Exam1_solutions

Exam1_solutions - ‘ -_..-—- Souffle/M 5 4 ._ ____ NAME...

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Unformatted text preview: ‘ -_..-—- Souffle/M 5 4 ._ ____ NAME email Rec. Instructor Rec. Hour 18.02A Exam 1 — Fall 2009 Friday Nov.13, 2009, 2:0543pm Directions: There are 6 pages, printed on one side. Answer all questions, using reverse side as needed. No books, calculators 07" notes. Fill out the above line, then proceed with the exam. Problem 1. (40 pts: 5+5+5+5+7+5+8 ) Consider the three vectors A :< 1,0, 2 >, B =< 0,1,3 > and C =< 2,1,1 > . a.) points) What is the engle between vectors {A + B) and C? (flit?) : (/,0,2)+ (0/3) : (LAX) (,2;+§)-5’ : (/,/,r).(2i/,/)E y /fld+E’//C—’/ ’(le'qlg ‘ 7153 'j —é— ‘2 C 575 H I Hedi/3 COS—Q- : 7 b) (5 points) What is the area of the parallelogram with sides A. and B? ._5 -..._$ ' .L .r’\ 4 x- ': Dfl'Ir‘q-fi'( Mir-e“. I1 r /_\ 2 /_ A A ‘- I .I - ~/ J . :£(——7_)"J37L/< / o 2 13 I 3 13> Digs/#4 {Ll/ea ':_‘ //? X t ‘i/ yf- 7 7t /' 2 c) (5 points) Write a unit vector perpendicular to both A and B. A /‘ F 4 Pi: 5X23; “ii—21+k r0 (:4— Due, I (continued on next page) 4 Problem 1. (...continued) d) (5 points) Write the equation of the plane containing vectors A,B and the origin (000)? ._-J '{khtofi'dn 11% (L fAJ/h a. [Alf/Z; MEI/“Mm / n/L/ : ("I g, C j ’\5 51x’l'47 1"- : Q'hif Mme 2U: ("2r_3,._7)':> ~2x—37+2: a:an fiL-Z'L (0/ 0/ 0) £5" /;\ 71—441/0/9111 fl (Likj7é 2* 0 _._-—- “b — 2x “37+ 2 : O ,3 7‘41 zebra/“4M e) (7 points) What is the volume of the parallelpiped with sides A, B and C? Via/Lame =j/5 53/32/=%/k;f/42/20/l/] MuLS T fusx'rrl/é 1L [ _2 + flyjj :_ : 6 U f) (5 points) Write the parametric equations of the line parallel to C and passing through the origin. "'— __—‘- H) oP—- 071 =- Cf :5 X227L, 7: 7‘, 2:74 g) (8 points) Deduce the minimum distance between this line and the point (0,1,0). [371 £41744; «(bifu’ficx jifinJ'flM-h 7/1"} /*\““- “"151 (‘0’ /' A 72L“ pzfi“): /(271.f,7")*/0;/,0)/1:Qflze /1’~~/)£_,,_713 t :47“ 327‘7L _7' fl.) U 4 flf/W/M'im Mid/m -— [‘2 O H Kc”. wild-urn ?(3 “é— a¥ MAM”: [DD/“7L I Problem 2. (20 pts: 5+5+5+5) a.) (5 points) Compute the determinant of Jaw: z/Zf/_ ,/ff—/+Z/;Da/ : 2‘4‘ [/3--Z)+Z(—9’) :- (BK/w? : —1 L0 1 4: 0 |—‘ 10 [\3 “j Jul fl 1‘2 b) (5 points) Compute the entries a and b of the inverse of A, F, #:4— ‘Jf C0 1!, ma. —' a ‘-= €h7LV‘7 /Z. I) 0/ fl 7- fiéjfifl 454%”7 (A 2') {‘76” 74w] :2— m'“ Mm «: /f’f/= 2 g ‘2 (-2, (bf/4 F, '1': (IL-i; (3, 2) 0f CVOIK‘LC‘AJ-r V1574}? . .24 t j (d) E/WL/flu) : :ZZ/Z*2 fl“; a : j A I: __ 2 . (continued on next page) 2 II Problem 2. ..continued) c) (5 points) Consider the linear system <3 g §)(§)=(§)3 From your calculation in (b), you should be able to deduce the value of one of m, y, or z without much extra work. Which of these do you actually know, and what is its value? -—I Léovg ,4 fl th }/0u X x 9; >4? '2 (7 ) t (f at- ‘2 /0) A“ “4“ 70% (M g )f % 3 pic 4?. 