exam1-review.pdf

# exam1-review.pdf - BASIC FORMULAS 1 Qeﬁnitign If a =(a1...

• Test Prep
• 18
• 100% (2) 2 out of 2 people found this document helpful

This preview shows pages 1–18. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

This preview has intentionally blurred sections. Sign up to view the full version.

This preview has intentionally blurred sections. Sign up to view the full version.

This preview has intentionally blurred sections. Sign up to view the full version.

This preview has intentionally blurred sections. Sign up to view the full version.

This preview has intentionally blurred sections. Sign up to view the full version.

This preview has intentionally blurred sections. Sign up to view the full version.

This preview has intentionally blurred sections. Sign up to view the full version.

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: BASIC FORMULAS 1. Qeﬁnitign, If a = (a1, a2, a3) and b = (b1, b2, b3) , then the dot product of a and b is the number a -b given by a-b =al-b1+ az-b2 + a3-b3. 2.|a|2=a-a = / a12 +a22 +a32. AM If 9 is the angle between the vectors a and b, then a - b = lallblcosa 6. Corollary. If 9 is the angle between the nonzero vectors a and b, then cos6= [2|]; . 7. Two vectors a and b are orthogonal if and only if a- b = 0. . . __ a - b 8.Scalarpro;ectzon of b onto a : compab — W . . . a- b Vector projection of b onto a : pI'OJab = Fa. a .EXERQSES 33. Show that the vector orthab = b -projab is orthogonal to a. m (orthab) -a = (b-projab) -a =b-a - (projab) -a =b-a- Illa—"gum =b-a — b-a =0. 43. Find the angle between a diagonal of a cube and one of its edges. 59mm The diagonal of the cube that begins at the origin and ends at ( 1, l, 1 ) has vector representation (1, 1, 1) . The angle 6) between this vector and the vector of the edge which also begins at the origin and runs alongx — axis is given by (1,1,1)'(1, 0, 0) = l l |(1,1,1)ll<1,0,0 >l J? ands08=cos_1(—)z 55°. J? cos6= ”L4 HIE QRQ§S ERQDDCT 5. 111m The vector a x b is orthogonal to both a and b. 6. I! hegrem If 9 is the angle between a and__b, then I a x b| =-|a| |b|sina TWTWO nonzero vectors a and b are parallel if and only if a x b = 0. 