Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 11.3 El producto escalar de dos vectores 791
63. Programación Dados los vectores y mediante sus componentes,
escribir un programa para herramienta de graficación
que calcule las componentes de la proyección de
en
64. Programación Usar el programa escrito en el ejercicio 63
para encontrar la proyección de en si y
Para pensar
En los ejercicios 65 y 66, usar la figura para determinar
mentalmente la proyección de u en v (se dan las coordenadas
de los puntos finales de los vectores en la posición estándar).
Verificar los resultados analíticamente.
En los ejercicios 67 a 70, encontrar dos vectores en direcciones
opuestas que sean ortogonales al vector u. (Las respuestas no son
únicas.)
71. Fuerza de frenado Un camión de 48 000 libras está estacionado
sobre una pendiente de 10° (ver la figura). Si se supone
que la única fuerza a vencer es la de la gravedad, hallar a) la
fuerza requerida para evitar que el camión ruede cuesta abajo y
b) la fuerza perpendicular a la pendiente.
Figura para 71 Figura para 72
72. Cables que soportan una carga Calcular la magnitud de la
proyección del cable OA en el eje z positivo como se muestra en
la figura.
73. Trabajo Un objeto es jalado 10 pies por el suelo, usando una
fuerza de 85 libras. La dirección de la fuerza es 60° sobre la horizontal
(ver la figura). Calcular el trabajo realizado.
Figura para 73 Figura para 74
74. Trabajo Un coche de juguete se jala ejerciendo una fuerza de
25 libras sobre una manivela que forma un ángulo de 20° con la
horizontal (ver la figura). Calcular el trabajo realizado al jalar el
coche 50 pies.
75. Trabajo Un carro se remolca usando una fuerza de 1 600 newtons.
La cadena que se usa para jalar el carro forma un ángulo de
25° con la horizontal. Encontrar el trabajo que se realiza al
remolcar el carro 2 kilómetros.
76. Trabajo Se tira de un trineo ejerciendo una fuerza de 100 newtons
en una cuerda que hace un ángulo de 25° con la horizontal.
Encontrar el trabajo efectuado al jalar el trineo 40 metros.
¿Verdadero o falso?
En los ejercicios 77 y 78, determinar si la
declaración es verdadera o falsa. Si es falsa, explicar por qué o
dar un ejemplo que demuestre que es falsa.
77. Si y entonces
78. Si y son ortogonales a entonces es ortogonal a
79. Encontrar el ángulo entre la diagonal de un cubo y una de sus
aristas.
80. Encontrar el ángulo entre la diagonal de un cubo y la diagonal
de uno de sus lados.
En los ejercicios 81 a 84, a) encontrar todos los puntos de intersección
de las gráficas de las dos ecuaciones; b) encontrar los
vectores unitarios tangentes a cada curva en los puntos de intersección
y c) hallar los ángulos (0° £ q £ 90°) entre las curvas en
sus puntos de intersección.
81.
82.
83.
84.
85. Usar vectores para demostrar que las diagonales de un rombo
son perpendiculares.
86. Usar vectores para demostrar que un paralelogramo es un rectángulo
si y sólo si sus diagonales son iguales en longitud.
87. Ángulo de enlace Considerar un tetraedro regular con los vértices
(k, 0, k) y donde es un número
real positivo.
a) Dibujar la gráfica del tetraedro.
b) Hallar la longitud de cada arista.
c) Hallar el ángulo entre cada dos aristas.
d) Hallar el ángulo entre los segmentos de recta desde el centroide
a dos de los vértices. Éste es el ángulo
de enlace en una molécula como o cuya estructura
es un tetraedro.
88. Considerar los vectores y
donde a > b. Calcular el producto escalar de los vectores
y usar el resultado para demostrar la identidad
89. Demostrar que
90. Demostrar la desigualdad de Cauchy-Schwarz
91. Demostrar la desigualdad del triángulo
92. Demostrar el teorema 11.6.
u v ≤ u v.
u v 2 u 2 v 2 2u v.
