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```angle this vector makes with the horizontal.
The vector has a length of approximately 5.2
centimeters and is at an approximate angle of 188°
with the horizontal.
Mid-Chapter Quiz: Lessons 8-1 through 8-3
Find the resultant of each pair of vectors using
either the triangle or parallelogram method.
State the magnitude of the resultant in
centimeters and its direction relative to the
horizontal.
3.
SOLUTION:
Translate s so that its tail touches the tip of p. Then
draw the resultant vector p + s as shown. Draw the
horizontal.
1.
SOLUTION:
Drawing may not be to scale.
Translate j so that its tail touches the tip of h. Then
draw the resultant vector h + j as shown. Draw the
horizontal.
Measure the length of p + s and then measure the
angle this vector makes with the horizontal.
The vector has a length of approximately 1.2
centimeters and is at an approximate angle of 330°
with the horizontal.
Drawing may not be to scale.
Measure the length of h + j and then measure the
angle this vector makes with the horizontal.
The vector has a length of approximately 1.2
centimeters and is at an approximate angle of 323°
with the horizontal.
2.
4.
SOLUTION:
Translate b so that its tail touches the tip of a. Then
draw the resultant vector a + b as shown. Draw the
horizontal.
Drawing may not be to scale.
SOLUTION:
Translate g so that its tail touches the tip of f. Then
draw the resultant vector f + g as shown. Draw the
horizontal.
Drawing may not be to scale.
Measure the length of f + g and then measure the
angle this vector makes with the horizontal.
The vector has a length of approximately 5.2
centimeters and is at an approximate angle of 188°
with the horizontal.
3.
Measure the length of a + b and then measure the
angle this vector makes with the horizontal.
The vector has a length of approximately 1.8
centimeters and is at an approximate angle of 102°
with the horizontal.
5. SLEDDING Alvin pulls a sled through the snow
with a force of 50 newtons at an angle of 35° with
the horizontal. Find the magnitude of the horizontal
and vertical components of the force.
SOLUTION:
Draw a vector to represent Alvin pulling the sled.
The vector can be resolved into a horizontal
component x and a vertical component y as shown.
Page 1
angle this vector makes with the horizontal.
The vector has a length of approximately 1.8
centimeters and is at an approximate angle of 102°
with the horizontal.
Mid-Chapter
Quiz: Lessons 8-1 through 8-3
5. SLEDDING Alvin pulls a sled through the snow
with a force of 50 newtons at an angle of 35° with
the horizontal. Find the magnitude of the horizontal
and vertical components of the force.
The magnitude of the horizontal component is about
41.0 newtons and the magnitude of the vertical
6. Draw a vector diagram of
c – 3d.
SOLUTION:
Draw a vector to represent Alvin pulling the sled.
The vector can be resolved into a horizontal
component x and a vertical component y as shown.
SOLUTION:
Rewrite the expression as the addition of two
vectors:
c − 3d =
vector
The horizontal and vertical components of the vector
form a right triangle. Use the sine or cosine ratios to
find the magnitude of each component.
c + (−3d). To represent
c, draw a
the length of c in the same direction as c.
To represent −3d, draw a vector 3 times as long as d
in the opposite direction from d.
Then use the triangle method to draw the resultant
vector.
The magnitude of the horizontal component is about
41.0 newtons and the magnitude of the vertical
6. Draw a vector diagram of
c – 3d.
Drawings may not be to scale.
Let
be the vector with the given initial and
terminal points. Write
as a linear
combination of the vectors i and j.
7. B(3, −1), C(4, −7)
SOLUTION:
First, find the component form of
.
SOLUTION:
Rewrite the expression as the addition of two
vectors:
c − 3d =
vector
c + (−3d). To represent
c, draw a
the length of c in the same direction as c.
To represent −3d, draw a vector 3 times as long as d
in the opposite direction from d.
Then rewrite the vector as a linear combination of
the standard unit vectors.
8. B(10, −6), C(−8, 2)
SOLUTION:
First, find the component form of
Page 2
.
the standard unit vectors.
