) mee he lne ( -) ( ) o ( ) () Fnd he ngle h he lne me wh he plne. [Rememe he equon of lne ] The pon of neeon mu f oh he equon of he plne ND he equon of he lne..e (ung he veo fom) [ ( ) ] ( ) ( ) ( ) / Suung no he equon of he lne gve he pon of neeon ( ) ( ) ( ) 6 () Fnd he ngle h he lne me wh he plne. The noml o he plne ( ) n. veo n he deon of he lne ( ) - ( ) n n o ( ) n n / ( ) / ( ) 6 o Engneeng Mhem.: Veo lge 7 () Fnd he pependul dne fom he pon P (-) o he plne The equon of he plne n veo fom ( ) veo noml o he plne n ( ) Hene he equon of lne pependul o he plne nd png hough P ( ) ( ) ( ) Th lne mee he plne when.e. ( ) ( ) ( ) ( ) ( ) ( 6 8) ( ) 9 Thu he lne mee he plne N wh o-odne ( ) ( 6 ) Hene he pependul dne PN [( ) ( ) ( 6 ) ] / un () Fnd he equon of he lne of neeon of he wo plne gven : nd In veo fom hee plne n e wen: ( ) ( ) The lne of neeon mu e pependul o oh he veo () nd (). Hene veo n he deon of he lne mu e n he deon ()() (-) To omplee he equon of he lne we need o fnd n one pon on he lne. Choong hen fom he Cen equon of he wo plne Hene fo - nd nd he pon ( - ) pon on he lne. The equon of he lne n hen e wen ( - ) ( -) Engneeng Mhem.: Veo lge 8
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