Classify the following coordinate systems as either right-handed or not right-handed.

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Lesson3 Assignment ­ Chapter 2.4­2.9 Due: 11:59pm on Sunday, July 17, 2016

You will receive no credit for items you complete after the assignment is due. Grading Policy

Video Solution Problem for Chapter 2 Sections 3, 4 ­ Summation of Coplanar Force Systems.

Watch the following video, then answer the question(s).

Part A

Determine the magnitude of the resultant force by adding the rectangular components of the three forces.

ANSWER:

Video Solution Problem for Chapter 2 Sections 3, 4 ­ Addition of Coplanar Force Systems.

Watch the following video, then answer the question(s).

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Part A

Determine the magnitude and direction of the resultant force.

ANSWER:

± Vector Math Practice

Learning Goal:

To review basic vector operations.

Let vectors , , and , where are the components of the vectors along the , , and axes, respectively; and let A, B and C be the magnitudes of , , and , respectively.

Part A ­ Addition of vectors

Calculate .

Express the components numerically to two significant figures separated by commas.

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ANSWER:

Part B ­ Magnitude of the vectors A, B, and C

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7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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What are the magnitudes of vectors , , and ?

Express your answers numerically to three significant figures separated by commas.

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ANSWER:

Part C ­ Dot product of the vectors A and B

Calculate .

Express your answer numerically to two significant figures.

ANSWER:

Part D ­ Angle between the vectors B and C

Determine the angle between and .

Express your answer numerically to three significant figures in radians.

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ANSWER:

Part E ­ Cross product of the vectors B and C

Calculate .

Express the components numerically to two significant figures separated by commas.

ANSWER:

Part F ­ Scalar triple product

Calculate .

Express your answer numerically to two significant figures.

ANSWER:

Right­Handed Cartesian Coordinate Systems

Learning Goal:

To identify and construct right­handed Cartesian coordinate systems.

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7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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The theory of vector algebra, especially the cross­product operation, is based upon using a right­handed system. Therefore, you should always work with a right­handed Cartesian coordinate system. These systems follow the right­hand rule. There are two equally valid methods of verifying whether a coordinate system is right­handed:

1. Using your right hand, extend your thumb in the positive direction of the z axis, and point your fingers in the direction of the x axis. If you can curl your fingers 90 degrees to match the y axis then you have a right­handed coordinate system.

2. Align your forearm with the x axis, point your fingers straight in the positive x direction, and position your wrist at the origin. Rotate your forearm and wrist around the x axis until you can curl your fingers 90 degrees to point in the direction of the positive y axis. If your thumb is pointing in the positive z axis direction then you have a right­handed coordinate system.

Part A

Classify the following coordinate systems as either right­handed or not right­handed.

Drag the appropriate coordinate systems to their respective bins.

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ANSWER:

Part B

Given the three coordinate systems, label the missing axes so that each is a right­handed coordinate system.

Drag the appropriate labels to their respective targets.

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7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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ANSWER:

Part C

Sometimes it is convenient to use a rotated coordinate system. For example, when analyzing a system that contains a tank of liquid, it is convenient to direct one of the axes downward from the liquid's surface.

Given three coordinate systems, each with a downward directed axis, label the missing axes so that each is a right­handed coordinate system.

Drag the appropriate labels to their respective targets.

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ANSWER:

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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± Cartesian Vectors

Learning Goal:

To understand Cartesian representations of vectors and how they can be determined from direction and magnitude; to learn how to represent vectors expressed as a product of the component magnitude and the unit direction vectors , , and ; and to learn how to represent a vector component as a function of the angle between the vector and the coordinate axis.

As shown, a pole is subjected to three forces: , , and . The force expressed in Cartesian vector form is . Force has magnitude and acts in the direction given by these direction angles: , , and to the x, y, and z axes, respectively. Force lies within the plane, its direction is given by the 3­4­5 triangle shown, and its magnitude is .

