Matrizes  Vetores e Geometria Analítica
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Matrizes Vetores e Geometria Analítica


DisciplinaGeometria Analítica e Sistemas Lineares137 materiais832 seguidores
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entre \ud449 e \ud44a e´ 60\u2218.
Determine, como combinac¸a\u2dco linear de \ud449 e \ud44a (\ud465\ud449 + \ud466\ud44a ):
(a) Um vetor \ud44b tal que \ud44b \u22c5 \ud449 = 20 e \ud44b \u22c5\ud44a = 5
(b) Um vetor \ud44b tal que \ud44b × \ud449 = 0¯ e \ud44b \u22c5\ud44a = 12.
Marc¸o 2010 Reginaldo J. Santos
210 Vetores no Plano e no Espac¸o
Exerc\u131´cios usando o MATLAB\u24c7
>> V=[v1,v2,v3] cria um vetor V, usando as componentes nume´ricas v1, v2, v3. Por
exemplo >> V=[1,2,3] cria o vetor \ud449 = (1, 2, 3);
>> subs(expr,x,num) substitui x por num na expressa\u2dco expr;
>> solve(expr) determina a soluc¸a\u2dco da equac¸a\u2dco expr=0;
Comandos nume´ricos do pacote GAAL:
>> V=randi(1,3) cria um vetor aleato´rio com componentes inteiras;
>> no(V) calcula a norma do vetor V.
>> pe(V,W) calcula o produto escalar do vetor V pelo vetor W.
>> pv(V,W) calcula o produto vetorial do vetor V pelo vetor W.
Comandos gra´ficos do pacote GAAL:
>> desvet(P,V) desenha o vetor V com origem no ponto P e >> desvet(V) desenha o vetor
V com origem no ponto \ud442 = (0, 0, 0).
>> po([P1;P2;...;Pn]) desenha os pontos P1, P2, ..., Pn.
>> lineseg(P1,P2,\u2019cor\u2019) desenha o segmento de reta P1P2.
>> eixos desenha os eixos coordenados.
>> box desenha uma caixa em volta da figura.
>> axiss reescala os eixos com a mesma escala.
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
3.2 Produtos de Vetores 211
>> rota faz uma rotac¸a\u2dco em torno do eixo \ud467.
>> zoom3(fator) amplifica a regia\u2dco pelo fator.
>> tex(P,\u2019texto\u2019) coloca o texto no ponto P.
3.2.17. Digite no prompt
demog21,
(sem a v\u131´rgula!). Esta func¸a\u2dco demonstra as func¸o\u2dces gra´ficas para vetores.
3.2.18. Coloque em duas varia´veis \ud449 e \ud44a dois vetores bi-dimensionais ou tri-dimensionais a seu
crite´rio.
(a) Use a func¸a\u2dco ilvijk(V) para visualizar o vetor V como uma soma de mu´ltiplos escalares
(combinac¸a\u2dco linear) dos vetores \ud456\u20d7, \ud457\u20d7 e \ud458\u20d7.
(b) Use a func¸a\u2dco ilpv(V,W) para visualizar o produto vetorial \ud449 ×\ud44a .
(c) Use a func¸a\u2dco ilproj(W,V) para visualizar a projec¸a\u2dco de \ud449 em \ud44a .
3.2.19. Use o MATLAB\u24c7 para resolver os Exerc\u131´cios Nume´ricos
Exerc\u131´cios Teo´ricos
3.2.20. Mostre que em um tria\u2c6ngulo iso´sceles a mediana relativa a` base e´ perpendicular a` base.
3.2.21. Mostre que o a\u2c6ngulo inscrito em uma semicircunfere\u2c6ncia e´ reto.
Sugesta\u2dco para os pro´ximos 2 exerc\u131´cios: Considere o paralelogramo \ud434\ud435\ud436\ud437. Seja \ud448 =
\u2212\u2192
\ud434\ud435
e \ud449 =
\u2212\u2192
\ud434\ud437. Observe que as diagonais do paralelogramo sa\u2dco \ud448 + \ud449 e \ud448 \u2212 \ud449 .
Marc¸o 2010 Reginaldo J. Santos
212 Vetores no Plano e no Espac¸o
3.2.22. Mostre que se as diagonais de um paralelogramo sa\u2dco perpendiculares enta\u2dco ele e´ um losango.
3.2.23. Mostre que se as diagonais de um paralelogramo te\u2c6m o mesmo comprimento enta\u2dco ele e´ um
reta\u2c6ngulo.
3.2.24. Se \ud449 \u22c5\ud44a = \ud449 \u22c5 \ud448 e \ud449 \u2215= 0¯, enta\u2dco \ud44a = \ud448?
3.2.25. Mostre que se \ud449 e´ ortogonal a \ud44a1 e \ud44a2, enta\u2dco \ud449 e´ ortogonal a \ud6fc1\ud44a1 + \ud6fc2\ud44a2.