63. Programming Given vectors and in component form,
write a program for a graphing utility in which the output is the
component form of the projection of
onto
64. Programming Use the program you wrote in Exercise 63 to
find the projection of onto for and
Think About It
In Exercises 65 and 66, use the figure to
determine mentally the projection of onto (The coordinates
of the terminal points of the vectors in standard position are
given.) Verify your results analytically.
65. 66.
In Exercises 67–70, find two vectors in opposite directions that
are orthogonal to the vector
(The answers are not unique.)
67. 68.
69. 70.
71. Braking Load A 48,000-pound truck is parked on a slope
(see figure). Assume the only force to overcome is that due to
gravity. Find (a) the force required to keep the truck from
rolling down the hill and (b) the force perpendicular to the hill.
Figure for 71 Figure for 72
72. Load-Supporting Cables Find the magnitude of the projection
of the load-supporting cable onto the positive axis as
shown in the figure.
73. Work An object is pulled 10 feet across a floor, using a force
of 85 pounds. The direction of the force is
above the
horizontal (see figure). Find the work done.
Figure for 73 Figure for 74
74. Work A toy wagon is pulled by exerting a force of 25 pounds
on a handle that makes a
angle with the horizontal (see
figure in left column). Find the work done in pulling the wagon
50 feet.
75. Work A car is towed using a force of 1600 newtons. The
chain used to pull the car makes a
angle with the horizontal.
Find the work done in towing the car 2 kilometers.
76. Work A sled is pulled by exerting a force of 100 newtons on
a rope that makes a
angle with the horizontal. Find the work
done in pulling the sled 40 meters.
True or False?
In Exercises 77 and 78, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
77. If and then
78. If and are orthogonal to then is orthogonal to
79. Find the angle between a cube’s diagonal and one of its edges.
80. Find the angle between the diagonal of a cube and the diagonal
of one of its sides.
In Exercises 81–84, (a) find all points of intersection of the
graphs of the two equations, (b) find the unit tangent vectors to
each curve at their points of intersection, and (c) find the angles
between the curves at their points of intersection.
81.
82.
83.
84.
85. Use vectors to prove that the diagonals of a rhombus are
perpendicular.
86. Use vectors to prove that a parallelogram is a rectangle if and
only if its diagonals are equal in length.
87. Bond Angle Consider a regular tetrahedron with vertices
and where is a positive
real number.
(a) Sketch the graph of the tetrahedron.
(b) Find the length of each edge.
(c) Find the angle between any two edges.
(d) Find the angle between the line segments from the centroid
to two vertices. This is the bond angle for a
molecule such as or where the structure of the
molecule is a tetrahedron.
88. Consider the vectors and
where
Find the dot product of the
vectors and use the result to prove the identity
89. Prove that
90. Prove the Cauchy-Schwarz Inequality
91. Prove the triangle inequality
92. Prove Theorem 11.6.
u v u v .
u v u v .
u v 2 u 2 v 2 2u v.
cos cos cos sen sen
.
> .
v cos , sen , 0 ,
u cos , sen , 0
PbCl 4 ,
CH 4
k 2, k 2, k 2
k
0, k, k ,
k, 0, k ,
k, k, 0 ,
0, 0, 0 ,
y x 3 1
y 1 2 x,
y x 2 1
y 1 x 2 ,
y x 13
y x 3 ,
y x 13
y x 2 ,
0 90
w.
u
v
w,
v
u
v w.
u 0,
u v u w
25
25
20
20°
60°
10 ft
85 lb
Not drawn to scale
60
z-
OA
(−5, −5, 20)
(10, 5, 20)
y
x
z
1000 kg
A
B
C
O
(5, −5, 20)
Weight = 48,000 lb
10°
10
u 4, 3, 6
u 3, 1, 2
u 9i 4j
u
1
4 i
3
2 j
u.