7 dry-e Cf-Z/ I fl 7:/-2+/—1)/3):-—§/ d) (5 points) Now let B be a 3 x 3 matrix with the property that BGHZ) Bet)- How many solutions does the homogeneous linear system B(§)=(il have? If it has a nontrivial solution, write down two distinct nontrivial solutions. [\3 I74 ALT /1:1 I'nf7le/[7I “4“”‘17’ jo/Wfia;\; - I“ /"W75fi(“/‘“'" I , / O W?) -/ I) : a Z I 0 Jo f; at. fo/oefiflih I A1 f)“ Ah7 amotfiqé z Mu/yt-i/i/i If {e O ( . Problem 3. (ZOpts: 5+5+5+5) Consider the parabolic trajectory described as a function of time t: r(t) = t2 i + t a) (5 points) Calculate the velocity v and acceleration a vectors. J 0/»: A V(_{/ : C]? 7: 2 Z,— 7‘- 25“): 3L? 1‘ d7 .2: 2! b) (5 points) Calculate the unit tangent vector T and unit normal vector N as functions of is. 37mm” : /l7(%J/: {fir/tug : {m Zx/Mt‘fil 7£*ng1_ m‘)‘ (/e - ‘ ‘.: .__ 4'— ' ’1‘ J 6:7,??? . T j hm / { T‘J) Mew-7! nOvWxu/ .’ “.2 .___ / I; k 1. 0 I r if ’L’ U C: o t A; at zrj) c) (5 points) Calculate the curvature K of the trajectory. X: H -_ X" I ( W74“: Id?!) = M (7th rK7’51J3 A4.“ x17“::)5<’: 2%,)(l’12 7=cfi> L/‘= ’17”=0 —-. f/OJ— 20) -— 2 fl ‘ 1‘ 2 r H “‘“T—"J: (2—?!)sz 1J3/1 “ +,)/ d) (5 points) For what value of t is the radius of curvature R of the trajectory a minimum, and What is its minimum value? Phi):— /J::L/¥)/ -: EL/V'I‘z-A,/—?/a WINK/w 2': M""‘“J’VIL1"‘-\ Him/Ix f:- O flff‘n..m-\\.Lvu_ l/fi/HQ -' : é Problem 4. (20pts: 10+5+5) I; L" 1‘ a 1. A single scull (row boat) advances in the < D, 1 > direction in the my—plane at a constant speed U. It is propelled by a pair of blades (oars) of radius R that sweep out a total angle 21/3 in the try—plane at a constant angular speed w. The angular position of the blade relative to the per endicular to the boat is iven b 6 = 3 sin wt. At time t z 0, the center of the boat is at the P g y 3 origin (0,0), and the tip of its right blade (the point P) is at (R, 0). a) (10 points) Find the parametric equations for the trajectory of the blade tip P. ..._- 0P: Org/+475 '5 F/i‘) (I A I t ‘f- Rfiajfi ( '1- F/f‘): Zffijé+7?605[—§I .5'.‘m¢o?‘]}+ Pfa'hf-g—SI‘HLJT‘JJ j X“): f ‘05 ? JD;wa 7/” :— + E) II‘Hg—Ix'nl—Uf] b) (5 points) Find the velocity v of the blade tip. "" a 6/? c/x A . ci/ ’7 (LN-h V“W=:¢7’+:ff1 M; A I *f 5111 J‘fm wrj. géo ('05 MIL r‘ A 7L [U + ECL*5[§J‘XHLJF{L]' gu Corarz‘JJ‘ 4 r. t‘ 'i W“ ‘ ' " ~ " I m 7* . . | . Z J "’L 3 £40 C03 6&7 [(05/‘§ JlnaJIQJ -Jm EJJJU—Jfit] “ifi .i “xxx, 7 ‘2 [Eu T4? 011/ c) (5 points) What is the maximum speed of the blade tip, and What is the direction of the velocity at the corresponding time? TEflflfiiATJON fliax J/a-erar{ waI'f—EJ (LI fMJ—er flufujé .6l : (j J ‘JAuk 74‘6‘mf/KL76CL‘k( am(( V0fi76‘dhzb/ S/J-EC/(IJ‘ aLr—L fl/JH&4//‘ 1):!qu 1‘ 2/?— ”L g: 2? LL) \_.———\,-—-—-— few “40% __,_____ K0 ...
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Exam1_solutions - ‘ -_..-—- Souffle/M 5 4 ._ ____ NAME...

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