Mid-Chapter Quiz: Lessons 8-1 through 8-3
8. B(10, −6), C(−8, 2)
11. MULTIPLE CHOICE Which of the following is
SOLUTION:
First, find the component form of
.
the component form of
with initial point A(–5, 3)
and terminal point B(2, −1)?
A
B
C
D
SOLUTION:
Then rewrite the vector as a linear combination of
the standard unit vectors.
9. B(1, 12), C(−2, −9)
SOLUTION:
First, find the component form of
Find the component form.
12. BASKETBALL With time running out in a game,
.
Rachel runs towards the basket at a speed of 2.5
meters per second and from half-court, launches a
shot at a speed of 8 meters per second at an angle of
36° to the horizontal.
Then rewrite the vector as a linear combination of
the standard unit vectors.
10. B(4, −10), C(4, −10)
SOLUTION:
First, find the component form of
.
a. Write the component form of the vectors
representing Rachel’s velocity and the path of the
ball.
b. What is the resultant speed and direction of the
shot?
SOLUTION:
a. Since Rachel is moving straight forward, the
component form of her velocity v 1 is
. Use
Then rewrite the vector as a linear combination of
the standard unit vectors.
11. MULTIPLE CHOICE Which of the following is
the component form of
with initial point A(–5, 3)
and terminal point B(2, −1)?
A
eSolutions
C
D
the magnitude and direction of the ball’s velocity v 2
to write this vector in component form.
The component form of the vector representing
Rachel’s velocity is
and the component
form of the vector representing the path of the ball is
Page 3
b. Add the algebraic vectors representing v 1 and v 2
The component form of the vector representing
Mid-Chapter
Quiz:
8-1
through 8-3
Rachel’s velocity
is Lessons
and the
component
form of the vector representing the path of the ball is
14. Q(1, −5), R(−7, 8)
SOLUTION:
b. Add the algebraic vectors representing v 1 and v 2
First, find the component form.
to find the resultant velocity, vector r.
Find the magnitude of the resultant.
Next, find the magnitude. Substitute x2 − x1 = −8 and
y2 − y 1 = 13 into the formula for the magnitude of a
vector in the coordinate plane.
The speed of the ball is about 10.2 meters per
second.
Find the resultant direction angle θ.
15. X(−3, −5), Y(2, 5)
SOLUTION:
First, find the component form.
The speed of the ball is about 10.2 meters per
second at an angle of about 27.6° with the horizontal.
Find the component form and magnitude of the
vector with each initial and terminal point.
13. A(−4, 2), B(3, 6)
SOLUTION:
First, find the component form.
Next, find the magnitude. Substitute x2 − x1 = 5 and
y2 − y 1 = 10 into the formula for the magnitude of a
vector in the coordinate plane.
Next, find the magnitude. Substitute x2 − x1 = 7 and
y2 − y 1 = 4 into the formula for the magnitude of a
vector in the coordinate plane.
16. P(9, −2), S(2, −5)
SOLUTION:
First, find the component form.
14. Q(1,Manual
8) by Cognero
−5), R(−7,
eSolutions
- Powered
SOLUTION:
First, find the component form.
Page 4
Next, find the magnitude. Substitute x2 − x1 = −7 and
y2 − y 1 = −3 into the formula for the magnitude of a
Mid-Chapter Quiz: Lessons 8-1 through 8-3
16. P(9, −2), S(2, −5)
SOLUTION:
18. u =
,v=
SOLUTION:
First, find the component form.
Next, find the magnitude. Substitute x2 − x1 = −7 and
y2 − y 1 = −3 into the formula for the magnitude of a
vector in the coordinate plane.
Find the angle θ between u and v to the
nearest tenth of a degree.
17. u =
,v=
19. u =
,v=
SOLUTION:
SOLUTION:
20. u =
,v=
SOLUTION:
18. u =
,v=
SOLUTION:
Page 5
Mid-Chapter Quiz: Lessons 8-1 through 8-3
20. u =
Find the dot product of u and v. Then
determine if u and v are orthogonal.
,v=
SOLUTION:
22.
SOLUTION:
Since
, u and v are not orthogonal.