8. The area of the parallelogram determined by a and b is equal to la X bl. 9W iO-L/ Emrcigg; @ .20. £7,440! 19mm cm r} vemiv VS orﬂmﬂom/ £0 batik 5~I¢ th [+S. {—3 -K —: <4,~1/-/>‘ So 1Lwo ww+ VaochrQ orééojoml 1%; 607,11, ffwg VerﬁoVS Qré :t 7% < 4/ 4/4). 29.(0>Fl‘m( 0 hop/zen) Vedor or%i»ujowa/ ~éo +142 plawe 1Lb/‘Otegk 1422 [30134753 [-9010, I); C7 ('ZJIJD) 20025), (b) Fwd £173 arm (273 {'Vi‘ﬁbf/Q PQR‘ Solth‘Lle) ——> L 5‘ [c ‘ 2 i; Z (—3; (60 WXPR; *3 ' 2’ 2/2L/{“(59/3+52 ’5 .1 L( {99 X. rm 2: <0, (72) ’9» Themaﬁom/ (0,157,’9> or (MEL Moh‘ZeFO ECG/0F Wulé-{pla 3%er Q9 {org—l) 1‘3 orélycgfohg} 7‘0 £149 plat/IQ eoﬂwx 2. ~_ *9 ‘9 _1 Lb) AWE, l5. [PmPRLE mpg m AAPQREERT‘. 35‘, Find ’Me VO/ume of. z‘ée para/Mk”; ® p60, Wan ng’gcem’f @0963 Pg; P/Q Qom/ PS where P62)/,0)) 0(2,3,2)) Hwy/J] 5756/). . ﬁ “—5 . .--;. Sofa—[(09) . Q: PO':{L{/,2/,2>) bzf—JE:<3,-Sf/>j C2P3<¢3\$Q 5‘5“ / V : if .g ..g_? 7; g. 0:; 32—.Hz/6J SO 2%? l/o/Muve 047. £92. para/(e/epz‘ped 1‘s /6 Cum am'z‘S‘ ‘5“. (a) Let" P be a pom-(L 1402‘ 014 H249. //'MéL Merl, pasges H; H) Lei; wéhe pawl: 49 (2140/ E, gl’IOVV ‘L’dQ‘! ‘Z’bf dISILQ HQ 0’ ¥POM1 {Ag pgl‘h‘g {0 +0 #12 (me L [3 ———a WIWE‘VZ. a: 51 QMO’ [2:09. n W? Area 1% 0.73. PQVWI/P/OgVQ/M pPT 2 [g E. A , ,. -..., ”W, T WW / A=Iawl amd . 1/" 7+: [CZ/"SE, :> O S R lGXb/:’Q(/R/ 2) ”TE, :1 (QXN: a, 10ka __ W :1 ﬂ 0?: )q/ ‘ VET a C 3' ta.” _ , ,( . ——I /, -(~Z C(Y b: l‘/ '2, "(/: ’ZS;_7/LJ/ l,7/~S+/,-S«/£ I -5; *7 ax b : <9,ng 7? >9 {akblrl/fl—Mwwﬂﬁ. q;_ (q) Lexé P be a pom; 140% 02 file [3/098 H10»! pagses zll’IVOL/ﬂ) {he pal‘m’g @Eaadéi Show Hzawf ﬁe 009th 0? 12m»: F +0 {be pfame. is d _.. {Q- (EXCM ’— [62be whera Q :. (SE) bib—:1 ahd 02453 0)) (136. H12 43%qu in pcﬁnL-(c) 7L0 f/‘W/ {446 dream/a 43“”? 1‘43 po/‘mL 296.21 51/) 710 Hm Mal/22 #vrouﬂ, {Ag poi/4149 QUJQQ) )Z(o/2,-o)) and \$(o,o,3). 96. 90414-(0M j (9 (a) V :1 [mKékCﬂ Md v; 7442/ ’2) 16% We)! 2-— M .—=.