MODULE 10
Vectors
Worksheet #1: Introduction to Vectors and Representing Vectors
Worksheet #2: Basic Applications of Vectors
Worksheet #3: Adding Vectors Geometrically
Worksheet #4: Adding Vectors in Component and Polar Forms
Worksheet #5: Scaling Vectors
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Introduction to Vectors and
Representing Vectors
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W#1
Most numbers we use in real life indicate a magnitude – the relative
size of a measureable attribute. For example, “16” might represent
“16 miles” and indicates a distance 16 times as long as 1 mile.
However, imagine we choose some reference point and decide to
measure a distance of 16 miles from that point. In which direction
should we measure? There are countless directions to choose from.
When we describe a distance of 16 miles, we often have a concrete
direction in mind. We might be looking at a map and determining
the distance between two cities, or we might be giving someone
directions, such as “Drive north on the freeway for 16 miles, then
take the 8th Street exit.” In short, when we think about what a
number represents or what we want others to think about the
number it’s common to involve a direction as well as a magnitude.
When we want to represent both a magnitude and a direction for a
quantity, we use a vector.
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We denote vectors graphically by an arrow. The length of the arrow
represents the quantity’s magnitude and the arrow points in the
quantity’s direction. For example, we can represent a distance of 4
miles measured directly north of some reference point as shown in
the graph on the left or a distance of 5 miles measured directly
southeast of some reference point as shown in the graph on the
right.
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For each vector described in Problems #1-6, complete the
following.
a. Pick any reference point on the graph from which to measure.
b. Draw the vector with the indicated magnitude and direction.
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1. 7 miles measured directly south of some reference point
2. 5 miles measured 10° west of north from some reference
point
10° west of north is equivalent to 100° measured counterclockwise
from the 3 o’clock position.
3. 4.5 miles measured π/3 radians north of east from some
reference point
π/3 radians north of east is equivalent to π/3 radians measured
counterclockwise from the 3 o’clock position.
4. 4 miles measured 20° south of east from some reference
point
20° south of east is equivalent to 340° measured counterclockwise
from the 3 o’clock position.
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5. 3 miles measured from some reference point along an
angle of 5π/4 radians counterclockwise from the 3 o’clock
position
6. 6.25 miles measured 30° south of west from some
reference point.
30° south of west is equivalent to 210° measured counterclockwise
from the 3 o’clock position.
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In Problem #1 we described the vector’s magnitude and
direction. To represent the quantity in a condensed form we
can use the polar form of the vector written as
,
indicating that the vector’s magnitude (length) is 7 miles in a
direction of 270° measured counterclockwise from the 3
o’clock position. (Note: We can also represent the vector as
using radians to measure the direction angle.) In
general, we can represent any vector with a magnitude of r
and a direction (measured as an angle counterclockwise from
the 3 o’clock position) of θ by
.
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We can also name vectors. For example, we could name the
vector in Problem #1 “v”. To make it clear that we are
representing the value of a vector quantity instead of simply a
number representing a magnitude we draw a small arrow over
the v as follows: . So we can say that
for
Problem #1.
Sometimes we might want to focus only on a vector’s
magnitude. To describe the magnitude of the vector we use the
notation
. So for Problem #1, we have
.
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7. Refer to Problems #2-6 to complete the following tasks.
a. Name the vectors in Problems #2-6, then represent
them in polar form.
Answers will vary based on the names students choose.
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7. Refer to Problems #2-6 to complete the following tasks.
b. Use vector magnitude notation to indicate the
magnitude of each vector in Problems #2-6.
Answers will vary based on the names students choose.
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When a vector is represented in polar form, such as
,
we have information about its magnitude and its direction, but
not about a reference point from which these values are
measured.
8. On the coordinate plane below, draw 5 vectors that could
all be represented by
.
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9. Consider the vector
representing a distance of 6
miles in a direction of 305° measured as an angle
counterclockwise from the 3 o’clock position.
a. How far does this vector displace to the east?
The horizontal displacement (displacement to the east) is
6cos(305o) ≈ 3.44 miles.
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9. Consider the vector
representing a distance of 6
miles in a direction of 305° measured as an angle
counterclockwise from the 3 o’clock position.
b. How far does this vector displace to the south?