(4, 6)
(3, −2)
u
v
y
−2 4 6
−2
2
4
6
−4 2 4 6
2
4
6
(−2, −3)
(4, 6)
u
v
y
v.
u
v 1, 3, 4 .
u 5, 6, 2
v
u
v.
u
v
u
11.3 The Dot Product of Two Vectors 791
63. Programming Given vectors and in component form,
write a program for a graphing utility in which the output is the
component form of the projection of
onto
64. Programming Use the program you wrote in Exercise 63 to
find the projection of onto for and
Think About It
In Exercises 65 and 66, use the figure to
determine mentally the projection of onto (The coordinates
of the terminal points of the vectors in standard position are
given.) Verify your results analytically.
65. 66.
In Exercises 67–70, find two vectors in opposite directions that
are orthogonal to the vector
(The answers are not unique.)
67. 68.
69. 70.
71. Braking Load A 48,000-pound truck is parked on a slope
(see figure). Assume the only force to overcome is that due to
gravity. Find (a) the force required to keep the truck from
rolling down the hill and (b) the force perpendicular to the hill.
Figure for 71 Figure for 72
72. Load-Supporting Cables Find the magnitude of the projection
of the load-supporting cable onto the positive axis as
shown in the figure.
73. Work An object is pulled 10 feet across a floor, using a force
of 85 pounds. The direction of the force is
above the
horizontal (see figure). Find the work done.
Figure for 73 Figure for 74
74. Work A toy wagon is pulled by exerting a force of 25 pounds
on a handle that makes a
angle with the horizontal (see
figure in left column). Find the work done in pulling the wagon
50 feet.
75. Work A car is towed using a force of 1600 newtons. The
chain used to pull the car makes a
angle with the horizontal.
Find the work done in towing the car 2 kilometers.
76. Work A sled is pulled by exerting a force of 100 newtons on
a rope that makes a
angle with the horizontal. Find the work
done in pulling the sled 40 meters.
True or False?
In Exercises 77 and 78, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
77. If and then
78. If and are orthogonal to then is orthogonal to
79. Find the angle between a cube’s diagonal and one of its edges.
80. Find the angle between the diagonal of a cube and the diagonal
of one of its sides.
In Exercises 81–84, (a) find all points of intersection of the
graphs of the two equations, (b) find the unit tangent vectors to
each curve at their points of intersection, and (c) find the angles
between the curves at their points of intersection.
81.
82.
83.
84.
85. Use vectors to prove that the diagonals of a rhombus are
perpendicular.
86. Use vectors to prove that a parallelogram is a rectangle if and
only if its diagonals are equal in length.
87. Bond Angle Consider a regular tetrahedron with vertices
and where is a positive
real number.
(a) Sketch the graph of the tetrahedron.
(b) Find the length of each edge.
(c) Find the angle between any two edges.
(d) Find the angle between the line segments from the centroid
to two vertices. This is the bond angle for a
molecule such as or where the structure of the
molecule is a tetrahedron.
88. Consider the vectors and
where
Find the dot product of the
vectors and use the result to prove the identity
89. Prove that
90. Prove the Cauchy-Schwarz Inequality
91. Prove the triangle inequality
92. Prove Theorem 11.6.
u v u v .
u v u v .
u v 2 u 2 v 2 2u v.
cos cos cos sen sen
.
> .
v cos , sen , 0 ,
u cos , sen , 0
PbCl 4 ,
CH 4
k 2, k 2, k 2
k
0, k, k ,
k, 0, k ,
k, k, 0 ,
0, 0, 0 ,
y x 3 1
y 1 2 x,
y x 2 1
y 1 x 2 ,
y x 13
y x 3 ,
y x 13
y x 2 ,
0 90
w.
u
v
w,
v
u
v w.
u 0,
u v u w
25
25
20
20°
60°
10 ft
85 lb
Not drawn to scale
60
z-
OA
(−5, −5, 20)
(10, 5, 20)
y
x
z
1000 kg
A
B
C
O
(5, −5, 20)
Weight = 48,000 lb
10°
10
u 4, 3, 6
u 3, 1, 2
u 9i 4j
u
1
4 i
3
2 j
u.