> 4:49,: 03> 61:63-92:44/2/0?)b:@3:<—l/0/3> 0910’ avg-73: <l/I/Lf> ’helﬂ -‘ l 0 ‘— -( 3 IO/ 1 I L/ Mia: <6j3).2> :5 {abe’ 36+9+V:7) (4*(55‘6/ L2: d: : 7 '49. Prove 952\$: é-meﬁ) :@—b)xa+(a—b)scbr amt/5m +0343 vab‘: O +ayb ﬁnch ~01— Ctxb *6 Mb —: Mama), 5. * ﬂ —— o Leé a» cauaapa» b«-fauszbg>/ woe/3)) @ SO bx C" :< bzeav—bgng5301~6I03/b1Cz—139_C1) 0970/ 61be H) = < mime-7, v-bacl) — agr/bgc,»blc3)) 0g C sz3 ~ [33 C?) ~03 (brat-5261)) oufbsm—b,cs)_qz[52(},bgpzp QXQXC)‘: < @262. +Q%Cs> be “ (ﬂak-{'03 53) C, ) (a; a; «mm? b; —- (Q3123 +0,b,)cz)(a,c,+qzcz)53 —(¢,b,+qzbz)g> MG) XC\:’— < (4;.C++GzCz+Qz (’3) E;- (mbﬁaz b2 +623L3)C’,2 (Q! Cl "’6 Q2 02 + 0-2 CS) bz’(0l I); +szz‘f43 b3) C2) (GM + 02 0.2 +Q3 Cay); v (a! bl +42bz+ (>353) 83> axéxc) = we) 19,) (a mm) (mm ‘<@‘5)[email protected]‘502 4.5m QXQKC) “1 @Q <50132ﬂ’35 “@‘b><C’/CZJ C3) Qv(be\ 226-83 [9 —— (a A) L 5L Use EA’Qra/‘Se 5‘0 {0 prove #2074 Cm (we) + beCx0)+Cquxb)-:o Piroajﬂ‘. 6‘2. Pro v2 ﬂea/3L Q . C b'C' , Cue? 5,617 PVODJQ; (Q Yb)“ (8X09) :@XJ)'[&X!9) :Q- (me/D: a .(wxc ~Q>o® d) :: (b‘cl)@«C)-Ké-C’) (4%): @X b5'(€ Xd) : o IQOQ [3°C (Mi Mi a (D 53- Sup/9036 ﬂoaz‘ a 7&0 C?) (a) M 42.5 ra-c2 0/025 ad ﬂew/aw aw 5:07 Solwgéfoio: My) 'E’K: u; :0 Am! ##3' (b) Ii 0X13: 0K0 doéa 1% ago/kw 249627552647 So/wxﬁtow: A/o Sofcesérom : Yes. f‘tﬁ- 9'5 :r C}- C :1“) albv-Q-C =Or>4oa—C):o QI'MCQ 7:) a r‘3 Perpeadrw/or 24-0 be) awe! s42. axb :axc :2) axb~QkC:Qx/E~CJ=O=D a‘ato 1'; 4/90 para/M ‘éo In~e_ Thus Sz‘hca 040 5an r9 502% pomﬂe/ qw/ PeVdpehOIFCLi/Qr 40 5’0, We ﬁat/e \$4.245 90 5:0. “9'9 E a WMS 0 UMQS CMO/ /OW€S (3 ’“1 Q+ZLV a Vecnizor ZWQY-f/Dh (of/«>761— F93 X2x0+a7§ y:‘yo 149+ 2:: area! ,_—— The; a 8% OIL/OMS awe (bf/(9M pan? magma awaﬁowg (9% My Mae L 745%qu 2%6 poiml Po (Xe/yD/ZD) 0(ch le/Q//€/ 7L0 Me veoﬁOV‘ VT—(q/[g/Q Eqw’ va/ue Oﬂﬂe paFQWQWLQV‘ ‘f‘ ngK 0, [OOH/ff (X/V/Z> 0”) L. [3:] M9.” :rzg-jz»?o p q f b a I e @?Lecn[fov/Q are Ca//eo/ symmézlm‘a QWQII‘WQ 04?? L- [ﬂ 7755 Moe SQfWPhYA ﬁéf’VDI/M r79 {0 git/em bay £172 Veaﬂor Efacﬂ/‘oh r“(113: (“9’1 +797 Oé‘ésl. i", (”Q A Flame M2 Space), i§ Ofﬂérm/néo/ b? (29 a goon/[é PDQ», Va, 20) “7 ﬁve p/mye 0120/ a vac/wk V7 #:07‘ {‘3 0W%¢9([email protected]/ 740' 7453 plat/72. s51 rum/4;) :0 V: 4X/y/Z> V6: <X9/V0/?0'>. [a “or :ho V7) 0 CX~-Xa> + b buys} + Q (2- 20‘) 2 Q Seq/6% (2quan 0427159 P/th QX+ by+ CZ ~— (0x0 +EVO+CZOB:U 3 “(+1“ +02 + 0/ :2 O C; (/‘PWQV 9 WWW / 7/ of: {We play Emmgle 7 Eff/2d a for-mu/q £0,» f4£_ @ aﬁ'smtu D {31mm a pox‘hé P/(A’UM/zJ 110 the plawa 0k+bV+CZ+dI=0 Soluﬁoh; La+ Po (X0, haze) be 014a? [029/0714 "17 [122 WW” [3/01/72 anal /€% béa {4e Vea‘éor‘ P'F,‘ 77m” [3 24X: *ko} yl'yo/ 21—20>) ﬂ :‘40/6/6> 1112/- /W)/ D: [COM/Iphb’: )n/ ., ' m ) 5.. 