The vertical displacement is 6sin(305°) ≈ −4.91
(indicating a displacement to the south of about 4.91
miles).
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10. Consider the vector
representing a distance of 4
miles in a direction of 170° measured as an angle
counterclockwise from the 3 o’clock position.
a. How far does this vector displace to the west?
The horizontal displacement is 4cos(170°) ≈ −3.94
(indicating a displacement to the west of about 3.94
miles).
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10. Consider the vector
representing a distance of 4
miles in a direction of 170° measured as an angle
counterclockwise from the 3 o’clock position.
b. How far does this vector displace to the north?
The vertical displacement (displacement to the north) is
4sin(170°) ≈ 0.69 miles.
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11. For any given vector
, how can we determine its
horizontal displacement and its vertical displacement?
The horizontal displacement is rcos(θ) and the vertical
displacement is rsin(θ). If rcos(θ) > 0 the horizontal
displacement is to the right and if rcos(θ) < 0 the horizontal
displacement is to the left. If rsin(θ) > 0 the vertical
displacement is up and if rsin(θ) < 0 the vertical displacement
is down.
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For each vector in polar form, give the component form of the
vector. (It is recommended that you make a quick sketch of the
vector first so that you can easily check that the component
form you provide is reasonable.)
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For each vector in polar form, give the component form of the
vector. (It is recommended that you make a quick sketch of the
vector first so that you can easily check that the component
form you provide is reasonable.)
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Basic Applications of Vectors
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In the previous examples we thought about vectors as
distances in certain directions, such as 5 miles in a direction of
60° measured counterclockwise from the 3 o’clock position.
However, the most common and useful applications of vectors
involve using them to represent speeds, forces, magnetic
fields, and even changes in population or economic indicators.
For example, if we imagine the positive y axis pointing north
and the positive x axis pointing east, we can represent an
object traveling 50 mph in a direction 10° west of north by
the vector
. The magnitude of the vector is
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In this example, the magnitude of the vector (its length on the
coordinate plane) doesn’t represent a distance but rather an
amount of speed. In such examples, longer vectors represent
objects with a greater speed while shorter vectors represent
objects traveling at slower speeds.
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1. A passenger jet takes off, climbing at an angle of 15°
relative to the horizontal. Suppose the speed of the jet is
200 mph at the moment it leaves the ground. The jet’s
overall speed can be thought of as the combination of both
its ground speed (the speed the airplane travels relative to
the ground) and its rate of climb (the rate at which the
airplane’s altitude changes).
a. Represent the airplane’s speed vector in polar form.
b. Draw a diagram showing the airplane’s speed vector
and vectors representing the airplane’s ground speed
and rate of climb.
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b. Draw a diagram showing the airplane’s speed vector
and vectors representing the airplane’s ground speed
and rate of climb.
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c. Find the jet’s ground speed to the nearest mile per hour.
d.
Find the jet’s rate of climb to the nearest mile per hour.
e. If the airplane continues at the angle and speed given for 2
minutes, what will be the airplane’s altitude above the
runway?
Two minutes is 2/60, or 1/30, of one hour. So the airplane’s
altitude above the runway will be about 51.76(1/30) = 1.73
miles, or about 1.73(5,280) = 9,110 feet.
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2. Consider the child’s wagon shown below. A parent is
pulling on the handle in the direction shown with a force of
250 Newtons.
(A Newton is unit of force. It represents the amount of force required
to accelerate a one kilogram mass at a rate of 1 m/s2. A Newton is
abbreviated N.)
a. Only the force in the horizontal direction goes to
moving the wagon along the sidewalk. How much
force is applied in the horizontal direction?
250cos(45°) = 176.78 Newtons (or 250cos(45°) =
250 (√2 / 2 ) = 125√2 Newtons)
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b. How can the parent increase the amount of force in the
horizontal direction without increasing the total amount of force
exerted on the wagon’s handle? Explain.
