copyright ©2010 carlson and oehrtman 1 module 10 vectors worksheet #2: basic applications of...

Copyright ©2010 Carlson and Oehrtman

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MODULE 10Vectors

Worksheet #2: Basic Applications of Vectors

Worksheet #1: Introduction to Vectors and Representing Vectors

Worksheet #4: Adding Vectors in Component and Polar Forms

Worksheet #3: Adding Vectors Geometrically

Worksheet #5: Scaling Vectors

Introduction to Vectors and Representing Vectors

Copyright@2008 Carlson, Oehrtman 2

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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

3. 4.5 miles measured π/3 radians north of east from some reference point

4. 4 miles measured 20° south of east from some reference point

10° west of north is equivalent to 100° measured counterclockwise from the 3 o’clock position.

π/3 radians north of east is equivalent to π/3 radians measured counterclockwise from the 3 o’clock position.

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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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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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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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.)

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.

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?

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.

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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.

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).

Answers will vary but should be close to 300 mph with a direction of about 320°.

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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.

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).

Answers will vary but should be close to 220 N with a direction of 100°.

Adding Vectors Geometrically


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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.)

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.

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.

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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.)

c. How can we geometrically represent the net force on the object when we combine the forces of the two people pushing?

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.

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 and explain what this tells us.

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 forma. 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 the force vector for the second person in polar form. The component form of the vectors are and .

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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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.

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?

b. What is polar form of the vector ? How does this compare to the polar form of the vector ?

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.

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 ?

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 magnitude is scaled by a factor of 3.5 without changing the direction.

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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copyright ©2010 carlson and oehrtman 1 module 10 vectors worksheet #2: basic applications of...

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