worksheet 2

# worksheet 2 - Math 200 Spring 2010 Worksheet 2 Name K a...

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Unformatted text preview: Math 200, Spring 2010 Worksheet 2 Name: K a; Section 12-3 Dot Product and the Angle Between Vectors Section 12-4 The Cross Product 1. Find a-b. (a) a=(—2,3),b=(—3,7) (b) a=4j—3k, b=2i-4j+6k at: = «av-<4,» “cf-2- : (W3—3EJ-(2'f-Hj'dl2) = (an—3) 4: 3-1 = 27 '5 0-2+~+(-H)+/—3J15}:-3Lr 2. Simplify the expression (v+w)o(v+w)—2v.w, (Tim «mm—2e; = r.(r+:)+e.(g+;) -33.; -—.I ha ._. .4 -J -—1 n.- —.3 .4 -_.3 "' u-v+u-u+w-u +u~w ~2u-w \ -—I-_|. 4—; —l ‘L -—I -—l ”‘ Mill +v-i» *Uw: 441ml} —-Du‘w 3. Find a-b if “all 2 Mb” = J6 , the angle between a and b is 45°. —’ 3-11: = “21:11me =3JZcosHL56= 352.123 =3J§c 5.20 4. Find the angle between the vectors a = (4,0,2) and b = (2,—1,0). 63\$ : (ﬂan-(24,03 , _ 8 = :r ‘— " ﬂa-f 5 “Elm M Sn Grros (:3) 237 5. Determine whether the given vectors are orthogonal, parallel, or neither. If neither, then specify whether the angle between them is acute or obtuse. (056: (a) u=(—3,9,6),v=(4,—12,—8) (b) u=i—j+2k, v=2i—j+k (c) u=(a,b,c),v=(—b,a,o) __| “J - \ "‘ v5 u 3 “la 3 .; so ﬂ Hi? “3‘5 11d “5”” "‘°’+‘f""’ ‘3; “ Kw MM) + Hex)“ a Also TIL-3:453: allﬂiu-‘n 2 *A-XU- 513=S>0- reufr mar :0 S in... H +9": . U f light-941‘ new-o.- “ Ml- .L and W __, .3 0 Us 0 one Qarﬁle \Mt- f m was. e chaotic”. Gaye page“ {S awh. 50 UL _L v 6. Pin the scalar and vector projections of :1 onto [1. (a) a=(1,2), b=(—4,1) A (b) a=i+j+k, b=i—-j+k r _..1__-_. "- __..|_-t+2 :_2 Cbm_;: 1' M '2 Lﬂ:'—lr (“91° ' C‘ W n u ' ml '57: 3 KPH ND: tr? “'7 h I u- .- _J h ‘ ‘- l - l : _._ ' h) .. —. 9m" “ ﬂ COW“ -511? ‘ 6?? m <67?) 4T7» ?'°J..o. - (an-p a. - P1- 7. Find the ecomposition a=an+al with respect to b, if _ 1 Ei _J D l ‘ \ _ __ . .__ __-_._. where a=(4,—1,o) and b=(0,1,1). d I 6‘3 \TE {3; 3; 3) -\ "'1 '-' ...s A 2—..— — -——- - .. Q" : Prom-’6' : (COMWQ) 8?: r L _ 1 < 01'1‘3 _ {OJ-t) 153 ._ ___"’ .. __ _ l .. a; - 0» at. — (s, new wort, :5 — «View 8. A tow truck drags a stalled car along a road. The chain makes an angle of 30° with the road and the tension in the chain is 1500 N. How much work is done by the truck in pulling the car 1 km? We. hose B“ : moo.“ ) [J F“ = JSGB N j 9 = 30° -—_’ _ d a a A d, m’lsﬂmo/ wwk W = F-D = HF): DD}! (036: 1500(l0w)(oi?oo : 1500600 13 :1 30 II ﬁll = ism“ : 759 000% 2 1 —— — 9. .. : _—. 9. Calculate the2><2 determinant 3 6 3'1 ' ills-('5') ' 55 ff— 6 —5 2 1 O '3 -2. 