If the parent lowers the angle of the handle so that its direction
measured counterclockwise from the 3 o’clock position is closer to
0° then the force applied in the horizontal direction will increase
even if the total force of 250 N is the same.
c. The vertical component of the force vector doesn’t help to move
the wagon and is generally overcome by the weight of the
wagon. However, if the force in the upward direction becomes
too great the wagon’s front wheels will lift off of the ground.
Suppose it takes 200 N to lift the wagon’s front wheel. How
much force in the direction of the handle will cause the front
wheels to lift off of the ground?
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c. The vertical component of the force vector doesn’t help to move
the wagon and is generally overcome by the weight of the
wagon. However, if the force in the upward direction becomes
too great the wagon’s front wheels will lift off of the ground.
Suppose it takes 200 N to lift the wagon’s front wheel. How
much force in the direction of the handle will cause the front
wheels to lift off of the ground?
Let r be the minimum total force in the 45° direction (the direction of
the handle) that will lift the front wheels off the ground. Then
So if the parent applies at least about 282.84 N (or 200√2 N) of force
in the direction of the handle then the wagon’s front wheels will lift
off of the ground.
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3. An airplane is flying with a heading and speed such that
• the airplane’s east/west position is changing at a speed of
237.8 mph east
• the airplane’s north/south position is changing at a speed
of 192.1 mph south
a. Draw the component vectors described, then draw the vector
representing the airplane’s overall speed and direction.
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b. Write the component form for each of the three vectors you
drew in part (a).
c.
Estimate the airplane’s overall speed and direction.
Answers will vary but should be close to 300 mph with a
direction of about 320°.
d. Use your knowledge of trigonometry to calculate the
airplane’s exact speed and direction (measured as an angle
counterclockwise from the 3 o’clock position).
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4. While landscaping his yard, David lifted a stone up and to his
left to change its location. In doing so, he applied the
equivalent of a force of 215 N up and 48 N to his left.
a. If the positive x-axis represents a direction directly to
David’s right and the positive y-axis represents a
direction directly up, draw the component vectors
described, then draw the vector representing the overall
force David applied to the rock and the direction in which
the force was applied.
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b. Write the component form for each of the three vectors you
drew in part (a).
c.
Estimate the polar form of the vector representing the force
David applied to the stone.
Answers will vary but should be close to 220 N with a direction of
100°.
d. Use your knowledge of trigonometry to calculate the exact
speed and direction (measured as an angle counterclockwise
from the 3 o’clock position) of the force vector in part (b).
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Adding Vectors Geometrically
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W#3
1. Suppose two people are pushing on the same object. The first
person pushes the object with a force of 230 N in a direction
due east while the second person pushes due north with a
force of 150 N.
a. Draw the vectors representing the force applied on the
object by each person.
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b. If the object were free to move in any direction, in which
direction will the object move? (Assume that no other forces
are acting upon the object.)
The object should move in a northeastern direction. Since the
person pushing east is applying more force to the object, the
direction will be more eastern than northern.
c. How can we geometrically represent the net force on the
object when we combine the forces of the two people
pushing?
d. Estimate the approximate magnitude and direction of the
resultant force vector after drawing the resultant force vector.
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d. Estimate the approximate magnitude and direction of the
resultant force vector after drawing the resultant force vector.
Answers will vary by student and by the accuracy of the
diagram but should be close to 275 N in a direction of about
30° measured counterclockwise from the 3 o’clock position.
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2. Suppose instead one person is pushing on the object with a
force of 200 N in the direction 30° north of west (150°
measured counterclockwise from the 3 o’clock position) and
the second person is pushing on the object with a force of
140 N in the direction 60° north of east (60° measured
counterclockwise from the 3 o’clock position).
a. Draw the vectors representing the force applied on the
object by each person.
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b. If the object were free to move in any direction, in which
direction will the object move? (Assume that no other
forces are acting upon the object.)
Both forces are being applied towards the north (instead of the
south), so the object will move in a northerly direction.
Whether the object moves towards the east or the west is more
difficult to determine.
c. How can we geometrically represent the net force on the
object when we combine the forces of the two people
pushing?
The same head-to-tail method of constructing the resultant
vector works here
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d. Estimate the approximate magnitude and direction of the
resultant force vector after drawing the resultant force
vector.