3.2.26. Demonstre que as diagonais de um losango sa\u2dco perpendiculares. (Sugesta\u2dco: mostre que
\u2212\u2192
\ud434\ud436 \u22c5
\u2212\u2192
\ud435\ud437= 0, usando o fato de que
\u2212\u2192
\ud434\ud435=
\u2212\u2192
\ud437\ud436 e \u2223\u2223
\u2212\u2192
\ud434\ud435 \u2223\u2223 = \u2223\u2223
\u2212\u2192
\ud435\ud436 \u2223\u2223.)
3.2.27. Sejam \ud449 um vetor na\u2dco nulo no espac¸o e \ud6fc, \ud6fd e \ud6fe os a\u2c6ngulos que \ud449 forma com os vetores \ud456\u20d7, \ud457\u20d7
e \ud458\u20d7, respectivamente. Demonstre que
cos2 \ud6fc + cos2 \ud6fd + cos2 \ud6fe = 1 .
(Sugesta\u2dco: cos\ud6fc = \ud449 \u22c5\u20d7\ud456\u2223\u2223\ud449 \u2223\u2223\u2223\u2223\u20d7\ud456\u2223\u2223 , cos \ud6fd =
\ud449 \u22c5\u20d7\ud457
\u2223\u2223\ud449 \u2223\u2223\u2223\u2223\u20d7\ud457\u2223\u2223 e cos \ud6fe =
\ud449 \u22c5\u20d7\ud458
\u2223\u2223\ud449 \u2223\u2223\u2223\u2223\u20d7\ud458\u2223\u2223 )
3.2.28. Demonstre que, se \ud449 e \ud44a sa\u2dco vetores quaisquer, enta\u2dco:
(a) \ud449 \u22c5\ud44a = 1
4
(\u2223\u2223\ud449 +\ud44a \u2223\u22232 \u2212 \u2223\u2223\ud449 \u2212\ud44a \u2223\u22232);
(b) \u2223\u2223\ud449 \u2223\u22232 + \u2223\u2223\ud44a \u2223\u22232 = 1
2
(\u2223\u2223\ud449 +\ud44a \u2223\u22232 + \u2223\u2223\ud449 \u2212\ud44a \u2223\u22232).
(Sugesta\u2dco: desenvolva os segundos membros das igualdades acima observando que
\u2223\u2223\ud449 +\ud44a \u2223\u22232 = (\ud449 +\ud44a ) \u22c5 (\ud449 +\ud44a ) e \u2223\u2223\ud449 \u2212\ud44a \u2223\u22232 = (\ud449 \u2212\ud44a ) \u22c5 (\ud449 \u2212\ud44a ))
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
3.2 Produtos de Vetores 213
3.2.29. Demonstre que se \ud449 e \ud44a sa\u2dco vetores quaisquer, enta\u2dco:
(a) \u2223\ud449 \u22c5\ud44a \u2223 \u2264 \u2223\u2223\ud449 \u2223\u2223 \u2223\u2223\ud44a \u2223\u2223;
(b) \u2223\u2223\ud449 +\ud44a \u2223\u2223 \u2264 \u2223\u2223\ud449 \u2223\u2223+ \u2223\u2223\ud44a \u2223\u2223;
(Sugesta\u2dco: mostre que \u2223\u2223\ud449 +\ud44a \u2223\u22232 = (\ud449 +\ud44a ) \u22c5 (\ud449 +\ud44a ) \u2264 (\u2223\u2223\ud449 \u2223\u2223+ \u2223\u2223\ud44a \u2223\u2223)2, usando o
item anterior)
(c)
\u2223\u2223\u2223 \u2223\u2223\ud449 \u2223\u2223 \u2212 \u2223\u2223\ud44a \u2223\u2223 \u2223\u2223\u2223 \u2264 \u2223\u2223\ud449 \u2212\ud44a \u2223\u2223.
(Sugesta\u2dco: defina \ud448 = \ud449 \u2212\ud44a e aplique o item anterior a \ud448 e \ud44a )
3.2.30. O produto vetorial e´ associativo? Justifique a sua resposta. (Sugesta\u2dco: experimente com os
vetores \ud456\u20d7, \ud457\u20d7, \ud458\u20d7)
3.2.31. Se \ud449 ×\ud44a = \ud449 × \ud448 e \ud449 \u2215= 0¯, enta\u2dco \ud44a = \ud448?
3.2.32. Demonstre que se \ud449 e \ud44a sa\u2dco vetores quaisquer no espac¸o, enta\u2dco
\u2223\u2223\ud449 ×\ud44a \u2223\u2223 \u2264 \u2223\u2223\ud449 \u2223\u2223 \u2223\u2223\ud44a \u2223\u2223.