Part A ­ Calculating the magnitude of a vector from its components

What is the magnitude of ?

Express your answer to three significant figures and include the appropriate units.

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ANSWER:

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Part B ­ Components of a vector

Determine the components of in the , , and directions.

Express your answers, separated by commas, to three significant figures.

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ANSWER:

Part C ­ Finding Cartesian components from right triangles

What are the Cartesian components of force ?

Express your answers, separated by commas, to three significant figures.

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ANSWER:

Cartesian Vector Addition

Learning Goal:

To find the resultant of multiple forces using Cartesian components and to determine the direction of this resultant from its components.

As shown, three forces act on the tip of a pole. . and forms the following angles with the x, y, and z axes, respectively: , , and . , forms the angle with the z axis, and forms the angle between the x axis and the projection of in the plane.

Part A ­ Resultant of adding F1 and F3

Find the resultant of adding and . Express your answer in component form.

Express your answers, separated by commas, to three significant figures.

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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You did not open hints for this part.

ANSWER:

Part B ­ Resultant of adding F1 and F2 and F3

Find the resultant of adding , , and . Express your answer in component form.

Express your answers, separated by commas, to three significant figures.

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ANSWER:

Part C ­ Direction of a resultant

For the given forces, , , and , find the three direction angles , , and between the resultant force found in Part B and the x, y, and z axes, respectively.

Express your answers, separated by commas, in degrees to three significant figures.

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ANSWER:

Part D ­ Find the direction angle α when given direction angles β and γ

As shown, a force of acts at the tip of a tower. Two of the force's direction angles are known, and , that define the angles between the force and the y and z axes, respectively. Find , the angle between the x axis and the force.

Express your answer in degrees to three significant figures.

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7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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ANSWER:

Video Solution Problem for Chapter 2 Sections 5, 6 ­ Force Resultant in 3­D.

Watch the following video, then answer the question(s).

Part A

Express each force in Cartesian vector form.

ANSWER:

Video Solution Problem for Chapter 2 Sections 5, 6 ­ Force Vector Coordinate Angles

Watch the following video, then answer the question(s).

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Part A

Specify the magnitude and direction angles , and of so that the resultant of the three forces acting on the post is . Note that lies in the x—y plane.

ANSWER:

± Vector Addition in a Unit Cell

Learning Goal:

To practice vector addition in a unit cell.

In nature, substances often possess a crystalline structure. The basic component of a crystal is the unit cell, such as the rectangular parallelepiped shown. (For this problem, take , , and as positive.)

When a question asks for a vector, express your answer as an ordered triplet. That is, a vector with length in the x direction, length in the y direction, and length in the z direction should be written as .

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Part A

What is , the vector from point O to point E?

Express your answer in terms of some or all the variables , , and .

ANSWER:

Part B

What is , the vector from point O to point S?

Express your answer in terms of some or all the variables , , and .

ANSWER:

Part C

What is , the vector from point O to point C?

Express your answer in terms of some or all the variables , , and .

ANSWER:

For the following questions , , and are the lengths of , , and , respectively.

Part D

What is the sum ?

Express your answer in terms of some or all the variables , , and .

You did not open hints for this part.

ANSWER:

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7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Part E

What is the length of vector ?

Express your answer in terms of some or all the variables , , and .

You did not open hints for this part.

ANSWER:

Part F

What is the length of vector ?

Express your answer in terms of some or all the variables , , and .

You did not open hints for this part.

ANSWER:

Part G

What is the length of vector ?

Express your answer in terms of some or all the variables , , and .

ANSWER:

Position Vectors

Learning Goal:

To find a position vector between two arbitrary points.

As shown, two cables connect three points. is below by a distance and connected to by a cable long. Cable forms an angle with the positive y axis. is above and the distances and are and , respectively.

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7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Part A ­ Position vector from A to B

Using the dimensions in the figure, find the position vector from to in component form.

Express your answers, separated by commas, to three significant figures.