Answers will vary by student and by the accuracy of the
diagram but should be close to 250 N in a direction of about
115° counterclockwise from the 3 o’clock position.
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3. Have you ever been driving a car when a gust of wind
pushes against the vehicle? The wind wants to push the car
in the direction it’s blowing, and you might have had to
correct the car’s direction with the steering wheel to keep
it in the correct lane. Since the car’s tires are in contact
with the ground the car is able to resist much of the
potential influence of the wind and the effect is often
relatively small.
When an airplane is flying and encounters wind, however,
it doesn’t have the benefit of ground contact to create
friction that can resist the effects of wind. Therefore an
airplane can easily be blown off course by a strong wind
unless the pilot corrects his course.
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a. Suppose an airplane is flying 215 mph with a heading
40° west of north (130° measured counterclockwise
from the 3 o’clock position) when it encounters a 50 mph
wind blowing steadily in a direction 10° east of south
(280° measured counterclockwise from the 3 o’clock
position). Using to represent the speed vector of the
airplane and
to represent the speed vector of the wind, draw and
in a head-to-tail configuration and the resultant vector .
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b. Estimate the magnitude and direction of
explain what this tells us.
and
Answers will vary but should be close to about 170 or 180 mph
in a direction of about 140° measured counterclockwise from
the 3 o’clock position.
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4. Three people are playing the Ring Game. In this game
each player grabs a large ring and attempts to pull it into
his/her own scoring section.
Suppose Player A pulls with a force of 200 N at an angle of
70°, Player B pulls with a force of 260 N at an angle of 200°,
and Player C pulls with a force of 215 N at an angle of 260°.
(All angles are measured from standard position according to
the diagram above.)
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a. Draw a diagram showing the force vectors described such
that each vector begins at the origin.
b. If all three players maintain the strength and direction of
their pulls, in what direction do you think the ring will
move? Why?
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b. If all three players maintain the strength and direction of
their pulls, in what direction do you think the ring will
move? Why?
Let’s break it down into two parts. First, let’s determine if the
ring will move to the right or the left. Two of the three players
are pulling to the left (as the diagram is oriented), so the ring
will likely move left. Also, since only one player is pulling with
most of his/her force to the left/right (Player B, to the left), it
should move towards the left. Second, let’s determine if the ring
will move up or down. Two of the three players are pulling
down (as the diagram is oriented). Player A and Player C’s
up/down pulls should mostly cancel out with Player B’s force in
the downward direction tipping the scales to result in the ring
moving down. Therefore, we should expect the ring to move
down and to the left (as the diagram is oriented).
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c. Redraw the vectors in a head-to-tail configuration and
draw the resultant vector.
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d. Estimate the direction and magnitude of the resultant
vector and explain what this tells us.
Answers will vary but should be close to about 240 N in a
direction of about 210°.
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5. Three workers are pulling on chains attached to a tree
stump trying to remove it from the ground. One worker is
pulling with a force of 400 N in a direction of 90°, the
second worker is pulling with a force of 250 N in a
direction of 0°, and the third worker is pulling with a
force of 300 N in a direction of 300° (all directions are
measured counterclockwise from the 3 o’clock position,
viewed from above looking down on the stump).
a. Draw a diagram showing the three force vectors
described in the problem.
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You may draw the diagram with all vectors beginning at the origin
or might draw them attached head to tail (both of these solutions
are shown below). (We let ,
, and
represent the first
worker’s, second worker’s, and third worker’s force vectors
respectively.)
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b. Draw the resultant vector of the combined force applied
to the stump by the three workers.
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c. Estimate the magnitude and direction of the resultant
vector of the combined force applied to the stump by the
three workers.
To estimate the magnitude students should compare the length
of the resultant vector to the scale on the axes. The resultant
force appears to be about 400 N in a direction of about 20°.
d. Explain why this arrangement is not the most efficient for
the workers to complete their task.
The first worker and the third worker are pulling in somewhat
opposite directions, so some of the force that each is applying is
being “cancelled out”. The most efficient setup is to have all of
the workers pulling in the same direction (or as close to the
same direction as possible).
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Consider the following vectors.