3 ..J_ a 10. Calculatethe 3x3determinant —-2 0 3 " ‘ l 3_‘\ ‘ 01 1' l\ \ 3 1 3 —1 ‘- —'1 +0 ~6: “5 11. Find the cross product axb, where a = (1,1,dl),b = (2,4,6) 994 L at 3*1'5 \\—-1 1‘1-é - 1" 'lH-L T [Hash {6-0-1113 + {Wu K = 102 ~33 ~22 12. Verify that ax b in Exercise 11 is orthogonal to both a and b. In: ‘3 a (cut-)w. = <10,—9)a‘>v<1,1)—n 2 “3.3.3,”; (Fiﬁ-1? = <to,—s,2>-<2,=+,£> : 2c- -22+w‘-o 13- Find llux v" and determine whether ux V is directed into the page or out of the page. 1&va = 11:11 Il‘Jil sin as" : 5( to) ‘ *5; = 2515‘; 13) “Hm 'f‘t‘jl'i'l" Lana hole) an? 13 clui‘ec‘l'ecl 'lh‘llﬁ ‘Hﬂf (page 14. Calculate (u—2v)x(u+2v)if uxv=<1,1,0). ("ti-ISHfEﬂTr) = (ti-m xi: + (Ti—13hr; ‘- ﬁxed -1‘6 {6" +11): 12—;- +31%}, = o +1R‘*§:“+2ﬁ,xﬁ~ c '= Li-kag = H‘<l,l,0):<‘+ﬁ3o) 15. Calculate the cross product (j—k)x(j+k) A (331%) NEH?) : (Hm; r (34%) 7‘“ 3 16. Sketch the parallelogram spanned by u = (1,1,1) and v = (0,0,4) and compute its area. A of: A n 111 =‘TL-‘t3 .J - “that = \\‘-t<1,~n®ll = 1 u 01-1421 : H‘Utl-i-HP : 4J3. 17. Given points P(2,1,5),Q(-1,3,4), and R(3,0,6) (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. —-—.h l .__I [995 : {-3’1J—J‘7 J PR 2 (ll-51> Ft decfor owﬂnagana! ‘14 445: (plane P, 9,444 9 ‘5 993 K Pl? : 4639;13 7~<lJ-|,\‘> :. ) :3 J h 2-\ A. A '\ l = L-t-lj-g-K = (1,1!) (b) Find the area of triangle PQR. g “A 5A The qnee. of; 414: Paralleleﬁdee-krminod 1), 1°93 and PR as U I 93 *1 R" 3 ll < 1,2,1)” 1 dl‘w-fuhh = J2 . _ .1. Hence, ‘Hne. ark. 0(- He +PI'Gn5/4: P Q R _ .1 LE 18. Calculate the volume of the parallelepiped spanned byu = (2, 2,1) , v = (1, 0, 3) , and w = (0, —4, 0). wloxiﬁ 2.1a % :1 ‘0’, a: (I o-‘t .-—..3 ‘- 3(11)—2-0+:(~4) 1 :20 Volume : '_' 1301 :10 19. Use the scalar triple product to determine whether the points A(1,3,2), B(3,—1,6), C (5,2,0), and D(3,6,—4) lie in the same plane. __‘ _'_ ._. _ Let c: -. 42 = <1;“-t,“*>jU = M "~' (“FINN w ‘ ’40 ' “432*” d . _' 4 : 2 'H’ "1 UL (uxwd \‘1—\1\: 1(\1)+L1(-‘20)+'7‘[H):0 1 3—6 50 ‘HM.’ Uclvmp 0's" "Hue ParkllElEPIPQd Ian‘l-‘Prmlh-Qd be! 1:) u;r Othd DJ 1‘5 0 hade Swat {454+ ‘l-lue trader-J lie m 1449. Same Flame: 4ft 20. Find the magnitude of the torque about P if a 36-lb force is applied as shown at the right. P 11?“: m = I415“: R. H’ ll-IA-I drawn gmm ’l’ne {Oath} P 'i'uo‘Hne [Dominik d- n-P’Jlicod-ien 0‘9 the 43mm? matte; q». Gin-ale one 190 _{t+s°1- 30°) : 105 wﬁl-k ‘er gen: wed-or. Thfr‘ﬁ'hor-e H 713’” z ‘3 FX 1:?“ ., “at "15“ “n G _.2...:....,_ 4ft e. . D \$34.50 : +43, ( 35) 91A 145 : 144111 Sm 75' mm ﬂab/"1‘ Homework #2 Read Sections 12.3, 12.4 1: 1°17 91-41. 6213311333) (Due 1/29) Do Section 12.3 Exercises: 1, 9, 13, 15, 23, 27, 39, 43, 47, 49, 53, 69 ' 0 Section 12.4 Exercises: 3,5, 11, 17, 21, 23, 37, 41, 43, 49, 53, 65 2 1‘35 Prepare for next class session: Read Sections 12.2 (pgs. 688-690), 12.5 ...
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worksheet 2 - Math 200 Spring 2010 Worksheet 2 Name K a...

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