3.2.33. Se \ud448 , \ud449 e \ud44a sa\u2dco vetores no espac¸o, prove que \u2223\ud448 \u22c5 (\ud449 ×\ud44a )\u2223 \u2264 \u2223\u2223\ud448 \u2223\u2223 \u2223\u2223\ud449 \u2223\u2223 \u2223\u2223\ud44a \u2223\u2223. (Sugesta\u2dco:
use o Teorema 3.2 na pa´gina 187 e o exerc\u131´cio anterior)
3.2.34. Mostre que \ud448 \u22c5 (\ud449 ×\ud44a ) = \ud449 \u22c5 (\ud44a × \ud448) = \ud44a \u22c5 (\ud448 × \ud449 ). (Sugesta\u2dco: use as propriedades do
determinante)
3.2.35. Mostre que
(a) (\ud6fc\ud4481 + \ud6fd\ud4482) \u22c5 (\ud449 ×\ud44a ) = \ud6fc\ud4481 \u22c5 (\ud449 ×\ud44a ) + \ud6fd\ud4482 \u22c5 (\ud449 ×\ud44a );
Marc¸o 2010 Reginaldo J. Santos
214 Vetores no Plano e no Espac¸o
(b) \ud448 \u22c5 [(\ud6fc\ud4491 + \ud6fd\ud4492)×\ud44a ] = \ud6fc\ud448 \u22c5 (\ud4491 ×\ud44a ) + \ud6fd\ud448 \u22c5 (\ud4492 ×\ud44a );
(c) \ud448 \u22c5 [\ud449 × (\ud6fc\ud44a1 + \ud6fd\ud44a2)] = \ud6fc\ud448 \u22c5 (\ud449 ×\ud44a1) + \ud6fd\ud448 \u22c5 (\ud449 ×\ud44a2).
(d) \ud448 \u22c5 (\ud449 ×\ud44a ) = \ud448 \u22c5 [(\ud449 + \ud6fc\ud448 + \ud6fd\ud44a )×\ud44a ].
(Sugesta\u2dco: use as propriedades dos produtos escalar e vetorial)
3.2.36. Prove a identidade de Lagrange
\u2223\u2223\ud449 ×\ud44a \u2223\u22232 = \u2223\u2223\ud449 \u2223\u22232\u2223\u2223\ud44a \u2223\u22232 \u2212 (\ud449 \u22c5\ud44a )2.
3.2.37. Mostre que a a´rea do tria\u2c6ngulo com ve´rtices (\ud465\ud456, \ud466\ud456), para \ud456 = 1, 2, 3 e´ igual a \u2223 det(\ud434)\u2223/2, em
que
\ud434 =
\u23a1
\u23a3 \ud4651 \ud4661 1\ud4652 \ud4662 1
\ud4653 \ud4663 1
\u23a4
\u23a6 .
(Sugesta\u2dco: Marque os pontos \ud4431 = (\ud4651, \ud4661, 1), \ud4432 = (\ud4652, \ud4662, 1), \ud4433 = (\ud4653, \ud4663, 1) e
\ud443 \u20321 = (\ud4651, \ud4661, 0). O volume do paralelep\u131´pedo determinado por \ud4431, \ud4432, \ud4433 e \ud443 \u20321 e´ dado por
\u2223
\u2212\u2192
\ud4431\ud443
\u2032
1 \u22c5
\u2212\u2192
\ud4431\ud4432 ×
\u2212\u2192
\ud4431\ud4433 \u2223. Mas, a altura deste paralelep\u131´pedo e´ igual a 1. Assim, o seu
volume e´ igual a` a´rea da base que e´ o paralelogramo determinado por \ud4431, \ud4432 e \ud4433. Observe
que
\u2212\u2192
\ud442\ud443 \u20321,
\u2212\u2192
\ud4431\ud4432 e
\u2212\u2192
\ud4431\ud4433 sa\u2dco paralelos ao plano \ud465\ud466.)
3.2.38. Sejam \ud4481, \ud4482 e \ud4483 tre\u2c6s vetores unita´rios mutuamente ortogonais. Se \ud434 = [ \ud4481 \ud4482 \ud4483 ] e´
uma matriz 3 × 3 cujas colunas sa\u2dco os vetores \ud4481, \ud4482 e \ud4483, enta\u2dco \ud434 e´ invert\u131´vel e \ud434\u22121 = \ud434\ud461.
(Sugesta\u2dco: mostre que \ud434\ud461\ud434 = \ud43c3.)
