Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 11.4 11.4 El producto The Cross vectorial Product de of of dos Two vectores Vectors en in el in Space espacio 799
11.4 The Cross Product of Two Vectors in Space 799
11.4 The Cross Product of Two Vectors in Space 7
Volumen
En los ejercicios 47 y encontrar el volumen of del
180 libras lb
In Exercises 47 and 48, find the volume of the
F F
180 lb
paralelepípedo parallelepiped with que tiene the given vértices vertices.
dados (ver las figuras).
B B
θ θθ
Volume In Exercises 47 and 48, find the volume of the
A
F
180 lb
Volume In Exercises 47 and 48, find the volume of
F
180 lb
47. parallelepiped 0, 0, 0, 0, 0 ,, 3, 3, with 0, 0, 0 , the , 0, 0, given 5, 5, 1 ,, vertices. 2, 2, 0, 0, 5
18 18 18 pulg in.
B θ
parallelepiped with the given vertices.
A
B θ
3, 5, , 5, 0, , 2, 5, , 5, 5, θ
3, 5, 1 5, 0, 5 2, 5, 6 5, 5, 6
θθ
FF
12 pulg 12 12 in.
A
47. 0, 0, 0 , 3, 0, 0 , 0, 5, 1 , 2, 0, 5
18 in.
47. 0, 0, 0 3, 0, 0 , 0, 5, 1 , 2, 0, 5
18 in.
48.
0, 3, 0, 0, 5, 0, 01 ,, 0, 5, 0, 4, 0, 4, 05 ,, 2, 3, 5, 3, 0, 60, 0 , , 5, , 5, 1, 61, 1, 1, 5
30° 30°
θ F 12 in.
3, 5, 1 5, 0, 5 , 2, 5, 6 , 5, 5, 6
θ F 12 in.
48. 0, 3, 0, 3, 4, 04, , 0 0, ,, 4, 1, 01, 5, , 5, 5 3, ,, 0, 4, 04, 1, , 1, 5 1, ,, 1, 4, 54, 5, 5, 5
O
48. 0, 0, 0 , 0, 4, 0 , 3, 0, 0 , 1, 1, 5
30°
15 pulg 15 15 in.
A A
30°
49. Si If If u3, 4, v 0 , 0
y1, u 5, 5 v ,
3, 4,
0, 0, 4, 1,
0 ,
¿qué what 5 ,
1,
se
5,
can puede 4, 5,
5 ,
you 5concluir 4,
conclude 1, 5 ,
acerca about
4, 5,
de
5
u
O
y v?
Figura Figure para for 39 39 O Figura Figure para for 40 40
15 in. A
and
v?
15 in. 49. If uA
v 0 y u v 0, what can you conclude about u
50. 50. Identificar Identify the los dot 49.
productos products If u
vectoriales that v are 0 yequal. u v
que Explain 0,
son iguales. your what
Explicar reasoning.
can you conclude abou
el
40. 40. Figure Optimización Optimization for 39
Figure Una A force fuerza of of
for 39de 180 180 Figure pounds libras for actúa acts 40 sobre on the Figure el soporte bracket
and v?
for 40 razonamiento. (Assume u, u, v, v, and (Suponer and w
are v?
que nonzero u, v vectors.) y w son vectores distintos de
mostrado shown in in en the la figure.
figura.
50. Identify the dot products that are equal. Explain your reasoning.
40. Optimization A force of 180 pounds acts on the bracket
40. Optimization A force of 180 pounds acts on the
a) cero.) 50. Identify the dot products that are equal. Explain your reasoni
bracket
a) u v w
b) b) v w u
a) (a) Determinar Determine el the vector vector \
AB AB \
\
y and el the vector vector F Fque representing representa the
(Assume u, v, and w are nonzero vectors.)
shown the figure.
la
(Assume u, v, and w are nonzero vectors.)
shown in the figure.
fuerza. force. (F ( F( estará will be en in in términos terms of of de .) .)
c)
.)
a) c) u u v v w
w
d) b) d) uv w w u v
(a) Determine the vector AB \
and the vector F representing the
a) u v w
b) v w u
(a) Determine the vector AB \
and the vector representing the
b) (b) Calcular Find the la magnitude del of of the momento moment respecto about
a by A evaluando evaluating
F e) e) u w v
f f ))
w v u
force. ( F will be in terms of .) A
c) u v w
d) u w v
force. ( will be in terms of .)
c) u v w
d) u w v
\ F
AB AB \
\
F. F ..
g) g) u v w
h) h)
u v
(b) Find the magnitude of the moment about A by evaluating e) u w v
f ) w v u
(b) Find the magnitude of the moment about by evaluating e) u w v
f ) w v u
c) (c) Usar Use A
AB el \
the resultado F . of of del part inciso (b) to to b) determine para determinar the magnitude la magnitud of of the
AB \ g) u v w
h) w u v
del moment momento when
cuando 30 ..
F .