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ANSWER:

Part B ­ Unit direction vector for line AB

For the position vector found in Part A, find the unit direction vector acting in the same direction. Express your answer in component form.

Express your answers, separated by commas, to three significant figures.

You did not open hints for this part.

ANSWER:

Part C ­ Position vector from A to C

The length of cable is , and the cable forms the angle with the y axis. Given this information and the dimensions provided in the figure, find the position vector from to . Express the position vector in component form.

Express your answers, separated by commas, to three significant figures.

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ANSWER:

Part D ­ Position vector from B to C

Find the position vector from to . Express your answer in component form.

Express your answers, separated by commas, to three significant figures.

You did not open hints for this part.

ANSWER:

Part E ­ Tension in the cable

A cable is attached between and . The cable is attached in such a way that it has a tension of along its length. The tensile force will attempt to pull toward . The force exerted on has a z component with a magnitude of . What is the magnitude of the tension in the cable, ?

Express your answer to three significant figures and include the appropriate units.

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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ANSWER:

± Position Vectors

Learning Goal:

To identify the correct notation for a point and a vector, determine the position vector of a point relative to another point, and calculate the corresponding unit vector.

Although vectors are often constructed from points, points are not vectors. Vectors are commonly constructed from either: (1) the origin to a point or (2) a starting point to an ending point.

Part A

As shown on the coordinate system, points A and B have the following distances from the origin: = 2.90 , = 2.50 , = 3.20 , = 1.00 , and = 1.70 . Which of the following statements correctly describes the location and position vector of point A from the origin? Use appropriate notation for the location and position vector of a point.

Check all that apply.

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ANSWER:

Part B

For the same two points A and B, what are the , , and components of the position vector that starts at A and ends at B?

Express your answers in feet to three significant figures separated by commas.

The location of A is given as A{2.90 + 2.50 } .

The location of A is given as A{2.90 + 0.00 + 2.50 } .

The position vector of A is given as {2.90 + 2.50 } .

The location of A is given as A(2.90, 0.00, 2.50) .

The position vector of A is given as {2.90, 0.00, 2.50} .

The position vector of A is given as {2.90 + 0.00 + 2.50 } .

The position vector of A is given as (2.90 , 2.50 ) .

The location of A is given as A{2.90, 0.00, 2.50} .

The location of A is given as A(2.90 , 2.50 ) .

The position vector of A is given as (2.90, 0.00, 2.50) .

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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ANSWER:

Part C

A wire is connected to two supports. The support on the wall has coordinates = 3.90 and = 3.90 . The support on the ground has coordinates = 1.30 and = 4.45 . What are the , , and components of the unit vector that describes the line of action from point A to point B?

Express your answers to three significant figures separated by commas.

You did not open hints for this part.

ANSWER:

Force Vector Directed along a Line

Learning Goal:

To find a force's directional components that act along a cable and to find the coordinates of a point on a cable that has a force vector applied to it.

Three­dimensional Cartesian force vectors are used throughout engineering mechanics. The generic force vector is represented as follows:

where is the force vector and , , and are the vector's , , and components, respectively. The force vector has a magnitude

. The vector's direction is described by a unit vector , where .

{ , , } =

, , =

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Part A

As shown, a force vector with a magnitude of = 36.0 is applied at point A and is directed toward point C. The distance is 8.00 and the distance is 4.50 . What are the , , and components of ?

Express your answers numerically in pounds to three significant figures separated by commas.

You did not open hints for this part.

ANSWER:

Part B

As shown, the cable OA exerts a force on point O of . If the cable's length is , what are the x, y, and z coordinates of A in terms of , , , and ,

where ?

Express your answers in terms of , , , and separated by commas.

You did not open hints for this part.

ANSWER:

Video Solution Problem for Chapter 2 Sections 7, 8 ­ Force Directed Along a Line.

Watch the following video, then answer the question(s).

, , =

(x, y, z) =

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Part A

Cable exerts a force of on the top of the flagpole. Determine the projection of this force along the positive axis of the pole.