Matrizes Vetores e Geometria Anal\u131´tica Marc¸o 2010
3.2 Produtos de Vetores 215
3.2.39. Sejam \ud448 = (\ud4621, \ud4622, \ud4623), \ud449 = (\ud4631, \ud4632, \ud4633) e \ud44a = (\ud4641, \ud4642, \ud4643). Prove a fo´rmula seguinte para o
produto vetorial duplo
\ud448 × (\ud449 ×\ud44a ) = (\ud448 \u22c5\ud44a )\ud449 \u2212 (\ud448 \u22c5 \ud449 )\ud44a,
seguindo os seguintes passos:
(a) Prove que
\ud448 × (\u20d7\ud456× \ud457\u20d7) = (\ud448 \u22c5 \ud457\u20d7)\u20d7\ud456\u2212 (\ud448 \u22c5 \ud456\u20d7)\u20d7\ud457
\ud448 × (\u20d7\ud457 × \ud458\u20d7) = (\ud448 \u22c5 \ud458\u20d7)\u20d7\ud457 \u2212 (\ud448 \u22c5 \ud457\u20d7)\ud458\u20d7
\ud448 × (\ud458\u20d7 × \ud456\u20d7) = (\ud448 \u22c5 \ud456\u20d7)\ud458\u20d7 \u2212 (\ud448 \u22c5 \ud458\u20d7)\u20d7\ud456
(b) Prove usando o item anterior e as propriedades do produto vetorial que
\ud448 × (\ud449 × \ud456\u20d7) = (\ud448 \u22c5 \ud456\u20d7)\ud449 \u2212 (\ud448 \u22c5 \ud449 )\u20d7\ud456
\ud448 × (\ud449 × \ud457\u20d7) = (\ud448 \u22c5 \ud457\u20d7)\ud449 \u2212 (\ud448 \u22c5 \ud449 )\u20d7\ud457
\ud448 × (\ud449 × \ud458\u20d7) = (\ud448 \u22c5 \ud458\u20d7)\ud449 \u2212 (\ud448 \u22c5 \ud449 )\ud458\u20d7
(c) Prove agora o caso geral usando o item anterior e as propriedades do produto vetorial.
3.2.40. (a) Prove que
[\ud434× (\ud435 × \ud436)] + [\ud435 × (\ud436 × \ud434)] + [\ud436 × (\ud434×\ud435)] = 0
(Sugesta\u2dco: use o exerc\u131´cio anterior).
(b) Mostre que se (\ud434× \ud436)×\ud435 = 0¯, enta\u2dco
\ud434× (\ud435 × \ud436) = (\ud434×\ud435)× \ud436,
ou seja, o produto vetorial e´, neste caso, associativo.
Marc¸o 2010 Reginaldo J. Santos
216 Vetores no Plano e no Espac¸o
Ape\u2c6ndice IV: Demonstrac¸a\u2dco do item (e) do Teorema 3.5 na pa´gina 196
Vamos dividir a demonstrac¸a\u2dco da distributividade do produto vetorial em relac¸a\u2dco a soma
\ud449 × (\ud44a + \ud448) = \ud449 ×\ud44a + \ud449 × \ud448 e (\ud449 +\ud44a )× \ud448 = \ud449 × \ud448 +\ud44a × \ud448
da seguinte forma:
(a) (\ud449 × \ud44a ) \u22c5 \ud448 > 0 se, e somente se, \ud449 , \ud44a e \ud448 satisfazem a regra da ma\u2dco direita, isto e´,
se o a\u2c6ngulo entre \ud449 e \ud44a e´ \ud703, giramos o vetor \ud449 de um a\u2c6ngulo \ud703 ate´ que coincida com \ud44a e
acompanhamos este movimento com os dedos da ma\u2dco direita, enta\u2dco o polegar vai apontar no
sentido de \ud448 .
(b) (\ud449 ×\ud44a ) \u22c5 \ud448 = \ud449 \u22c5 (\ud44a × \ud448), ou seja, pode-se trocar os sinais × e \u22c5 em (\ud449 ×\ud44a ) \u22c5 \ud448 .
(c) \ud449 × (\ud44a + \ud448) = \ud449 ×\ud44a + \ud449 × \ud448 e (\ud449 +\ud44a )× \ud448 = \ud449 × \ud448 +\ud44a × \ud448 .
Provemos, agora, os tre\u2c6s \u131´tens acima.
(a) Como vemos na Figura 3.25 na pa´gina 221 \ud449,\ud44a e \ud448 satisfazem a regra da ma\u2dco direita se, e
somente se, 0 < \ud703 < \ud70b/2, ou seja, cos \ud703 > 0, em que \ud703 e´ o a\u2c6ngulo entre \ud449 ×\ud44a e \ud448 . Como,
(\ud449 ×\ud44a ) \u22c5 \ud448 = \u2223\u2223\ud449 ×\ud44a \u2223\u2223\u2223\u2223\ud448 \u2223\u2223 cos \ud703, enta\u2dco \ud449,\ud44a e \ud448 satisfazem a regra da ma\u2dco direita