WRITING g) u ABOUT v w CONCEPTS
h) w u v
(c) Use the result of part (b) to determine the magnitude of the
(c) Use the result of part (b) to determine magnitude of the
d) (d) Usar Use el the resultado of of del part inciso (b) to to b) determine para determinar the angle
el when ángulo
the
51. Define the cross product of of vectors u
and
v. v.
moment when 30 .
WRITING ABOUT CONCEPTS
when 30 .
WRITING ABOUT CONCEPTS
cuando magnitude la magnitud of of the moment del momento is is maximum. es máxima. At that A angle, ese ángulo, what
Desarrollo 52. State the geometric de conceptos
properties of of the cross product.
(d) Use the result of part (b) determine the angle when the
51. Define the cross product of vectors u and v.
¿cuál is is the es relationship relación (d) entre between Use the los result the vectores vectors of part F y F
(b) to ¿Es determine \
lo Is que it the se angle when 51. Define the cross product of vectors u and v.
AB and \ ? AB \ ?
Is it what
magnitude of the moment is maximum. At that angle, what 51. 53. 52. Definir If State If the the magnitudes el geometric producto of of vectorial properties two vectors de of los are the vectores doubled, cross product. u y how v. will the
esperaba? you expected? ¿Por qué Why sí magnitude o or or por why qué not?
of no? the moment is maximum. At that angle, what 52. State the geometric properties of the cross product.
is the relationship between the vectors F and AB
magnitude of of the cross product of of the vectors change? Explain.
\
? Is it what 52.
is the relationship between the vectors F and AB \ ?
53. Dar
Is it
If
what
the las magnitudes propiedades of geométricas two vectors del are producto doubled, vectorial. how will the
e) (e) Usar Use you una expected? a herramienta graphing Why utility de or why graficación to to not? graph para the representar function for la función
la magnitud of del momento respecto a A para 0° £ . q 53. Si magnitude las magnitudes of the cross
the
53. If the magnitudes of two vectors are doubled, how will th
you expected? Why or why not?
£
magnitude
de product dos vectores of the vectors
of the cross
se duplican, change?
product of
¿cómo Explain.
magnitude of the moment about A for 0 180 . Find
the vectors
se
change? Explain
(e) Use a graphing utility graph the function for the
180°. the zero Hallar of of el the cero (e) function Use de la a función in in graphing the given el utility domain. dominio to Interpret dado. graph Inter-
the
modificará la magnitud del producto vectorial de los vectores?
Explicar.
function CAPSTONE
for the
magnitude of the moment about A for 0 180 . Find
pretar meaning el significado of of the zero magnitude del in in cero the context en of el the contexto of moment of the problem. del about problema. A for 0 180 . Find
the zero of the function in the given domain. Interpret the CAPSTONE
54. The vertices of of a triangle in in space are
x 1 , 1 , y 1 , 1 , zz 11 ,,
x 2 , 2 , y 2 , 2 , zz 22 ,,
the zero of the function in the given domain. Interpret the
CAPSTONE
En los ejercicios 41 a 44, calcular
meaning of the .
3 , 3 , z 3 .
to In Exercises meaning 41–44, of the find zero u in v the w context . of the problem.
and x 3 , y 3 , z 3 . Explain how to find a vector perpendicular
u
zero
.
in the context of the problem.
v w
Para
54. The
to to the discusión
vertices of a triangle in space are x
triangle.
54. The vertices of a triangle 1 , y 1 , z
in 1 , x
space 2 , y
are 2 , z
x 2 ,
1 y 1 , z 1 , x 2 , y 2 , z
and x 2
41. In Exercises u
ii
41–44, find u v 42. w . u 1, 1, 1, 1, 1
3 , y 3 , z 3 . Explain how to find a vector perpendicular
and x Explain how to find a vector perpendicula
41. In Exercises 41–44, 42. find u v w .
3 , y 3 , z 3 .
u i
u 1, 1, 1
54. Los to the vértices triangle. de un triángulo en el espacio son x j
2, 1, to the triangle.
1 , y 1 , z 1 ,
41. v
v
u j
j
i
42. v
41. u i v
u 2,
2,
1, 1,
1,
1, 0
0
1
x 2 , y 2 , z 2 , y x 3 , y 3 , z 3 . Explicar cómo encontrar un vector
42. u 1, 1, 1
w
k
w 0, 0, 0, 0, 1
True perpendicular or or False?
In al Exercises triángulo.
55–58, determine whether the
w
v
k
j
v j w
v
0,
2,
0,
1,
1
0
v 2, 1, 0 statement is is true or false. If If it it is is false, explain why or give an
43. u 2, 0, 2, 0, 43. u
w
2,
k2, 0, 1
44. u
0, 1 w k 44. u
w 2,
2,
0, 0,
0,
0, 0
0
1
True or False? In Exercises 55–58, determine whether the
w 0, 0, 1 ¿Verdadero example that o falso? shows True it it En is or is false.
los False? ejercicios In Exercises 55 a 58, determinar 55–58, determine si la whether
v 0, 0, 3, 3, 0
v 1, 1, 1, 1, 1
statement is true or false. If it is false, explain why or give an
43.
v
u
0,
2,
3,
0,
0
1
44.