For #6-11, draw the given resultant vector. (Hint: You may
want to use a ruler and a protractor to accurately represent
the vectors, or use another sheet of paper to trace the
vectors.)
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Adding Vectors in Component
and Polar Form
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1. While running errands, suppose you make your first stop 3.5
miles east and 2 miles north of your house. You then travel
an additional 4 miles east and 5.2 miles north before making
your second stop.
a. Represent the displacement from your house to your first
stop and from your first stop to your second stop using
vectors written in component form. Assume that the
positive horizontal axis points east and the positive
vertical axis points north.
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b. Draw the vectors you defined in part (a) in a head-to-tail
configuration.
c. What is the total displacement from your house to your
second stop?
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c. What is the total displacement from your house to your
second stop?
7.5 miles east and 7.2 miles north, or in vector component
form
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d. How can we determine the total displacement vector in
component form without drawing the given vectors?
We can add the east direction displacements of the two vectors
and the north direction displacements. In other words, the total
displacement is 3.5 + 4 = 7.5 miles east and 2 + 5.2 = 7.2 miles
north, or in vector component form
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2. An airplane is traveling with a speed and heading such that
its distance to the south is increasing at a rate of 300 mph
and its distance to the east is increasing at a rate of 200
mph. The airplane then encounters a 45 mph wind blowing
steadily directly west.
a. Represent the speed vectors for the airplane and the
wind in component form. Assume that the positive
horizontal axis points east and the positive vertical axis
points north.
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b. Draw the vectors you defined in part (a) in a head-to-tail
configuration.
c. What is the resultant speed vector of the airplane under the
effects of the wind (assuming the pilot does not correct his
course)?
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c. What is the resultant speed vector of the airplane under the
effects of the wind (assuming the pilot does not correct his
course)?
155 mph east and 300 mph south, or
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d. How can we determine the resultant speed vector in
component form without drawing the given vectors?
We can add the east/west components of the speed vectors and
the north/south components of the speed vectors. That is,
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3. If
and
are two vectors given in
component form, what is the component form of the
resultant vector
?
4. Consider the following vectors in component form.
Represent each of the following resultant vectors in
component form. You might wish to make a sketch of the
vectors involved to check your answer.
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4. Consider the following vectors in component form.
Represent each of the following resultant vectors in
component form. You might wish to make a sketch of the
vectors involved to check your answer.
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5. Does the technique for finding the resultant vector in
component form work when we have more than two vectors
being added together? Explain your reasoning and/or
demonstrate your thinking using an example.
Yes. In component form, we are simply keeping track of the net
horizontal displacement and the net vertical displacement. So when
we have any number of vectors in component form adding up the
horizontal components of each vector will tell us the net horizontal
displacement and adding up the vertical components of each vector
will tell us the net vertical displacement. Since the components
indicate their direction by using positive and negative numbers,
adding components with different signs automatically accounts for
vectors acting in different directions
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6. Thus far we’ve established that if we represent two vectors
in component form, such as
and
,
that the resultant vector is
.
Does the same technique work for two vectors written in
polar form? For example, suppose
and
. Is it true that
? Explain your reasoning.
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7. Consider two vectors in polar form
a. Draw the vectors in a head-to-tail configuration, then
draw the resultant vector
.
b. Estimate the magnitude and direction of the resultant
vector.
Answers will vary. The magnitude is about 7.5 to 8 with a
direction of about 30° (measured counterclockwise from
the 3 o’clock position).
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c. How can we determine the horizontal and vertical
displacement for the resultant vector exactly?
Earlier in the module we learned how to convert the polar
form of a vector to component form by
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d. How can we determine the exact magnitude and direction
for the resultant vector?
Earlier in the module we learned how to convert from the
component form of a vector to its polar form using either
depending on whether the horizontal displacement was positive
or negative.
Since the horizontal displacement is positive in this case,
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8. In Worksheet 3 we examined the following context.
Suppose one person is pushing on the object with a force of
200 N in the direction 30° north of west (150° measured
counterclockwise from the 3 o’clock position) and the second
person is pushing on the object with a force of 140 N in the
direction 60° north of east (60° measured
counterclockwise from the 3 o’clock position).