ANSWER:

Video Solution Problem for Chapter 2 Sections 7, 8 ­ Force Directed Along a Structural Member.

Watch the following video, then answer the question(s).

Part A

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Determine the magnitude and direction angles of , so that the resultant of the two forces acts upward along the axis of the pole and has a magnitude of .

ANSWER:

Force Acting along a Line

Learning Goal:

To determine Cartesian force components from magnitude and determine position and resultant vectors by summing Cartesian components.

As shown, a 2.9­ ball is suspended at point D inside a box with dimensions , , and . The ball is held in place by three cables anchored at points A, B, and C on the surface of the box.

If point D is the origin of the Cartesian coordinate system, point A is located at , point B is located at , and point C is located at .

Part A ­ Determining Cartesian components of force

The tension in cable DA has a magnitude of . Find the Cartesian components of tension , which is directed from D to A.

Express your answers, separated by commas, to three significant figures.

You did not open hints for this part.

ANSWER:

7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Part B ­ Determining an unknown force

If the tensions in cables DA and DB are and , respectively, what is the tension in cable DC?

Express your answer to three significant figures and include the appropriate units.

You did not open hints for this part.

ANSWER:

± Vector Dot Product

Learning Goal:

To understand the rules for computing dot products.

Let vectors , , and .

Part A ­ Dot product of the vectors A and B

Calculate .

Express your answer numerically.

You did not open hints for this part.

ANSWER:

Part B ­ Angle between the vectors A and B

What is the angle between and ?

Express your answer numerically to three significant figures in radians.

You did not open hints for this part.

ANSWER:

Part C ­ Dot product of two vectors multiplied by constants

Calculate .

Express your answer numerically.

ANSWER:

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7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Part D ­ Multiplication of a dot product by a scalar

Calculate .

Express your answer numerically.

ANSWER:

Part E ­ Dot product of a vector and a scalar?

Which of the following can be computed?

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ANSWER:

Let and be different vectors with lengths and , respectively.

Part F ­ Dot product of a vector with itself

Calculate .

Express your answer in terms of .

You did not open hints for this part.

ANSWER:

Part G ­ Dot product of two perpendicular vectors

If and are perpendicular, calculate .

Express your answer numerically.

You did not open hints for this part.

ANSWER:

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7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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Part H ­ Dot product of two parallel vectors

If and are parallel, calculate .

Express your answer in terms of and .

You did not open hints for this part.

ANSWER:

Dot Product

Learning Goal:

To use the dot product to find the components of forces acting in arbitrary directions.

A three­legged structure is shown below. It is attached to the ground at points = , , and = , , and connected to the ceiling at point = , , . The three legs connect at point = , 2.42, , where two forces, and , act. Force is given by

; has magnitude and direction angles , , and for the x, y, and z axes, respectively.

Part A ­ Finding the Cartesian components of a force described by direction angles

Find the Cartesian components of force acting in the x, y, and z directions given , , , and . Recall that is the angle between the vector and the x axis, is the angle between the vector and the y axis, and is the angle between the vector and the z axis.

Express your answers, separated by commas, to three significant figures.

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ANSWER:

Part B ­ Finding the angle between forces

Find the angle between forces and .

Express your answer to three significant figures in degrees.

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7/14/2016 Lesson3 Assignment - Chapter 2.4-2.9

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ANSWER:

Part C ­ Determining the force along a member

Determine the magnitude, , acting along member , due to the applied force .

Express your answer to three significant figures and include the appropriate units.

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ANSWER:

Part D ­ Finding the component of a force perpendicular to a direction

Given , find the component of that acts perpendicular to member such that the vector addition of the perpendicular and parallel components of with respect to DA equals . Express your answer in component form.

Express your answers, separated by commas, to three significant figures.

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ANSWER:

Score Summary: Your score on this assignment is 0.0%. You received 0 out of a possible total of 17 points.