43. 44.
declaración es verdadera statement o falsa. is true Si or es false. falsa, If explicar it is false, por explain qué o why or give
u 2, 0, 1 v
u
1,
2,
1,
0,
1
0
u 2, 0, 0 55. example It It is is that possible shows to to it find is false. the cross product of of two vectors in in a
w 0, 0, 0, 2, dar un ejemplo que example demuestre that shows que es it falsa. is false.
w
v
0,
0, 0,
0,
3, 0,
1
01
w
v 0, 3, 0 w
v
0,
1, 0,
2,
1, 2,
2
12
v 1, 1, 1 two-dimensional coordinate system.
55. It is possible to find the cross product of two vectors in a
Volume
w In 0, Exercises 0, 1
55. Es posible encontrar 55. It el is producto possible vectorial find the de dos cross vectores product en of un two vectors in
w45 and 0, 46, 0, 1use the w triple 0, 2, scalar 2 product w 0, to to 2, 2 56. If two-dimensional If u
and v
are vectors coordinate space system. that are nonzero and nonparallel,
Volumen find the volume En los of of ejercicios the parallelepiped 45 y 46, having usar el adjacent triple producto edges
u, u, sistema then
u de v coordenadas v two-dimensional u. u. bidimensional. coordinate system.
Volume In Exercises 45 and 46, use the triple scalar product to 56. If u and v are vectors in space that are nonzero and nonparallel,
escalar v, v, and
w. para encontrar Volume el volumen In Exercises del paralelepípedo 45 and 46, use que the triple tiene scalar 56. 57. product Si If If u y tov 0
son and vectores u56. vIf uen and u el vw,
espacio are then
vectors vque in w. son space distintos that are de nonzero cero y and nonparal
find the volume of the parallelepiped having adjacent edges u, then u v v u.
como aristas adyacentes find the u, volume v y w. of the parallelepiped having adjacent edges no paralelos, u, entonces then u v v u.
45. v,
and u w. ii jj
46.
u 1, 1, 3, 3, 1
58. If If 0, 0, u v u w,
and u v u w,
then
v w.
v, and w.
57. If u 0 and u v u w, then v w.
45. u 46.
57. Si y u 57. v If u w, 0 and entonces u vv uw.
w, then v w.
v ijj jk
uv 1, 0, 0, 3, 6, 6, 1
45. u i j
46. u 1, 3, 61
45. u i j
46. u 1, 3, 1 58. In 58. Exercises If u 0,
Si 0, 59–66, u v
58. prove u w,
If u w, the and
y 0, uproperty u u v
v uof of uthe w,
w,
and entonces cross then
u product.
v w.
v v uw.
w, then v w.
w j i i k
vw 0, 6, 4, 4, 6
v j k
v 0, 6, 60, 0, 4
w i k v j k w 4, 0, 4 v 0, 6, 6 En 59. In Exercises los u ejercicios v 59–66, w u
In Exercises a prove 66, v demostrar the property u w
z
59–66, la prove propiedad of the cross
the property del product. producto
w i zk
w zz
4, 0, 4
of the cross product.
w i k
w 4, 0, 4vectorial.
60.
c u v cu v u cv
z
z
59. u v w u v u w
z
z
59. u v w u v u w
22
z
66
v
z 59. 61. 60.
uc u v u v 0w cu u v v u u cv w
6 4 v
60. c u v cu v u cv
2
4
60. 62. 61.
cu v v w cuu vv uw
2
6 v
u u 0
cv
v
2
422
6
u
v
61. u u 0
w
4
4 61. 63. 62.
u u v is orthogonal w0
u to vto both w u
and
v. v.
v
y
62. u v w u v w
4 u
y
w v
6
y
8
v
4 2 6 2 62. 64. v w 0
if if and u only v if if wu
and v
are scalar multiples of of each
1w
1
y
u
u
2 w 4 w
8
63. u v is orthogonal to both u and v.
2
6
y u
63. u v is orthogonal to both u and v.
22
y
8 y
other.
1 u 2
x
4 6
y
w
8
63. 64. u y ves ortogonal 0 if and only tanto if a u
and como v are a v. scalar multiples of each
4
x
6
y
8
64. u v 0
if and only if u
and v are scalar multiples of ea
2x
1 u 2
w
65. Prove that u v u v if u and v are orthogonal.
2
1 u 2
w 64. uother.
v 0 si y sólo other. si u y v son múltiplos escalares uno del otro.
x
2
66. that
u w u w v u v w.
x
x
65. 65. Demostrar Prove that que u
65. uv Prove vu that u v if
uv
u and si v u vy are vu son orthogonal.
vortogonales.
if u and v are orthogonal.
67. 66. Demostrar Prove Prove Theorem that
uque
66. u11.9.
v Prove vw that wu u u wv wvu w u v w.
u vw. w v u v w.
67.
67.
Demostrar
Prove Theorem
el teorema
11.9.
11.9.
67. Prove Theorem 11.9.
30.