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Test 1 Date: January 24, 2008 Clearly show all work for full credit. 1. (4 points) Find an equation of the sphere with center (6,5,2) that passes through the
point (4,5,2). r9: (47*!)1 (5:93 t 02:02)? 2 q (Mr + (w); + ma)”: H 2. (4 points) Find the center and radius of the sphere given by x2 + y2 + 22 = 4x — 2y. ”Kam ‘ix +va+9v til =0 Cac 933% (v HY“ 22‘: "lt/ (ﬁnder " ((2, ﬁl) O.) Raoul; : (‘3 jg; Page 2 of 5 3. (4 points each) Given a = 3i — 2k and b = i — j + k , compute each of the following: M K . N h  a, _, 7
(a) 2a—3b 2 (@1174 ‘11?) A (EX33) r313): 3C7+ 3.} “' '7k (b) lal >— ~ Ci+o+LI :: J73
(C) M) '2 3m + 0(«0 .9210): I
(d) axb L J k .42 K]: “5 <~QJ”SJ“3> (e) the angle 6’ between a and b > \ r O
_ 5V L .. ——L———‘ :1» Q = 808
403(91 ‘“ 1'2? x (f?! ““ (JE)LJ‘3’>
(t) the scalar projection of 3 onto b «a
8, v) 53‘ ‘0 ,— J...
O a = “.7“ ~—
”‘ Pt: I I If?
(g) the vector projection of a onto b 1/ >
. f7 cQIkP~7a _ ’1. r: < "’— v
Praia: r——.%—Itl>’ Iii) 32323
i? l L
h th d' to sin f : 3i
() e 1recrnco eso a Cu__g(o<\_ $3
6.0; C 63 = 0
C05 (20 1‘ :% 4 (4 points) Find the volume of the parallelepiped determined by vectors a= <1,1,—,=1>b <1, ,,11> andc=<,11,1>. 2:7 =21 LJkiJ '~)11*11~I ﬁxedi to“: J! [114‘ (”UK 8:) <11'l>v<§lao>L1I
V: I LII LI 5'3 ‘5 , , .
chc, : {'o?)';)0> Page 3 of5 5. (8 points) A pilot is steering a plane in the direction of N3 O°E at a speed of 200 mph.
The wind is blowing at a speed of 10 mph from the direction N75°E. Find the true
course and ground speed of the plane. ”7: PL.“ vafbr : <aw coneo“) , 20c. siutcc>°)>
.___. < ioo, mfg» «.1 4 (00) 573.31»
Cf) v.34 vac/(w :— (JOCosU‘iS‘),105i¥1(‘i€$’°)‘7 res Ilia/wk  03+ q], C3=< 610.34, {70.52)
R taizei: "3"? => 6:: $910
0' 1 ‘10»8: arm lift: (540.3qafl70‘692 24— 143.“: TM CawﬁSa': ~91an True Speed 1 [€3,043 mink 6. (2 points each) Describe, in words, the intersection of the graph of x2 + 4 y2 — 222 = —8 with each of the following planes, and also state the equation of
each intersection. (a) the yzplane (70:0)
)Lypcrimxek’: L‘VQ‘ QED—T‘g (b) the xz—plane (4/16)
I Q ’3 .—
kyper’iaetea' ’X  92' ~ 8)
(C) the xyplane (2:0)
Ito ihiﬂsaebh S‘nce» X9+qﬁor~3 Las no SchgftbA
(d) the planezz—Z the, Pm“: (O) O} ‘Q\ Snag,
XD+HYQ“A(”Q)D=‘S( =57 XQrLlﬁf‘l: O :2) 'x:O:'~1 (e) the planez= 3
eﬂf’osgi 12+ 71122(3332 ~8 :3 XQ+LIYQ :: IO Page 4 of5 7. (4 points) For r(t) =t2 i+ t 11 j+ ln(t k, ﬁnd the domain of r and compute limr(t)
3:; £50 11*“
Domain: P05 61139 Peta/2&1 nméers excepé [1“: W564, {
M [a Ida—.20 . :1 £1 _ .L
SW, A z: :1, L; ,9, N. @1111» 12 ,
1:9! to” 4i 6?! £113., [57 7—366134‘Ja30> 8. (4 points) Write the rectangular equation 2 — x2 + y2 — y using spherical coordinates. “Z: (‘2 ’y Z: €EOS(¢) F: €S¢‘n(¢)y
"ftrsierQl 1‘ (355161951316) :2 ECU; (Cb) ":2 €9s‘_na(®) “‘" F$iin(¢\$ir\(e§ 9. (4 points) Change the point with spherical coordinate (22 2\/—,—— 37f 2,”) to cylindrical coordinates. Sm“), (Z): 1%: (ﬁe. {Dairié I: On (2‘4 Xy/>2€&4:¢, ) .i, Z: O 5.11% F: P 1 (mag49133:?) I" 10. (2 points each) For the line 1n 3 space through the points:(6, 1,3) and Q”,(2 4 ,,5) determine 31—)
(a) avector equation of the hne l7 rm< —‘I 3 52>
(Agra? P: (e, 8‘33 F3651: {(9,150 + t <q,3,8> (b) parametric equations for the line ’X: é*"{6
NY: (it—36
Z: “3+31i: (c) symmetric equations for the line Page 5 of5 ll. (5 points) Find an equation of the plane in R3 that passes through the point (2,8,10)
and is perpendicular to the line x =1+t , y 2 2t , z = 4 —— 3t. 7?: if: <i)91*3‘> 2 (W) 7+ 90,40 ‘ECzI’Oh : o I 12. (5 points) Determine whether the lines L1 and L2 given below are parallel, skew, or
intersecting. If they are intersecting, ﬁnd the point of intersection. L1: x=l+2t,y=2t,z=4—3t ‘07: 4;,Q)3\/
L2: x=6t,y=l+6t,z=5—9t V3: <G,é,”“i> 1):: 33? 33> L. // La 13. (5 points) Find an equation of the plane in R3 that passes through the point (1 ,—l,1)
and contains the line with symmetric equations x = 2 y = 32 . P: (0,0,0) [We (2: (‘e)'3,a\ m Page m at [at
L&é [2: (23") i» a.) ~$i ,
:3:— PE? '2 < 93;» x7”: PR: < 1,4, D
W: Cfxvzz 5,"‘i)”4>
U859 P ‘1 (0)0’03 :
wal‘ixj "q? :0 l4. (5 points) Find the point at which the line with parametric equations
x=1+2t, y =4t, z =2~3t intersects theplane x+2y—z+l =0. M+Q~pe+l =0 :2» (ItatlraCHQ(av3t3tlTO
:> :35 :0 .—~.:> £20 H l
a (xi—ER (l ...
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 Spring '08
 Paris
 Calculus

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