(4, 6)
(3, −2)
u
v
y
−2 4 6
−2
2
4
6
−4 2 4 6
2
4
6
(−2, −3)
(4, 6)
u
v
y
v.
u
v 1, 3, 4 .
u 5, 6, 2
v
u
v.
u
v
u
11.3 The Dot Product of Two Vectors 791
63. Programming Given vectors and in component form,
write a program for a graphing utility in which the output is the
component form of the projection of
onto
64. Programming Use the program you wrote in Exercise 63 to
find the projection of onto for and
Think About It
In Exercises 65 and 66, use the figure to
determine mentally the projection of onto (The coordinates
of the terminal points of the vectors in standard position are
given.) Verify your results analytically.
65. 66.
In Exercises 67–70, find two vectors in opposite directions that
are orthogonal to the vector
(The answers are not unique.)
67. 68.
69. 70.
71. Braking Load A 48,000-pound truck is parked on a slope
(see figure). Assume the only force to overcome is that due to
gravity. Find (a) the force required to keep the truck from
rolling down the hill and (b) the force perpendicular to the hill.
Figure for 71 Figure for 72
72. Load-Supporting Cables Find the magnitude of the projection
of the load-supporting cable onto the positive axis as
shown in the figure.
73. Work An object is pulled 10 feet across a floor, using a force
of 85 pounds. The direction of the force is
above the
horizontal (see figure). Find the work done.
Figure for 73 Figure for 74
74. Work A toy wagon is pul
on a handle that makes a
figure in left column). Find
50 feet.
75. Work A car is towed us
chain used to pull the car m
Find the work done in towi
76. Work A sled is pulled by
a rope that makes a
ang
done in pulling the sled 40
True or False?
In Exercises
statement is true or false. If
example that shows it is false.
77. If and
78. If and are orthogonal to
79. Find the angle between a c
80. Find the angle between the
of one of its sides.
In Exercises 81–84, (a) find
graphs of the two equations, (
each curve at their points of in
between the cu
81.
82.
83.
84.
85. Use vectors to prove tha
perpendicular.
86. Use vectors to prove that a
only if its diagonals are eq
87. Bond Angle Consider a
real number.
(a) Sketch the graph of the
(b) Find the length of each
(c) Find the angle between
(d) Find the angle between
to two
molecule such as
molecule is a tetrahedr
88. Consider the vecto
wher
vectors and use the result t
89. Prove that
90. Prove the Cauchy-Schwar
91. Prove the triangle inequalit
92. Prove Theorem 11.6.
u v 2 u
cos cos cos
v cos , sen , 0 ,
CH 4
k 2, k 2, k 2
k, 0, k ,
k, k, 0 ,
0, 0, 0 ,
y x 3
y 1 2 x,
y x 2 1
y 1 x 2 ,
y x 1 3
y x 3 ,
y x 13
y x 2 ,
0 90
v
u
u
u v u w
25
20°
60°
10 ft
85 lb
Not drawn to scale
60
z-
OA
(−5, −5, 20)
(10, 5, 20)
y
x
z
1000 kg
A
B
C
O
(5, −5, 20)
Weight = 48,000 lb
10°
10
u 4, 3, 6
u 3, 1, 2
u 9i 4j
u
1
4 i
3
2 j
u.
(4, 6)
(3, −2)
u
v
y
−2 4 6
−2
2
4
6
−4 2 4 6
2
4
6
(−2, −3)
(4, 6)
u
v
y
v.
u
v 1, 3, 4 .
u 5, 6, 2
v
u
v.
u
v
u
11.3 The Dot P
63. Programming Given vectors and in component form,
write a program for a graphing utility in which the output is the
component form of the projection of
onto
64. Programming Use the program you wrote in Exercise 63 to
find the projection of onto for and
Think About It
In Exercises 65 and 66, use the figure to
determine mentally the projection of onto (The coordinates
of the terminal points of the vectors in standard position are
given.) Verify your results analytically.
65. 66.
In Exercises 67–70, find two vectors in opposite directions that
are orthogonal to the vector
(The answers are not unique.)
67. 68.
69. 70.