Find the exact magnitude and direction of the resultant force
vector, then compare your answer to the estimate you
provided in #2d on Worksheet 3.
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We begin by writing the vectors involved. Let
represent the force vector for the first person and
force vector for the second person in polar form.
The component form of the vectors are
and
.
the
The component form of the resultant vector is
Since the horizontal displacement of the resultant vector is negative,
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9. Explain the general process for finding the magnitude and
direction of a resultant vector
in polar form
when given two vectors
and
in polar
form.
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10. In Worksheet 3 we examined the following context.
Three people are playing the Ring Game. In this game each
player grabs a large ring and attempts to pull it into his/her
own scoring section. Suppose Player A pulls with a force of
200 N at an angle of 70°, Player B pulls with a force of 260
N at an angle of 200°, and Player C pulls with a force of
215 N at an angle of 260°. (All angles are measured from
standard position according to the diagram above.)
Find the exact magnitude and direction of the resultant force
vector, then compare your answer to the estimate you
provided in #4d on Worksheet 3.
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Since the horizontal displacement of the resultant vector is negative,
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11. Consider the following vectors in polar form.
Represent each of the following resultant vectors in polar form.
Caution! It’s recommended that you make a quick sketch of the given
vectors and the resultant vector first so that you can make sure that the
magnitude and direction for the resultant vector you provide are
reasonable.
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W#4
11. Consider the following vectors in polar form.
Represent each of the following resultant vectors in polar form. Caution! It’s
recommended that you make a quick sketch of the given vectors and the resultant
vector first so that you can make sure that the magnitude and direction for the
resultant vector you provide are reasonable.
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Scaling Vectors
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When we scale a vector, we change its magnitude by some
factor without changing its direction. To denote the process,
we use the following notation. If we want to triple the
magnitude of a vector without changing its direction, we
write . If we want to make the magnitude 10 times as large
without changing the direction, we write
.
In general, if we want to scale the magnitude of a vector by
some factor k where k > 0 without changing its direction, we
write
.
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1. An airplane is traveling at 210 mph with a heading 40°
north of east. Its speed vector is represented in polar form
by
a. Suppose we scale this vector by a factor of 2 (i.e., we
create the vector
). What does this represent in the
context of the situation?
The magnitude of a speed vector represents the speed of the
object. Therefore, when we scale the magnitude by a factor
of 2, the object’s speed is doubling. The airplane would be
traveling 420 mph.
b. What is polar form of the vector ? How does this
compare to the polar form of the vector ?
The speed doubles but the direction remains the same.
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c. What is the component form of the vector ? How does
this compare to the component form of vector ?
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2. Suppose two people are pushing on an object at the same
time. The first person pushes with a force of 140 N in a
direction of 80° measured clockwise from the 3 o’clock
position. The second person pushes with a force of 230 N in
a direction of 125° measured counterclockwise from the 3
o’clock position.
a. Represent the force vectors described in both polar and
component forms, then find the resultant force vector and
represent it in both polar and component forms.
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a. Represent the force vectors described in both polar and
component forms, then find the resultant force vector and
represent it in both polar and component forms.
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b. Suppose the first person increases the force he applies to the
object by 30% while the second person decreases the force he
applies by 10%. (Neither changes the direction of his push.)
Write the component and polar forms for the new force vectors.
c. Find the resultant force vector under the new conditions
described in part (b). Write the resultant vector in both polar and
component forms.
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3. Consider the vector
. If the vector is scaled by a
factor of 3.5, what is the polar form of
?
The magnitude is scaled by a factor of 3.5 without changing the
direction.
4. Consider the vector
represented in component
form. If the vector is scaled by a factor of 0.4, what is the
component form of
?
The components are scaled by a factor of 0.4.
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5. Consider the vectors below given in component form.
Write the component form for each of the following
vectors.
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6. Consider the vectors below given in polar form.
Write the polar form for each of the following vectors.
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9. Explain the impact of multiplying a vector by a negative
scalar.
Multiplying a vector by a scalar k where k < 0 changes the
magnitude of the vector by a factor of | k | and changes the
direction to be opposite of the original direction (changes the
direction by 180° or π radians).
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