Gabarito Lista 2
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Transcript of Gabarito Lista 2
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GABARITO LISTA 3 DE AVE
1) a) OCvu = rr b) OCuv = rr c) OCuv = rr 2 d) OCvu = rr 32
ur
vr
O A
B C
ur
vr
O A
B C
O ur2 A
B C
ur2
vr
vr3
O
B C
A
-
2 a) cba r
rr 24 b) cba r
rr ++ c) )(2 cab rr
r +
ar br
cr
cr
ar4
br
2
cba rrr 24
ar
br cr
cba rrr ++
cr
ar ca rr +
ca rr +
br
2
)(2 cab rrr +
)( ca rr +
-
3) a) vu rr e b) vu rr e c) vu rr - e
ur
vr
060
ur
vr
0120
ur
vr
0120
ur
vr
060
-
d) vu rr 3 e 2 4)
:,
:
setemABvetorocomovvetorodoConsideran
Soluo
r
)2,1()2,1(),(
)3,1()5,2(),()3,1(),()5,2(
==
+==
==
Byxyx
yxABv
ABvrr
ur2
vr3
060
-
5)
a) wuwvu rrrrr =+ 231)(4
=
=
=
=
=+
=+
215,
215
430,
430
34)10,10(
31)2,6()12,16(
)2,6(31)3,4(4
)1,3(231))2,1()1,3((4
: vemequao, na dados vetoresos doSubstituin:
w
w
ww
ww
ww
Soluo
r
r
rr
rr
rr
b) )34(2)2(3 uwuvw rrrrr =
==
==
=
511,
523
)11,23(5)5,5()6,18(83
))6,18(8()1,3()4,2((3))1,3(34(2))1,3()2,1(2(3
: vemequao, na dados vetoresdos valoresos doSubstituin:
w
www
wwww
Soluo
r
rrr
rrrr
-
6)
)30,5()24,4()6,9(43
)6,1(4)2,3(343
))1,3()5,2((4))5,2()3,1((343
)(4)(343
43
:
===
==
CBBA
CBBA
CBBA
CBBACBBA
CBBA
Soluo
7)
34
3443
343
49
)3,49)4,3(
3,49)4,3(
:
=
==
==
=
==
aLogo
aa
aaaa
a
vauSoluo
rr
43
4343
433
49
)4,3(3,49
)4,3(3,49
:
=
==
===
=
=
bLogo
bb
bbbb
b
ubvSoluo
rr
-
8)
1446)2(4
:se-obtem64
em obtido valor o Com2189
6424104
64)2(1252
641252
)4,52()6,12(
)1,5()4,2()6,12(
:
11
1
21
22
21
21
21
21
21
212121
21
21
===+
=+
==
=+=
=+=
=+=+=
+=+=
kkk
kk
kkkkkk
kkxkk
kkkk
kkkk
kkvkukw
Soluorrr
9)
D(4,-4):Assim
431422
)3,2()1,2(
)0,1()3,1(),()1,2(
=====
==
=
yyxx
yx
yxBADC
BADC
-
10)
)21P(3,2,-
:Assim2121
153342
)2,5,4()1,3,2(
),,()2,5,4()1,3,2(),,(
:
====+==
=+
==
=
zzz
yyyxxx
zyxzyx
zyxzyxPBAP
PBAP
Soluo
11)
6)P(14,-10,-:Assim
69310122
14151)9,12,15()3,2,1(
)3,4,5(3)3,2,1()3,2,1()0,2,4((3)3,2,1(),,(
)(33
:
====
==+=+
=+=
==
zzyy
xxzyx
zyxzyx
ABAPABAP
Soluo
12)
)1,1,1()3,3,3(3
)1,7,3()4,10,6(2)4,10,6(2)1,7,3(
:
==
=+=+
vvvv
vvSoluo
rr
rrrr
-
13)
3422
: vem1 em valor este Com2a
: vem2 De(3) 144(2) 402
(1) 42)4,02,2()14,4,4(
)4,0,2()1,2,1()14,4,4(
:
2
2
1
21
1
21
21121
21
2211
==+
=
==+=+
++=
+=+=
aa
aaaaa
aaaaa
aavavaw
Soluorrr
14)
===
===
==
=
=
=
293
323
231
32.1
32m46m
bm)am,(6m,(4,1,-3)
b)a,m(6,(4,1,-3):Assim . m onde
vm.u:condio arespeitar devem eles paralelos, sejam vetoresdois que Para
:
bbbm
aaam
Soluo
rr
-
15)
.colineares so no C e B A, pontos os scoordenada as todaspara a verificadfoi nok razo a Como412
21
41
12
21xk
:condio a Aplicando)4,1,2()0,1,3()4,0,1(BC
)1,2,1()1,1,2()0,1,3(
BC e AB b)
.colineares so C e B A, pontos os scoordenada as todaspara igual k razo a Como43
43
43
43
86
43xk
:condio a Aplicando)4,8,4()3,1,2()1,7,2(BC
)3,6,3()0,5,1()3,1,2(
BC e AB a)
: vetoresosdefinir se-pode Assim .xk sse paralelos sero eles
),,(xv e ),,(xu vetoresos dados que determine vetoresentre oparalelism de condioA
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
222111
=========
======
=========
======
=====
kkzz
yy
x
BC
ABAB
kkzz
yy
x
BC
ABAB
zz
yy
x
zyzy rr
16)
C(-3,13,7):ser C ponto o Assim
13363393
14
3189393
32-
vem; termos,ltimo o com segundo o e termoltimo e primeiro o se-Igualando93
14
32-xk
:condio a Aplicando)9,1,3()2,1,3()7,,(BC
)3,4,2()2,1,3()1,5,1(
BC e AB a)
: vetoresosdefinir se-pode Assim
.x
k
:anterior exerccio do pensamento de linha mesma a Seguindo:
2
1
2
1
2
1
2
1
2
1
2
1
===
===
======
======
===
bbb
aaa
bak
zz
yy
x
babaBC
ABAB
zz
yy
x
Soluo
-
17)
ADkkzz
yy
x
ADAD
BC
CDkkzz
yy
x
CDCD
AB
ADCD
//BC11122
11
22-xk
)2,1,2()1,0,4()3,1,2(
)2,1,2()3,1,5()5,2,3(BC
//AB1112
21
11
1xk
)2,1,1()5,2,3()3,1,2(
)2,1,1()1,0,4()3,1,5(AB
//BC e //AB
:sejaou opostos, lados os entre oparalelism decondio arespeitar se-deve ABCD, amoparalelogr um formem dados vrticesos que Para
2
1
2
1
2
1
2
1
2
1
2
1
===============
============
===
18)
(-5,-1,-4)P' Logo-4z04z-1y0y1-5x05x
(0,0,0)4)zy,5,1(x
(0,0,0)(-1,0,-3)-z)y,(x,(-1,0,-3)-(3,1,-2)(0,0,0)P'-AA-P
P'. determinar deseja se onde0"AP
:queafirmar se-pode Assim .AP' vetor um sejaou ,AP vetor ao simtrico vetor no se-pensando obtidoser
poder P ponto do simtrico ponto o Assim .AP vetor o definemA e P pontos os quepensar se-Deve
:Soluo
==+==+==+
=+++
=+=+
=+ rAP