'W” D: laxﬁbwwz, + 0” “1 9 EXeVCfseg CD 57 M‘mo/ q (man/br- eym‘ﬁ‘ot: 4516/ parqmééé- we, a¢wq¥fmv§ 190;» 1L4Q 01:8. Ila/29%; ﬁe, por'm’ (I) 0/ 6) awe? perpeha/mcz/Or- {0 21-68.. plahe. Y+ 3V1PZ 25‘. SM.) 24' like, perpewdrcubk 110 ﬁle. git/eta plame Mg Hoe. 9Qme dz‘r‘edl‘oh (PC a hoVVMQ/ Veasﬂor {o 1149 p/amg/ Sac-£2 a9 H:<I)'3,I>. SO Po:(//@/.6> of V:' .; L+3s+ K ﬁmd ref—[+614‘ Them 0 Vet/720V QWHW’ ’53 r : (1+9 6, + 3% 5 ﬁnals, 0140/ WWW-{W eCf/umlr’D‘vS 0"? X:[+‘l:) V2311) 236%. 15-. Find a {mailer QWQWOA 730;» ﬂvQ hue. Sean/120\$ {gram (2,4,?) 140 {if/6M). 4‘0‘ ‘ « 80[L{l [4 r0: 2L”‘S+/-/IL J n:g[\+6j‘(t'=> y—(ﬂ :1 0,1.) K, + —£— I", “:92{/—2L)C~{l*22)£n+(“ha/4+ (47"?) L [email protected]¢)4f + + /c. :(2 L‘ ‘3' +Lm)+ 15(2L‘+7j_3k) Osi—é: I. ‘9' De‘vLeVW/V’e Whévl‘her 7%6 ll‘heg L1 ma/ @ L2 are party/la} Stein/j 0r z‘mtp‘rggcgpyy‘ f4): H7209 "“{e-VSQaIL/ 42/774 {he [Do/“14% aﬂ iméersedim. LI; 11%.. Y—3 __ 2.4 F“ ’-—- .._.._—.—. l ‘2, *- “ ____ ’- 3 '— -7" SD/LHQOVI ,‘ 3 face + #% :1 “:37 J {42 mag Cami/N para/{eL L. :_ X:.2++) y:3—.2¥, 2=I~3+ 490/ L2 : y:5+g_ / yr—LHZS) Z:Z+7:<‘. Mug; #0!” H22 Ham 40 ”#83906 ﬂue Mme QW‘M 2+{;3+3) g”‘2~[:—’7’+3§/ /*31£::2'1‘7S M’qu‘ﬁ be 9a+r§£red Simul—éqheougéf. 2+{2-("P’23f5' L/+Z+::6+2S ;> 7:114? A l 322+ =—'—f+?>§ 3 ”21‘ “94/439 y :g‘g :) 3': J—H—r 3+! 3') += 2 cm 0/ (. 3~2 :2—F7' / [~5‘:"5ﬂ / "S\:"§\. Met/roe x:s+/:L/ / yr—u+3:-// 2:92_7:_3: 4M4 Ié l; M; :0; (two—53f E1740? 4% QWQIZI’DW of 49%? p/QMQ. @ 29. 7772 b/Qhe %‘ka~/ péK‘SéQ {’Arou/Z‘Za zééa pofml PDQ/D 01/20? COM/m?) téée ”be (973 [MerseCzL/‘ma 011 £46 p/ames Kan—2:2 14 A (“WM-fizz) Veaﬂor 7401» £42 #172 0—K M ‘(Lei’falfzbg {S Q: ”(kl/izr< (/ l/—-/> >(<:2,»//3)>:<,,e)..§j—3>J aw! a 1‘8 POPa/le/ +0 'HQ ales/"rial p/Omé. Saki-mg, kc.o/ ~2~6e 2%471-(‘099 Oaﬁ £412 [9/4669 #8111461) %-0 V— 2:2 Wad _y+.3?:/ mag; SOILNHO"? Vz—i emu 2; g 90 c; por'la7( 04/, {—52 {fl/72. I‘S (0/:E/é) ama/ 01004480" Veceﬂoo’ pal/af/e/ H) @e p/moe i‘S <~b‘~§:f~£>. 