71. Braking Load A 48,000-pound truck is parked on a slope
(see figure). Assume the only force to overcome is that due to
gravity. Find (a) the force required to keep the truck from
rolling down the hill and (b) the force perpendicular to the hill.
Figure for 71 Figure for 72
72. Load-Supporting Cables Find the magnitude of the projection
of the load-supporting cable onto the positive axis as
shown in the figure.
73. Work An object is pulled 10 feet across a floor, using a force
of 85 pounds. The direction of the force is
above the
horizontal (see figure). Find the work done.
Figure for 73 Figure for 74
74. Work A toy wagon is pulled by exerting a force of 25 pounds
on a handle that makes a
angle with the horizontal (see
figure in left column). Find the work done in pulling the wagon
50 feet.
75. Work A car is towed using a force of 1600 newtons. The
chain used to pull the car makes a
angle with the horizontal.
Find the work done in towing the car 2 kilometers.
76. Work A sled is pulled by exerting a force of 100 newtons on
a rope that makes a
angle with the horizontal. Find the work
done in pulling the sled 40 meters.
True or False?
In Exercises 77 and 78, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
77. If and then
78. If and are orthogonal to then is orthogonal to
79. Find the angle between a cube’s diagonal and one of its edges.
80. Find the angle between the diagonal of a cube and the diagonal
of one of its sides.
In Exercises 81–84, (a) find all points of intersection of the
graphs of the two equations, (b) find the unit tangent vectors to
each curve at their points of intersection, and (c) find the angles
between the curves at their points of intersection.
81.
82.
83.
84.
85. Use vectors to prove that the diagonals of a rhombus are
perpendicular.
86. Use vectors to prove that a parallelogram is a rectangle if and
only if its diagonals are equal in length.
87. Bond Angle Consider a regular tetrahedron with vertices
and where is a positive
real number.
(a) Sketch the graph of the tetrahedron.
(b) Find the length of each edge.
(c) Find the angle between any two edges.
(d) Find the angle between the line segments from the centroid
to two vertices. This is the bond angle for a
molecule such as or where the structure of the
molecule is a tetrahedron.
88. Consider the vectors and
where
Find the dot product of the
vectors and use the result to prove the identity
89. Prove that
90. Prove the Cauchy-Schwarz Inequality
91. Prove the triangle inequality
92. Prove Theorem 11.6.
u v u v .
u v u v .
u v 2 u 2 v 2 2u v.
cos cos cos sen sen
.
> .
v cos , sen , 0 ,
u cos , sen , 0
PbCl 4 ,
CH 4
k 2, k 2, k 2
k
0, k, k ,
k, 0, k ,
k, k, 0 ,
0, 0, 0 ,
y x 3 1
y 1 2 x,
y x 2 1
y 1 x 2 ,
y x 13
y x 3 ,
y x 13
y x 2 ,
0 90
w.
u
v
w,
v
u
v w.
u 0,
u v u w
25
25
20
20°
60°
10 ft
85 lb
Not drawn to scale
60
z-
OA
(−5, −5, 20)
(10, 5, 20)
y
x
z
1000 kg
A
B
C
O
(5, −5, 20)
Weight = 48,000 lb
10°
10
u 4, 3, 6
u 3, 1, 2
u 9i 4j
u
1
4 i
3
2 j
u.
(4, 6)
(3, −2)
u
v
y
−2 4 6
−2
2
4
6
−4 2 4 6
2
4
6
(−2, −3)
(4, 6)
u
v
y
v.
u
v 1, 3, 4 .
u 5, 6, 2
v
u
v.
u
v
u
11.3 The Dot Product of Two Vectors 791
PbCl 4 ,
CH 4
k2, k2, k2
k
0, k, k,
k, k, 0,
0, 0, 0,
y x 3 1
y 1 2 x,
y x 2 1
y 1 x 2 ,
y x 13
y x 3 ,
y x 13
y x 2 ,
w.
u v
w,
v
u
v w.
u 0,
u v u w
20°
60°
10 pies
85 libras
No está dibujado a escala
(−5, −5, 20)
(10, 5, 20)
y
x
z
1 000 kg
A
B
C
O
(5, −5, 20)
Peso = 48 000 libras
10°
v 1, 3, 4.
u 5, 6, 2
v
u
v.
u
v
u
63. Programming Given vectors and in component form,
write a program for a graphing utility in which the output is the
component form of the projection of
onto
64. Programming Use the program you wrote in Exercise 63 to
find the projection of onto for and
Think About It
In Exercises 65 and 66, use the figure to
determine mentally the projection of onto (The coordinates
of the terminal points of the vectors in standard position are
given.) Verify your results analytically.
65. 66.
In Exercises 67–70, find two vectors in opposite directions that
are orthogonal to the vector
(The answers are not unique.)
67. 68.
69. 70.
71. Braking Load A 48,000-pound truck is parked on a slope
(see figure). Assume the only force to overcome is that due to
gravity. Find (a) the force required to keep the truck from
rolling down the hill and (b) the force perpendicular to the hill.
Figure for 71 Figure for 72
72. Load-Supporting Cables Find the magnitude of the projection
of the load-supporting cable onto the positive axis as
shown in the figure.
73. Work An object is pulled 10 feet across a floor, using a force
of 85 pounds. The direction of the force is
above the
horizontal (see figure). Find the work done.
Figure for 73 Figure for 74
74. Work A toy wagon is pulled by exerting a force of 25 pounds
on a handle that makes a
angle with the horizontal (see
figure in left column). Find the work done in pulling the wagon
50 feet.
75. Work A car is towed using a force of 1600 newtons. The
chain used to pull the car makes a
angle with the horizontal.
Find the work done in towing the car 2 kilometers.
76. Work A sled is pulled by exerting a force of 100 newtons on
a rope that makes a
angle with the horizontal. Find the work
done in pulling the sled 40 meters.
True or False?
In Exercises 77 and 78, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
77. If and then
78. If and are orthogonal to then is orthogonal to
79. Find the angle between a cube’s diagonal and one of its edges.
80. Find the angle between the diagonal of a cube and the diagonal
of one of its sides.
In Exercises 81–84, (a) find all points of intersection of the
graphs of the two equations, (b) find the unit tangent vectors to
each curve at their points of intersection, and (c) find the angles
between the curves at their points of intersection.
81.
82.
83.
84.
85. Use vectors to prove that the diagonals of a rhombus are
perpendicular.
86. Use vectors to prove that a parallelogram is a rectangle if and
only if its diagonals are equal in length.
87. Bond Angle Consider a regular tetrahedron with vertices
and where is a positive
real number.
(a) Sketch the graph of the tetrahedron.
(b) Find the length of each edge.
(c) Find the angle between any two edges.
(d) Find the angle between the line segments from the centroid
to two vertices. This is the bond angle for a
molecule such as or where the structure of the
molecule is a tetrahedron.
88. Consider the vectors and
where
Find the dot product of the
vectors and use the result to prove the identity
89. Prove that
90. Prove the Cauchy-Schwarz Inequality
91. Prove the triangle inequality
92. Prove Theorem 11.6.
u v u v .
u v u v .
u v 2 u 2 v 2 2u v.
cos cos cos sen sen
.
> .
v cos , sen , 0 ,
u cos , sen , 0
PbCl 4 ,
CH 4
k 2, k 2, k 2
k
0, k, k ,
k, 0, k ,
k, k, 0 ,
0, 0, 0 ,
y x 3 1
y 1 2 x,
y x 2 1
y 1 x 2 ,
y x 13
y x 3 ,
y x 13
y x 2 ,
0 90
w.
u
v
w,
v
u
v w.
u 0,
u v u w
25
25
20
20°
60°
10 ft
85 lb
Not drawn to scale
60
z-
OA
(−5, −5, 20)
(10, 5, 20)
y
x
z
1000 kg
A
B
C
O
(5, −5, 20)
Weight = 48,000 lb
10°
10
u 4, 3, 6
u 3, 1, 2
u 9i 4j
u
1
4 i
3
2 j
u.
(4, 6)
(3, −2)
u
v
x
y
−2 4 6
−2
2
4
6
−4 2 4 6
2
4
6
(−2, −3)
(4, 6)
u
v
x
y
v.
u
v 1, 3, 4 .
u 5, 6, 2
v
u
v.
u
v
u
11.3 The Dot Product of Two Vectors 791
63. Programming Given vectors and in component form,
write a program for a graphing utility in which the output is the
component form of the projection of
onto
64. Programming Use the program you wrote in Exercise 63 to
find the projection of onto for and
Think About It
In Exercises 65 and 66, use the figure to
determine mentally the projection of onto (The coordinates
of the terminal points of the vectors in standard position are
given.) Verify your results analytically.
65. 66.
In Exercises 67–70, find two vectors in opposite directions that
are orthogonal to the vector
(The answers are not unique.)
67. 68.
69. 70.
71. Braking Load A 48,000-pound truck is parked on a slope
(see figure). Assume the only force to overcome is that due to
gravity. Find (a) the force required to keep the truck from
rolling down the hill and (b) the force perpendicular to the hill.
Figure for 71 Figure for 72
72. Load-Supporting Cables Find the magnitude of the projection
of the load-supporting cable onto the positive axis as
shown in the figure.
73. Work An object is pulled 10 feet across a floor, using a force
of 85 pounds. The direction of the force is
above the
horizontal (see figure). Find the work done.
Figure for 73 Figure for 74
74. Work A toy wagon is pulled by exerting a force of 25 pounds
on a handle that makes a
angle with the horizontal (see
figure in left column). Find the work done in pulling the wagon
50 feet.
75. Work A car is towed using a force of 1600 newtons. The
chain used to pull the car makes a
angle with the horizontal.
Find the work done in towing the car 2 kilometers.
76. Work A sled is pulled by exerting a force of 100 newtons on
a rope that makes a
angle with the horizontal. Find the work
done in pulling the sled 40 meters.
True or False?
In Exercises 77 and 78, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
77. If and then
78. If and are orthogonal to then is orthogonal to
79. Find the angle between a cube’s diagonal and one of its edges.
80. Find the angle between the diagonal of a cube and the diagonal
of one of its sides.
In Exercises 81–84, (a) find all points of intersection of the
graphs of the two equations, (b) find the unit tangent vectors to
each curve at their points of intersection, and (c) find the angles
between the curves at their points of intersection.
81.
82.
83.
84.
85. Use vectors to prove that the diagonals of a rhombus are
perpendicular.
86. Use vectors to prove that a parallelogram is a rectangle if and
only if its diagonals are equal in length.
87. Bond Angle Consider a regular tetrahedron with vertices
and where is a positive
real number.
(a) Sketch the graph of the tetrahedron.
(b) Find the length of each edge.
(c) Find the angle between any two edges.
(d) Find the angle between the line segments from the centroid
to two vertices. This is the bond angle for a
molecule such as or where the structure of the
molecule is a tetrahedron.
88. Consider the vectors and
where
Find the dot product of the
vectors and use the result to prove the identity
89. Prove that
90. Prove the Cauchy-Schwarz Inequality
91. Prove the triangle inequality
92. Prove Theorem 11.6.
u v u v .
u v u v .
u v 2 u 2 v 2 2u v.
cos cos cos sen sen
.
> .
v cos , sen , 0 ,
u cos , sen , 0
PbCl 4 ,
CH 4
k 2, k 2, k 2
k
0, k, k ,
k, 0, k ,
k, k, 0 ,
0, 0, 0 ,
y x 3 1
y 1 2 x,
y x 2 1
y 1 x 2 ,
y x 1 3
y x 3 ,
y x 1 3
y x 2 ,
0 90
w.
u
v
w,
v
u
v w.
u 0,
u v u w
25
25
20
20°
60°
10 ft
85 lb
Not drawn to scale
60
z-
OA
(−5, −5, 20)
(10, 5, 20)
y
x
z
1000 kg
A
B
C
O
(5, −5, 20)
Weight = 48,000 lb
10°
10
u 4, 3, 6
u 3, 1, 2
u 9i 4j
u
1
4 i 3
2 j u.
(4, 6)
(3, −2)
u
v
y
−2 4 6
−2
2
4
6
−4 2 4 6
2
4
6
(−2, −3)
(4, 6)
u
v
y
v.
u
v 1, 3, 4 .
u 5, 6, 2
v
u
v.
u
v
u
11.3 The Dot Product of Two Vectors 791
n vectors and in component form,
graphing utility in which the output is the
he projection of
onto
the program you wrote in Exercise 63 to
of onto for and
ercises 65 and 66, use the figure to
projection of onto (The coordinates
f the vectors in standard position are
ts analytically.
66.
two vectors in opposite directions that
ctor
(The answers are not unique.)
68.
70.
,000-pound truck is parked on a
slope
the only force to overcome is that due to
force required to keep the truck from
and (b) the force perpendicular to the hill.
Figure for 72
bles
Find the magnitude of the projection
ng cable onto the positive axis as
pulled 10 feet across a floor, using a force
direction of the force is
above the
). Find the work done.
Figure for 74
74. Work A toy wagon is pulled by exerting a force of 25 pounds
on a handle that makes a
angle with the horizontal (see
figure in left column). Find the work done in pulling the wagon
50 feet.
75. Work A car is towed using a force of 1600 newtons. The
chain used to pull the car makes a
angle with the horizontal.
Find the work done in towing the car 2 kilometers.
76. Work A sled is pulled by exerting a force of 100 newtons on
a rope that makes a
angle with the horizontal. Find the work
done in pulling the sled 40 meters.
True or False?
In Exercises 77 and 78, determine whether the
statement is true or false. If it is false, explain why or give an
example that shows it is false.
77. If and then
78. If and are orthogonal to then is orthogonal to
79. Find the angle between a cube’s diagonal and one of its edges.
80. Find the angle between the diagonal of a cube and the diagonal
of one of its sides.
In Exercises 81–84, (a) find all points of intersection of the
graphs of the two equations, (b) find the unit tangent vectors to
each curve at their points of intersection, and (c) find the angles
between the curves at their points of intersection.
81.
82.
83.
84.
85. Use vectors to prove that the diagonals of a rhombus are
perpendicular.
86. Use vectors to prove that a parallelogram is a rectangle if and
only if its diagonals are equal in length.
87. Bond Angle Consider a regular tetrahedron with vertices
and where is a positive
real number.
(a) Sketch the graph of the tetrahedron.
(b) Find the length of each edge.
(c) Find the angle between any two edges.
(d) Find the angle between the line segments from the centroid
to two vertices. This is the bond angle for a
molecule such as or where the structure of the
molecule is a tetrahedron.
88. Consider the vectors and
where
Find the dot product of the
vectors and use the result to prove the identity
89. Prove that
90. Prove the Cauchy-Schwarz Inequality
91. Prove the triangle inequality
92. Prove Theorem 11.6.
u v u v .
u v u v .
u v 2 u 2 v 2 2u v.
cos cos cos sen sen
.
> .
v cos , sen , 0 ,
u cos , sen , 0
PbCl 4 ,
CH 4
k 2, k 2, k 2
k
0, k, k ,
k, 0, k ,
k, k, 0 ,
0, 0, 0 ,
y x 3 1
y 1 2 x,
y x 2 1
y 1 x 2 ,
y x 13
y x 3 ,
y x 13
y x 2 ,
0 90
w.
u
v
w,
v
u
v w.
u 0,
u v u w
25
25
20
20°
scale
60
z-
OA
(−5, −5, 20)
(10, 5, 20)
y
x
z
1000 kg
A
B
C
O
(5, −5, 20)
0 lb
10
u 4, 3, 6
u 9i 4j
u.
(4, 6)
(3, −2)
u
v
y
−2 4 6
−2
2
4
6
v.
u
u 5, 6, 2
v
u
v.
u
v
u
11.3 The Dot Product of Two Vectors 791