23.
SOLUTION:
Since
, u and v are not orthogonal.
24.
SOLUTION:
21. MULTIPLE CHOICE If u =
and w =
F −18
G −2
H 15
J 38
,v=
, find (u ⋅ v ) + (w ⋅ v ).
,
Since
, u and v are not orthogonal.
25.
SOLUTION:
SOLUTION:
Since
, u and v are orthogonal.
26. WAGON Henry uses a wagon to carry
newspapers for his paper route. He is pulling the
wagon with a force of 25 newtons at an angle of 30°
with the horizontal.
Find the dot product of u and v. Then
determine if u and v are orthogonal.
22.
SOLUTION:
eSolutions
Manual - Powered
by Cognero
v are not
Since
, u and
23.
orthogonal.
a. How much work in joules is Henry doing when
he pulls the wagon 150 meters?
b. If the handle makes an angle of 40° with the
horizontal and he pulls the wagon the same distance
with the same force, is Henry doing more or less
Page 6
SOLUTION:
Mid-Chapter Quiz: Lessons 8-1 through 8-3
, u and v are orthogonal.
Since
26. WAGON Henry uses a wagon to carry
newspapers for his paper route. He is pulling the
wagon with a force of 25 newtons at an angle of 30°
with the horizontal.
If the handle makes an angle of 40° with the
horizontal, Henry is doing about 2872.7 joules of
work. Therefore, Henry is doing less work.
Find the projection of u onto v. Then write u as
the sum of two orthogonal vectors, one of which
is the projection of u onto v.
27. u =
,v=
SOLUTION:
Find the projection of u onto v .
a. How much work in joules is Henry doing when
he pulls the wagon 150 meters?
b. If the handle makes an angle of 40° with the
horizontal and he pulls the wagon the same distance
with the same force, is Henry doing more or less
SOLUTION:
a. Use the projection formula for work. The
magnitude of the projection of F onto
is
The magnitude of the directed
distance
is 150.
To write u as the sum of two orthogonal vectors,
start by writing u as the sum of two vectors w1 and
w2, or u = w1 + w2. Since one of the vectors is the
projection of u onto v , let w1 = projvu and solve for
Henry is doing about 3247.6 joules of work pulling
the wagon.
w2.
b. Use the projection formula for work. The
magnitude of the projection of F onto
is
The magnitude of the directed
distance
is 150.
Thus,
If the handle makes an angle of 40° with the
horizontal, Henry is doing about 2872.7 joules of
work. Therefore, Henry is doing less work.
28. u =
.
,v=
SOLUTION:
Find the projection of u onto v .
Find the projection of u onto v. Then write u as
the sum of two orthogonal vectors, one of which
is the projection of u onto v.
27. u =
,v=
SOLUTION:
Find the projection of u onto v .
Page 7
Thus,
.
Mid-Chapter Quiz: Lessons 8-1 through 8-3
28. u =
Thus,
29. u =
,v=
.
,v=
SOLUTION:
SOLUTION:
Find the projection of u onto v .
Find the projection of u onto v .
To write u as the sum of two orthogonal vectors,
start by writing u as the sum of two vectors w1 and
To write u as the sum of two orthogonal vectors,
start by writing u as the sum of two vectors w1 and
w2, or u = w1 + w2. Since one of the vectors is the
projection of u onto v , let w1 = projvu and solve for
w2, or u = w1 + w2. Since one of the vectors is the
projection of u onto v , let w1 = projvu and solve for
w2.
w2.
Thus,
Thus,
.
30. u =
.
,v=
SOLUTION:
29. u =
,v=
Find the projection of u onto v .
SOLUTION:
Find the projection of u onto v .
Page 8
Mid-Chapter
Quiz: Lessons 8-1. through 8-3
Thus,
30. u =
,v=
SOLUTION:
Find the projection of u onto v .
To write u as the sum of two orthogonal vectors,
start by writing u as the sum of two vectors w1 and
w2, or u = w1 + w2. Since one of the vectors is the
projection of u onto v , let w1 = projvu and solve for
w2.
Thus,