777214 0} horma/ Vebﬂal/ 7L0 #42 [0/6572 [9 h 2 <21'5; '3> X ¢"/ ’-3:_;’é> r<~.2, 9,—55 01004 0m ewaéfﬂh 0%. 1%? plat/2e 1‘s —.2 (ML/ML ‘f (Y’Z) .—?(2-/)~_:0 or -,Z/\’+‘fV—— £72 ~2~—5 +8 :0 *ZX’Inyv‘32. ’2 :O (W ___\ 3% Where \$098 {\$32 [ma {imouyé' (99/) (3 0/140! (Lb-27.2) Wei/320\$ #2 [9/6th )mc—W2267 §0/u%10h" V0241/0/i>) V:<3/._z//> :2.) <Y/Y,‘Z?:: (1)0, I> +~Z~<3/__2/(> A) ’w X2‘f3‘é) YZw2~LI Z:/+\é — pamwﬂm‘c Q‘WCdL/OL/lg Di II‘VIQ. :::> X+ n 2 :— 6 / (If 3%) + (—27.1) +f+z< 20’ “'9 ,2 +2+ —:.. -:> 2+: / )14-22. Whoa (7) 403) I‘S z‘ﬁze bow-04 0‘78 r‘m’mm 41"014 0.13 ll‘loe WI‘ILL; +43 [3/th 40. PM}! cm egamlr‘m; ﬂap» age [0/th magi-9kg), Qﬂ 0/! loo/“mi: ﬂew Qhe egL/fe/zrllom! 43er #22 pmng (OZ/573‘) and (—43%). Co/LMI'OM [9/an 1L9q+ posceg Mikey-£3 MI‘O/pO/“mi -- Za—é, *“ >r=-~——-/» y» '2 J ( ”2/ V/ 3) th per/pécydfcq/op £0 h =<~8, *2/40 M9 Ggwcasifﬂh S"+3 A.- 5:31.“? 2g?” / Z,— Z” «3 ()(+2) —-2 {WA --’7’ (252) :0 '8’X’2V -sz 4 l/ :0 w w +22 ~—2 :0. 5‘5, 520w Hag-g4 {ﬁe dc’swhce loo/ﬂee” 1‘42 (3') pqmﬂel p/aozeg ay+by+cz+dlzo aw? améwcuc/ :0 [AS D” Idl"012[ I! 42+ 51+ 02 W The Mrsﬁmcq 56.4109/2qu 521/0 pal/q- He} p/cmes f\$ Ilka Sam/2 as 448 0097(0th bah/€2.14 6%- por‘rré 014 owe 0% Hz? p/cmeg and {/1312 0%8 1" P/OIVE. [.24 Pa :: (K9, ya]. 20) be. a [901‘ :4 0h %&2 pf 4012. git/€14 baa. QX—Fbwaerq Them aXoF’bﬁ-«L CZo~€0ptro Ghd ﬁe dfﬁ‘Qﬂoéﬁ bei—wee Po 62m»! {48 p/ame gin?» 55L OXPIOV+CZ1L 092. :0 K D: ’W: (”dlw’zl gaudy V92+b2+61 f (22106??? W 5‘5 Fit/Id 'é'L/e aﬂ'S‘MMC/Q bah/Pet; 1152 9({901/ (9) [Meg Walk POVQOMZ‘KVc‘Q eg/aé‘é‘aeﬂ x:1+£/ YZ/‘(éiz/ 2—2275 W62 yrﬁﬁaggj yssu/Vg/ 22—24 69. SW; I13 7.1mm #1499 at? 9/59va Mean cw; be View/PM (:49 lyméz U: 751/0 Pam/fe/ fig/awe: mm 30 We drum/me ﬂew/2852 ﬂ/e Skew 1;ng Wow/cl be ﬁve 9&51/28 49 14/179 é/A‘MW Mia/e?» #1292 pGI’D/fé/A p/W/QQ, U? Caz/mac? WOW/1401’ V€O¥0V <40 2%? ,b/Qhég ﬁlm/491C 56 h :2 V, m; : <1/6/2>X<£//S}6>:<6J~2/3‘>_ 98%{Lt'tnda %:o (2070/ 9:0 gr‘i/QS %&e pom? (bl/Q) am? (0.57-2). 50 €k~2y+32+0€, :20 ~»/\$_é p/cm{ 21d 3 @X*2)/+32+O/? :‘O ”A 2 pkg/”‘9 6 v2+3~0 ~ch( :0 :9 (1/114/ 6 “/0 ~44 1c 0/2 2:0 :9 0/2: /0 mm” Cu ...
View Full Document

• Spring '17
• Review, Exam

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern