ch 01 vector addition and vector components

7/6/2014 Ch 01 Vector Addition and Vector Components

http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2952267 1/41

Ch 01 Vector Addition and Vector Components

Due: 4:00pm on Monday, July 7, 2014

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

Exercise 1.36

Part A

Find the magnitude of the vector represented by the pair of components -8.90 , 2.70 .

ANSWER:

Correct

Part B

Let the direction of a vector be the angle that the vector makes with the +x-axis, measured counterclockwise from

that axis. Find the direction of the vector .

ANSWER:

Correct

Part C

Find the magnitude of the vector represented by the pair of components -6.00 , -1.70 .

ANSWER:

Correct

Part D

Find the direction of the vector . Let the direction of a vector be the angle that the vector makes with the +x-axis,

measured counterclockwise from that axis.

4

DN

5

DN

9.30 DN

163

4

N

5

N

6.24 N

Typesetting math: 22%



ANSWER:

Correct

Part E

Find the magnitude of the vector represented by the pair of components 8.50 , -3.00 .

ANSWER:

Correct

Part F

Find the direction of the vector . Let the direction of a vector be the angle that the vector makes with the +x-axis,

measured counterclockwise from that axis.

ANSWER:

Correct

Exercise 1.43

Part A

Write the vector in the figure in terms of the unit vectors and .

196

4

LN

5

LN

9.01 LN

341

%

?

&

?



ANSWER:

Correct

Part B

Write the vector in the figure in terms of the unit vectors and .

ANSWER:

Correct

Part C

Use unit vectors to express the vector , where .

ANSWER:

Correct

Part D

= %? &?

%

?

&

?

= %? &?

= %? &?



Find the magnitude of .

ANSWER:

Correct

Part E

Find the direction of .

ANSWER:

Correct

Vector Magnitude and Direction Conceptual Question

A man out walking his dog makes one complete pass around a perfectly square city block. He starts at point A andwalks clockwise around the block.

Let be the displacement vector from A to B, be the displacement vector from B to C, etc.

Part A

Which of the following vectors is equal to ?

Hint 1. Determining a vector

19.2 N

51.2 counterclockwise from +x-axis

.

"#

.

#$

.

"#

"#



Recall that is a vector representing the displacement of the man and his dog as they walk from point A

to point B. This vector has a magnitude equal to one block and a direction along the positive x axis.

Hint 2. Equal vectors

Two vectors are equal if they have the same magnitude and the same direction.

ANSWER:

Correct

Recall that, for vectors to be equal, they must have the same magnitude and direction.

Part B

Which of the following vectors is equal to ?

ANSWER:

Correct

Part C

Which of the following vectors is equal to \vec r_{AB}-\vec r_{DA}?

Hint 1. Determining the difference of two vectors

\vec r_{AB}-\vec r_{DA} can be determined by adding the vector \texttip{\vec{r}_{\rm AB}}{r_AB_vec} to the

vector pointing opposite to \texttip{\vec{r}_{\rm DA}}{r_DA_vec}. Thus \vec r_{AB}-\vec r_{DA} looks like this:

.

"#

only

only

only

All of the above

None of the above

.

#$

.

$%

.

%"

.

\texttip{\vec{r}_{\rm BC}}{r_BC_vec} only

\texttip{\vec{r}_{\rm CD}}{r_CD_vec} only

\texttip{\vec{r}_{\rm DA}}{r_DA_vec} only

All of the above

None of the above



Carefully perform the vector addition in each of the options and compare the resultant vectors to the oneshown above.

ANSWER:

Correct

Exercise 1.42

Part A

Given the vector \vec A = 4.00\hat i + 7.00\hat j , find the magnitude of the vector.

ANSWER:

Correct

Part B

Given the vector \vec B = 5.00\hat i -2.00\hat j , find the magnitude of the vector.

-(\vec r_{CD}+\vec r_{DA}) only

\vec r_{AB}+\vec r_{BC} only

\vec r_{BC}-\vec r_{CD} only

All of the above

None of the above

8.06



ANSWER:

Correct

Part C

Write an expression for the vector difference \vec A - \vec B using unit vectors.

Express your answer in terms of unit vectors.

ANSWER:

Correct

Part D

Find the magnitude of the vector difference \vec A - \vec B.

ANSWER:

Correct

Part E

Find the direction of the vector difference \vec A - \vec B.

ANSWER:

Correct

Part F

In a vector diagram show {\vec A}, {\vec B}, and {\vec C} = {\vec A} - {\vec B}.

Draw the vectors starting at the black dot. Both the orientation and length of your vectors will be graded.Use "vector info" button to see the angle and length of your vectors.

5.39

\vec A - \vec B = -1.00 \hat{i}+9.00 \hat{j}

9.06

96.3 {^\circ} counterclockwise from +\texttip{x}{x} direction



ANSWER:

Correct

Resolving Vector Components with Trigonometry

Often a vector is specified by a magnitude and a direction; forexample, a rope with tension \texttip{\vec{T}}{T_vec} exerts aforce of magnitude \texttip{T}{T} in a direction 35 \^circ north ofeast. This is a good way to think of vectors; however, tocalculate results with vectors, it is best to select a coordinatesystem and manipulate the components of the vectors in thatcoordinate system.

Part A



Find the components of the vector \texttip{\vec{A}}{A_vec} with length \texttip{a}{a} = 1.00 and angle \alpha=10.0

{\rm \^circ} with respect to the x axis as shown.

Enter the x component followed by the y component, separated by a comma.

Hint 1. What is the x component?

Look at the figure shown.

\texttip{\vec{A}_{\mit x}}{A_vec_x} points in the

positive x direction, so \texttip{A_{\mit x}}{A_x} is

positive. Also, the magnitude |A_x| is just the

length {\rm OL = OM} \cos(\alpha).

ANSWER:

Correct

Part B

Find the components of the vector \texttip{\vec{B}}{B_vec} with length \texttip{b}{b} = 1.00 and angle \beta=10.0

{\rm {^\circ}} with respect to the x axis as shown.


Hint 1. What is the x component?

The x component is still of the same form, that is, L\cos(\theta).

ANSWER:

\texttip{\vec{A}}{A_vec} = 0.985,0.174

\texttip{\vec{B}}{B_vec} = 0.985,0.174



Correct

The components of \texttip{\vec{B}}{B_vec} still have the same form, that is,

\left( L \cos(\theta),L \sin(\theta) \right), despite \texttip{\vec{B}}{B_vec}'s placement with respect to the y axis

on the drawing.

Part C

Find the components of the vector \texttip{\vec{C}}{C_vec} with length \texttip{c}{c} = 1.00 and angle \phi = 35.0

{\rm {^\circ}} as shown.


Hint 1. Method 1: Find the angle that \texttip{\vec{C}}{C_vec} makes with the positive x axis

Angle \texttip{\phi }{phi} = 0.611 differs from the other two angles because it is the angle between the vector

and the y axis, unlike the others, which are with respect to the x axis. What is the angle that \texttip{\vec{C}}{C_vec} makes with the positive x axis?

Express your answer numerically in degrees.

ANSWER:

Hint 2. Method 2: Use vector addition

Look at the figure shown.

1. \vec{C} = \vec{C}_x + \vec{C}_y.

2. |\vec{C}_x| = {\rm length(QR)} = c \sin(\phi).

3. \texttip{C_{\mit x}}{C_x}, the x component

of \texttip{\vec{C}}{C_vec} is negative, since

\texttip{\vec{C}_{\mit x}}{C_vec_x} points in

the negative x direction.Use this information to find \texttip{C_{\mit x}}{C_x}. Similarly, find

\texttip{C_{\mit y}}{C_y}.

ANSWER:

125

\texttip{\vec{C}}{C_vec} = -0.574,0.819



Correct

Geometric vs Componentwise Vector Addition

Learning Goal:

To understand that adding vectors by using geometry and by using components gives the same result, and thatmanipulating vectors with components is much easier.

Vectors may be manipulated either geometrically or using components. In this problem we consider the addition of twovectors using both of these two methods.

The vectors \texttip{\vec{A}}{A_vec} and \texttip{\vec{B}}{B_vec} have lengths \texttip{A}{A} and \texttip{B}{B},respectively, and \texttip{\vec{B}}{B_vec} makes an angle \texttip{\theta }{theta} from the direction of \texttip{\vec{A}}{A_vec}.

Vector addition using geometry

Vector addition using geometry is accomplished by putting the tail of one vector (in this case \texttip{\vec{B}}{B_vec}) onthe tip of the other (\texttip{\vec{A}}{A_vec}) and using thelaws of plane geometry to find the length \texttip{C}{C}, andangle \texttip{\phi }{phi}, of the resultant (or sum) vector,\vec{C}=\vec{A}+\vec{B}:

1. C=\sqrt{A 2^ +B 2^ -2 A B \cos(c)},

2. \large{\phi = \sin {^-1}\left(\frac{B\sin(c)}{C}\right).}

Vector addition using components

Vector addition using components requires the choice of a coordinate system. In this problem, the x axis is chosenalong the direction of \texttip{\vec{A}}{A_vec} . Then the x and y components of \texttip{\vec{B}}{B_vec} are B\cos(\theta)and B\sin(\theta) respectively. This means that the x and y components of \texttip{C}{C} are given by

3. C_x = A + B\cos(\theta),

4. C_y = B\sin(\theta).



Part A

Which of the following sets of conditions, if true, would show that the expressions 1 and 2 above define the samevector \texttip{\vec{C}}{C_vec} as expressions 3 and 4?

Check all that apply.

ANSWER:

Correct

To show that the two pairs of expressions (for \texttip{C}{C} and \texttip{\phi }{phi} and for

\texttip{C_{\mit x}}{C_x} and \texttip{C_{\mit y}}{C_y}) define the same vector \texttip{\vec{C}}{C_vec} you can

show that any of the sets of conditions listed above are met except

They give the same length and x component for \texttip{\vec{C}}{C_vec}.

They give the same length and y component for \texttip{\vec{C}}{C_vec}.

If you consider just the first set of conditions, showing that the two sets of expressions have the same lengthand x component will imply that the y component has the correct magnitude. However, there is no way to knowthe sign (i.e., direction) of the y component. To show that the pairs of expressions given above define the samevector \texttip{\vec{C}}{C_vec}, we would need to show that they give the same length and the same x and y

components. We will do this in the questions that follow.

Part B

The two pairs of expressions give the same length and direction for \texttip{\vec{C}}{C_vec}.

The two pairs of expressions give the same length and x component for \texttip{\vec{C}}{C_vec}.

The two pairs of expressions give the same direction and x component for \texttip{\vec{C}}{C_vec}.

The two pairs of expressions give the same length and y component for \texttip{\vec{C}}{C_vec}.

The two pairs of expressions give the same direction and y component for \texttip{\vec{C}}{C_vec}.

The two pairs of expressions give the same x and y components for \texttip{\vec{C}}{C_vec}.



We begin by investigating whether the lengths are the same.Find the length of the vector \texttip{\vec{C}}{C_vec} starting from the components given in Equations 3 and 4.

Express \texttip{C}{C} in terms of \texttip{A}{A}, \texttip{B}{B}, and \texttip{\theta }{theta}.

Hint 1. Apply the Pythagorean theorem

You are given the x and y components in Equations 3 and 4. Simply square these, add them, and take thesquare root (i.e., apply the Pythagorean theorem) to find the length of vector \texttip{C}{C}.

ANSWER:

Correct

Part C

The following reasons might explain why the equation for the length just obtained using components is the same asthe answer obtained using geometry (Equation 1 above):

1. \texttip{c}{c} and \texttip{\theta }{theta} are supplementary angles, that is, \theta + c=\pi.

2. The cosine function satisfies \cos(\alpha) = - \cos(\pi - \alpha).

3. Cosine is an even function of its argument, so the extra negative sign in one expression does notmatter.

Which of these reasons is/are necessary to show that \sqrt{A 2^ +B 2^ -2 A B \cos(c)} = \sqrt{A 2^ +B 2^ +2 A B \cos(\theta)}?

ANSWER:

Correct

Having shown that the length of \texttip{\vec{C}}{C_vec} obtained using either geometrical addition OR componentwiseaddition are the same, all that remains is to show that any one of the other conditions from part A is satisfied. In the lasttwo parts of the problem, we'll show that the y component of \texttip{\vec{C}}{C_vec} determined geometrically is equal tothat given above.

\texttip{C}{C} = \sqrt{\left(A+B {\cos}{\theta}\right) {^2}+\left(B {\sin}{\theta}\right) {^2}}

1 only

2 only

3 only

1 and 2

2 and 3

1 and 3

1 and 2 and 3



Part D

We begin by finding the y component of \texttip{\vec{C}}{C_vec} from its length and the angle it makes with the x

axis, that is, from geometry.

Express the y component of \texttip{\vec{C}}{C_vec} in terms of \texttip{C}{C} and \texttip{\phi }{phi}.

ANSWER:

Correct

Part E

Now express the y component of \texttip{\vec{C}}{C_vec} just found by using the geometrical approach in terms of

\texttip{\theta }{theta} rather than \texttip{\phi }{phi}.

Express \texttip{C_{\mit y}}{C_y} in terms of \texttip{A}{A}, \texttip{B}{B} and \texttip{\theta }{theta}.

Hint 1. Law of sines

The law of sines can be used to relate \sin{(\phi)} and \sin{(c)}.

Express \sin{(\phi)} in terms of \sin{(c)}, \texttip{C}{C}, and \texttip{B}{B}.

ANSWER:

Hint 2. Supplementary angles

Angles \texttip{\theta }{theta} and \texttip{c}{c} are supplementary, which means that \theta + c = \pi. What

is the relationship of \sin(x) and \sin(\pi-x)?

ANSWER:

\texttip{C_{\mit y}}{C_y} = C {\sin}\left({\phi}\right)

\large{\frac{\sin{(\phi)}}{B}} = \large{\frac{{\sin}\left(c\right)}{C}}

\texttip{C_{\mit y}}{C_y} = B {\sin}\left({\pi}-{\theta}\right)



Correct

This is the same as Equation 4 above, obtained using components. Thinking in terms of components, theresult is fairly obvious: The y component of \texttip{\vec{C}}{C_vec} is equal to the the y component of

\texttip{\vec{B}}{B_vec} since \texttip{\vec{A}}{A_vec} has no y component in the chosen coordinate system.

At this point you have actually shown that the two vectors are equal by showing that the overall lengths areequal, and also that the y components are equal. You would only have to argue that the x component ispositive to complete the proof of equality.

Nevertheless, we are asking you to complete the algebra to show that the x components are equal as well.

Part F

Now find the x component of \texttip{\vec{C}}{C_vec}.

Express your answer in terms of \texttip{C}{C} and \texttip{\phi }{phi} only.

ANSWER:

Correct

Components of Vectors

Shown is a 10 by 10 grid, with coordinate axes x and y .The grid runs from -5 to 5 on both axes. Drawn on this grid arefour vectors, labeled \texttip{\vec{A}}{A_vec} through\texttip{\vec{D}}{D_vec}. This problem will ask you variousquestions about these vectors. All answers should be indecimal notation, unless otherwise specified.

Part A

What is the x component of \texttip{\vec{A}}{A_vec}?

\texttip{C_{\mit x}}{C_x} = C {\cos}\left({\phi}\right)



Express your answer to two significant figures.

Hint 1. How to derive the component

A component of a vector is its length (but with appropriate sign) along a particular coordinate axis, the axesbeing specfied in advance. You are asked for the component of \texttip{\vec{A}}{A_vec} that lies along the x

axis, which is horizontal in this problem. Imagine two lines perpendicular to the x axis running from the head(end with the arrow) and tail of \texttip{\vec{A}}{A_vec} down to the x axis. The length of the x axis between

the points where these lines intersect is the x component of \texttip{\vec{A}}{A_vec}. In this problem, the x

component is the x coordinate at which the perpendicular from the head of the vector hits the origin (becausethe tail of the vector is at the origin).

ANSWER:

Correct

Part B

What is the y component of \texttip{\vec{A}}{A_vec}?

Express your answer to the nearest integer.

ANSWER:

Correct

Part C

What is the y component of \texttip{\vec{B}}{B_vec}?


Hint 1. Consider the direction

Don't forget the sign.

ANSWER:

\texttip{A_{\mit x}}{A_x} = 2.5

\texttip{A_{\mit y}}{A_y} = 3

\texttip{B_{\mit y}}{B_y} = -3



Correct

Part D

What is the \texttip{x}{x} component of \texttip{\vec{C}}{C_vec}?


Hint 1. How to find the start and end points of the vector components

A vector is defined only by its magnitude and direction. The starting point of the vector is of no consequenceto its definition. Therefore, you need to somehow eliminate the starting point from your answer. You can runtwo perpendiculars to the x axis, one from the head (end with the arrow) of \texttip{\vec{C}}{C_vec}, and

another to the tail, with the x component being the difference between x coordinates of head and tail(negative if the tail is to the right of the head). Another way is to imagine bringing the tail of \texttip{\vec{C}}{C_vec} to the origin, and then using the same procedure you used before to find the

components of \texttip{\vec{A}}{A_vec} and \texttip{\vec{B}}{B_vec}. This is equivalent to the previous method,

but it might be easier to visualize.

ANSWER:

Correct

The following questions will ask you to give both components of vectors using the ordered pairs method. In this method,the x component is written first, followed by a comma, and then the y component. For example, the components of\texttip{\vec{A}}{A_vec} would be written 2.5,3 in ordered pair notation.

The answers below are all integers, so estimate the components to the nearest whole number.

Part E

In ordered pair notation, write down the components of vector \texttip{\vec{B}}{B_vec}.

Express your answers to the nearest integer.

ANSWER:

Correct

Part F

\texttip{C_{\mit x}}{C_x} = -2

\texttip{B_{\mit x}}{B_x}, \texttip{B_{\mit y}}{B_y} = 2,-3



In ordered pair notation, write down the components of vector \texttip{\vec{D}}{D_vec}.

Express your answers to the nearest integer.

ANSWER:

Correct

Part G

What is true about \texttip{\vec{B}}{B_vec} and \texttip{\vec{D}}{D_vec}? Choose from the pulldown list below.

ANSWER:

Correct

Exercise 1.40

In each case, find the x- and y-components of vector \vec A.

Part A

\vec A = 4.40 \hat i - 6.30 \hat j

Enter your answers numerically separated by a comma. Express your answers using three significantfigures.

ANSWER:

Correct

Part B

\vec A = 19.2 \hat j - 5.91 \hat i

Enter your answers numerically separated by a comma. Express your answers using three significant

\texttip{D_{\mit x}}{D_x}, \texttip{D_{\mit y}}{D_y} = 2,-3

They have different components and are not the same vectors.

They have the same components but are not the same vectors.

They are the same vectors.

A_x, A_y = 4.40,-6.30



figures.

ANSWER:

Correct

Part C

\vec A =- 14.0 \hat i+ 21.6 \hat j

Enter your answers numerically separated by a comma. Express your answers using three significantfigures.

ANSWER:

Correct

Part D

\vec A =5.0\hat B, where \hat B = 8 \hat i - 2 \hat j

Enter your answers numerically separated by a comma. Express your answers using one significantfigure.

ANSWER:

Correct

Vector Addition and Subtraction

In general it is best to conceptualize vectors as arrows in space, and then to make calculations with them using theircomponents. (You must first specify a coordinate system in order to find the components of each arrow.) This problemgives you some practice with the components.

Let vectors \vec{A} = (1, 0, -3), \vec{B} = (-2, 5, 1), and \vec{C} = (3, 1, 1). Calculate the following, and express youranswers as ordered triplets of values separated by commas.

Part A

ANSWER:

A_x, A_y = -5.91,19.2

A_x, A_y = -14.0,21.6

A_x, A_y = 40,-10



Correct

Part B

ANSWER:

Correct

Part C

ANSWER:

Correct

Part D

ANSWER:

Correct

Part E

ANSWER:

Correct

Part F

ANSWER:

\vec{A} - \vec{B} = 3,-5,-4

\vec{B}-\vec{C} = -5,4,0

- \vec{A} + \vec{B} - \vec{C} = -6,4,3

3\vec{A} -2\vec{C} = -3,-2,-11

- 2\vec{A} +3\vec{B} -\vec{C} = -11,14,8



Correct

Vector Addition: Geometry and Components

Learning Goal:

To understand how vectors may be added using geometry or by representing them with components.

Fundamentally, vectors are quantities that possess both magnitude and direction. In physics problems, it is best tothink of vectors as arrows, and usually it is best to manipulate them using components. In this problem we consider theaddition of two vectors using both of these methods. We will emphasize that one method is easier to conceptualize andthe other is more suited to manipulations.

Consider adding the vectors \texttip{\vec{A}}{A_vec} and \texttip{\vec{B}}{B_vec}, which have lengths \texttip{A}{A} and\texttip{B}{B}, respectively, and where \texttip{\vec{B}}{B_vec} makes an angle \texttip{\theta }{theta} from the direction of\texttip{\vec{A}}{A_vec}.

In vector notation the sum is represented by

\vec{C}=\vec{A} + \vec{B}.

Addition using geometry

Part A

Which of the following procedures will add the vectors \texttip{\vec{A}}{A_vec} and \texttip{\vec{B}}{B_vec}?

ANSWER:

2\vec{A} - 3(\vec{B} -\vec{C}) = 17,-12,-6



Correct

It is equally valid to put the tail of \vec{A} on the arrow of \vec{B}; then \vec{C} goes from the tail of \vec{B} to

the arrow of \vec{A}.

Part B

Find \texttip{C}{C}, the length of \texttip{\vec{C}}{C_vec}, the sum of \texttip{\vec{A}}{A_vec} and

\texttip{\vec{B}}{B_vec}.

Express \texttip{C}{C} in terms of \texttip{A}{A},

\texttip{B}{B}, and angle \texttip{\theta }{theta},

using radian measure for known angles.

Hint 1. Law of cosines

The law of cosines relates the lengths of the sides of any triangle of sides \texttip{A}{A}, \texttip{B}{B}, and

\texttip{C}{C}. Using the geometric notation where \texttip{c}{c} is the angle opposite side \texttip{C}{C}:

C 2^=A 2^ + B 2^ - 2AB \cos(c).

Hint 2. Interior and exterior angles

Note that \texttip{\theta }{theta} is an exterior angle that is the supplement of angle \texttip{c}{c}.

Express \texttip{\theta }{theta} in terms of \texttip{c}{c} and relevant constants such as

\texttip{\pi }{pi}, using radian measure for known angles.

ANSWER:

Put the tail of \vec{B} on the arrow of \vec{A}; \vec{C} goes from the tail of \vec{A} to the arrow of \vec{B}

Put the tail of \vec{A} on the tail of \vec{B}; \vec{C} goes from the arrow of \vec{B} to the arrow of \vec{A}

Put the tail of \vec{A} on the tail of \vec{B}; \vec{C} goes from the arrow of \vec{A} to the arrow of \vec{B}

Calculate the magnitude as the sum of the lengths and the direction as midway between \vec{A} and \vec{B}.



ANSWER:

Correct

Part C

Find the angle \texttip{\phi }{phi} that the vector \texttip{\vec{C}}{C_vec} makes with vector \texttip{\vec{A}}{A_vec}.

Express \texttip{\phi }{phi} in terms of \texttip{C}{C} and any of the quantities given in the problem

introduction ( \texttip{A}{A}, \texttip{B}{B}, and/or \texttip{\theta }{theta}) as well as any necessary

constants. Use radian measure for known angles. Use asin for arcsine

Hint 1. Law of sines

Although this angle can be determined by using the law of cosines with \texttip{\phi }{phi} as the angle, this

results in more complicated algebra. A better way is to use the law of sines, which in this case is

\large{\frac{\sin(\phi)}{B}=\frac{\sin(c)}{C}=\frac{\sin(a)}{A}}.

ANSWER:

Correct

Addition using vector components

Part D

To manipulate these vectors using vector components, we must first choose a coordinate system. In this casechoosing means specifying the angle of the x axis. The y axis must be perpendicular to this and by convention isoriented \pi/2 radians counterclockwise from the x axis.

Indicate whether the following statement is true or false:There is only one unique coordinate system in which vector components can be added.

ANSWER:

\texttip{\theta }{theta} = {\pi}-c

\texttip{C}{C} = \sqrt{A {^2}+B {^2}-2AB {\cos}\left({\pi}-{\theta}\right)}

\texttip{\phi }{phi} = \large{{\asin}\left(\frac{B {\sin}\left({\pi}-{\theta}\right)}{C}\right)}



Correct

Part E

The key point is that you are completely free to choose any coordinate system you want in which to manipulate thevectors. It is a matter of convenience only, and so you must consider which orientation will simplify finding thecomponents of the given vectors and interpreting the results in that coordinate system to get the required answer.Considering these factors, and knowing that you are going to be required to find the length of \texttip{\vec{C}}{C_vec}

and its angle with respect to \texttip{\vec{A}}{A_vec}, which of the following orientations simplifies the calculation the

most?

ANSWER:

Correct

Part F

Find the components of \texttip{\vec{B}}{B_vec} in the coordinate system shown.

Express your answer as an ordered pair: xcomponent, y component ; in terms of \texttip{B}{B} and \texttip{\theta }{theta}. Use radian

measure for known angles.

ANSWER:

true

false

The x axis should be oriented along \vec{A}

The x axis should be oriented along \vec{B}

The x axis should be oriented along \vec{C}



Correct

Part G

In the same coordinate system, what are the components of \texttip{\vec{C}}{C_vec}?

Express your answer as an ordered pair separated by a comma. Give your answer in terms of variablesdefined in the introduction ( \texttip{A}{A}, \texttip{B}{B}, and \texttip{\theta }{theta}). Use radian measure

for known angles.

ANSWER:

Correct

This should show you how easy it is to add vectors using components. Subtraction is similar except that thecomponents must be subtracted rather than added, and this makes it important to know whether you arefinding \vec{D}=\vec{A} - \vec{B} or \vec {E}=\vec{B} - \vec{A}. (Note that \vec{E}=-\vec{D}.)

Although adding vectors using components is clearly the easier path, you probably have no immediate picturein your mind to go along with this procedure. Conversely, you probably think of adding vectors in the way we'vedrawn the figure for Part B.

This justifies the following maxim: Think about vectors geometrically; add vectors using components.

Adding and Subtracting Vectors Conceptual Question

Six vectors (A to F) have the magnitudes and directions indicated in the figure.

B_x,B_y = B {\cos}\left({\theta}\right),B {\sin}\left({\theta}\right)

C_x,C_y = A+B {\cos}{\theta},B {\sin}{\theta}



Part A

Which two vectors, when added, will have the largest (positive) x component?

Hint 1. Largest x component

The two vectors with the largest x components will, when combined, give the resultant with the largest xcomponent. Keep in mind that positive x components are larger than negative x components.

ANSWER:

Correct

Part B

Which two vectors, when added, will have the largest (positive) y component?

Hint 1. Largest y component

The two vectors with the largest y components will, when combined, give the resultant with the largest ycomponent. Keep in mind that positive y components are larger than negative y components.

ANSWER:

Correct

Part C

Which two vectors, when subtracted (i.e., when one vector is subtracted from the other), will have the largestmagnitude?

C and E

E and F

A and F

C and D

B and D

C and D

A and F

E and F

A and B

E and D



Hint 1. Subtracting vectors

To subtract two vectors, add a vector with the same magnitude but opposite direction of one of the vectors tothe other vector.

ANSWER:

Correct

Exercise 1.37

A disoriented physics professor drives a distance 3.30{\rm km} north, then a distance 4.65{\rm km} west, and then adistance 1.20{\rm km} south.

Part A

Find the magnitude of the resultant displacement, using the method of components.

ANSWER:

Correct

Part B

Find the direction of the resultant displacement, using the method of components.

ANSWER:

Correct

Vector Components--Review

A and F

A and E

D and B

C and D

E and F

5.10 {\rm km}

65.7 \^circ west of north



Learning Goal:

To introduce you to vectors and the use of sine and cosine for a triangle when resolving components.

Vectors are an important part of the language of science, mathematics, and engineering. They are used to discussmultivariable calculus, electrical circuits with oscillating currents, stress and strain in structures and materials, andflows of atmospheres and fluids, and they have many other applications. Resolving a vector into components is aprecursor to computing things with or about a vector quantity. Because position, velocity, acceleration, force,momentum, and angular momentum are all vector quantities, resolving vectors into components is the most importantsk ill required in a mechanics course.

The figure shows the components of \texttip{\vec{F}}{F_vec},\texttip{F_{\mit x}}{F_x} and \texttip{F_{\mit y}}{F_y}, along thex and y axes of the coordinate system, respectively. Thecomponents of a vector depend on the coordinate system'sorientation, the key being the angle between the vector andthe coordinate axes, often designated \texttip{\theta }{theta}.

Part A

The figure shows the standard way of measuring theangle. \texttip{\theta }{theta} is measured to the vector

from the x axis, and counterclockwise is positive.

Express \texttip{F_{\mit x}}{F_x} and

\texttip{F_{\mit y}}{F_y} in terms of the length of the

vector \texttip{F}{F} and the angle

\texttip{\theta }{theta}, with the components

separated by a comma.

ANSWER:

\texttip{F_{\mit x}}{F_x}, \texttip{F_{\mit y}}{F_y} = F {\cos}{\theta},F {\sin}{\theta}



Correct

In principle, you can determine the components of any vector with these expressions. If \texttip{\vec{F}}{F_vec}

lies in one of the other quadrants of the plane, \texttip{\theta }{theta} will be an angle larger than 90 degrees (or

\pi/2 in radians) and \cos(\theta) and \sin(\theta) will have the appropriate signs and values.

Unfortunately this way of representing \texttip{\vec{F}}{F_vec}, though mathematically correct, leads to

equations that must be simplified using trig identities such as

\sin(180 \^circ + \phi) = -\sin(\phi)

and

\cos(90 \^circ + \phi) = -\sin(\phi).

These must be used to reduce all trig functions present in your equations to either \sin(\phi) or \cos(\phi).

Unless you perform this followup step flawlessly, you will fail to recoginze that

\sin(180 \^circ + \phi) + \cos(270 \^circ - \phi) = 0,

and your equations will not simplify so that you can progress further toward a solution. Therefore, it is best toexpress all components in terms of either \sin(\phi) or \cos(\phi), with \texttip{\phi }{phi} between 0 and 90

degrees (or 0 and \pi/2 in radians), and determine the signs of the trig functions by knowing in which quadrant

the vector lies.

Part B

When you resolve a vector \texttip{\vec{F}}{F_vec} into components, the components must have the form

\vert \vec{F}\vert\cos(\theta) or \vert\vec{F}\vert\sin(\theta). The signs depend on which quadrant the vector lies in,

and there will be one component with \sin(\theta) and the other with \cos(\theta).

In real problems the optimal coordinate system is often rotated so that the x axis is not horizontal. Furthermore,most vectors will not lie in the first quadrant. To assign the sine and cosine correctly for vectors at arbitrary angles,you must figure out which angle is \texttip{\theta }{theta} and then properly reorient the definitional triangle.

As an example, consider the vector \texttip{\vec{N}}{N_vec} shown in the diagram labeled "tilted axes," where you

know the angle \texttip{\theta }{theta} between \texttip{\vec{N}}{N_vec} and the y axis.

Which of the various ways of orienting the definitional triangle must be used to resolve \texttip{\vec{N}}{N_vec} into

components in the tilted coordinate system shown? (In the figures, the hypotenuse is orange, the side adjacent to \texttip{\theta }{theta} is red, and the side opposite is yellow.)



Indicate the number of the figure with the correct orientation.

Hint 1. Recommended procedure for resolving a vector into components

First figure out the sines and cosines of \texttip{\theta }{theta}, then figure out the signs from the quadrant

the vector is in and write in the signs.

Hint 2. Finding the trigonometric functions

Sine and cosine are defined according to the following convention, with the key lengths shown in green: Thehypotenuse has unit length, the side adjacent to \texttip{\theta }{theta} has length \cos(\theta), and the side

opposite has length \sin(\theta). The colors are chosen to remind you that the vector sum of the two

orthogonal sides is the vector whose magnitude is the hypotenuse; red + yellow = orange.



ANSWER:

Correct

Part C

Choose the correct procedure for determining the components of a vector in a given coordinate system from thislist:

ANSWER:

Correct

Part D

1

2

3

4

Align the adjacent side of a right triangle with the vector and the hypotenuse along a coordinate directionwith \theta as the included angle.

Align the hypotenuse of a right triangle with the vector and an adjacent side along a coordinate directionwith \theta as the included angle.

Align the opposite side of a right triangle with the vector and the hypotenuse along a coordinate directionwith \theta as the included angle.

Align the hypotenuse of a right triangle with the vector and the opposite side along a coordinate directionwith \theta as the included angle.



The space around a coordinate system is conventionally divided into four numbered quadrants depending on thesigns of the x and y coordinates . Consider the followingconditions:

A. x > 0, y > 0

B. x > 0, y < 0

C. x < 0, y > 0

D. x < 0, y < 0

Which of these lettered conditions are true in which thenumbered quadrants shown in ?

Write the answer in the following way: If A weretrue in the third quadrant, B in the second, C in thefirst, and D in the fourth,enter "3, 2, 1, 4" as your response.

ANSWER:

Correct

Part E

Now find the components \texttip{N_{\mit x}}{N_x} and \texttip{N_{\mit y}}{N_y} of \texttip{\vec{N}}{N_vec} in the

tilted coordinate system of Part B.

Express your answer in terms of the length of the vector \texttip{N}{N} and the angle

\texttip{\theta }{theta}, with the components separated by a comma.

ANSWER:

1,4,2,3

\texttip{N_{\mit x}}{N_x}, \texttip{N_{\mit y}}{N_y} = -N {\sin}{\theta},N {\cos}{\theta}



Correct

Vector Addition Ranking Task

Six vectors (\texttip{\vec{a}}{a_vec} through \texttip{\vec{f}}{f_vec}) have the magnitudes and directions indicated in thefigure.

Part A

Rank the vector combinations on the basis of their magnitude.

Rank from largest to smallest. To rank items as equivalent, overlap them.

Hint 1. Adding vectors graphically

To add two vectors together, imagine sliding one vector (without rotating it) until its tail coincides with the tipof the second vector. The sum of the two vectors, termed the resultant vector \texttip{\vec{R}}{R_vec}, is the

vector that goes from the tail of the first vector to the tip of the second vector. The magnitude of the resultant,|\vec R|, is determined by the sum of the squares of its x and y components, that is,

|\vec R| = \sqrt {R_x 2^ + R_y 2^}.

ANSWER:



Correct

Part B

Rank the vector combinations on the basis of their angle, measured counterclockwise from the positive x axis.Vectors parallel to the positive x axis have an angle of 0 \^circ . All angle measures fall between 0 \^circ and

360 \^circ.

Rank from largest to smallest. To rank items as equivalent, overlap them.

Hint 1. Angle of a vector

The angle of a vector \angle \vec V is to be measured counterclockwise from the x axis, with the x axis as 0

\^circ. The following vectors are at the angles listed and are shown on the graph below.

\angle \vec V_1 = 0 \^circ







Notice that the magnitude of the vector is irrelevant when determining its angle

ANSWER:

Correct

Exercise 1.41



Write each vector in the figure in terms of the unit vectors \hati and \hat j.

Part A

ANSWER:

Correct

Part B

ANSWER:

Correct

Part C

ANSWER:

Correct

Part D

ANSWER:

\vec A = 0 \hat{i},-8 \hat{j}

\vec B = 7.50 \hat{i}+13.0 \hat{j}

\vec C = -10.9 \hat{i}-5.07 \hat{j}



Correct

PSS 1.3 Vector Addition

Learning Goal:

To practice Problem-Solving Strategy 1.3 Vector Addition.

Three people pull horizontally on ropes attached to a post, as shown in the figure. The total force that they exert on thepost is zero. One of them pulls directly north with\texttip{F_{\rm 1}}{F_1} = 500{\rm N} . Another pulls with\texttip{F_{\rm 2}}{F_2} = 400{\rm N} in a direction 60.0 {^\circ}west of north. Find the magnitude and direction of the force\texttip{\vec{F}_{\rm 3}}{F_3_vec} exerted by the third personso that the sum of the three forces is zero.

Problem-Solving Strategy: Vector addition

IDENTIFY the relevant concepts: Decide what your target variable is. It may be the magnitude of the vector sum, the direction, or both.

SET UP the problem using the following steps: Draw the individual vectors being summed and the coordinate axes being used. In your drawing, place the tail of the firstvector at the origin of the coordinates; place the tail of the second vector at the head of the first vector; and so on. Drawthe vector sum \texttip{\vec{R}}{R_vec} from the tail of the first vector to the head of the last vector. Use your drawing tomake rough estimates of the magnitude and direction of \texttip{\vec{R}}{R_vec}; you'll use these estimates later tocheck your calculations.

EXECUTE the solution as follows:

1. Find the x and y components of each individual vector, and record your results in a table. Be particularlycareful with signs: Some components may be positive, and some may be negative.

2. Add the individual x components algebraically, including signs, to find \texttip{R_{\mit x}}{R_x}, the x

component of the vector sum. Do the same for the y components to find \texttip{R_{\mit y}}{R_y}.

3. Then, the magnitude \texttip{R}{R} and direction \texttip{\theta }{theta} of the vector sum are given by

R=\sqrt {R_x 2^ +R_y 2^} \qquad \theta={\rm arctan}{R_y \over R_x}.

EVALUATE your answer:

\vec D = -7.99 \hat{i}+6.02 \hat{j}



Check your results for the magnitude and direction of the vector sum by comparing them with the rough estimates youmade from your drawing. Keep in mind that the magnitude \texttip{R}{R} is always positive and that \texttip{\theta }{theta} is measured from the +x axis. The value of \texttip{\theta }{theta} that you find with a calculator may be thecorrect one, or it may be off by 180 degrees. You can decide by examining your drawing.

IDENTIFY the relevant concepts

Force is a vector quantity, so vector addition must be used to add the three forces acting on the post. This problemrequires you to find both the magnitude and direction of one of the forces, such that the total vector sum is zero.

SET UP the problem using the following steps

Part A

Add vectors to the diagram below to indicate the vector sum of the three forces acting on the post: {\vec R}={\vec F_1}+{\vec F_2}+{\vec F_3}=0. Use the conventional choice of coordinates, with the +x axis as east

and the +y axis as north.

Draw vectors \texttip{\vec{F}_{\rm 1}}{F_1_vec}, \texttip{\vec{F}_{\rm 2}}{F_2_vec}, and

\texttip{\vec{F}_{\rm 3}}{F_3_vec} on the diagram below such that {\vec F_1}+{\vec F_2}+{\vec F_3}=0.

Draw \texttip{\vec{F}_{\rm 1}}{F_1_vec} starting from the origin of the axes provided. Each unit on the

graph is 100 \rm N.

Hint 1. How to approach the problem

You should draw the three vectors corresponding to the forces applied to the post, adding them head to tailas outlined in the Set Up step of the strategy. You have enough information to straightforwardly draw thevectors \texttip{\vec{F}_{\rm 1}}{F_1_vec} and \texttip{\vec{F}_{\rm 2}}{F_2_vec}. By considering that the total

force exerted on the post is zero, you can determine how to draw the unknown force \texttip{\vec{F}_{\rm 3}}{F_3_vec}.

Hint 2. How to draw the vector of the force exerted by the third person

The total force on the post is zero: {\vec F_1}+{\vec F_2}+{\vec F_3}=0. When a set of vectors sum to zero,

the head of the last vector must be located at the tail of the first vector. \texttip{\vec{F}_{\rm 3}}{F_3_vec} is

therefore drawn from the head of \texttip{\vec{F}_{\rm 2}}{F_2_vec} to the tail of

\texttip{\vec{F}_{\rm 1}}{F_1_vec}.

ANSWER:



Correct

EXECUTE the solution as follows

Part B

Calculate the magnitude of the force exerted by the third person.

Express the magnitude of \texttip{\vec{F}_{\rm 3}}{F_3_vec} in newtons to three significant figures.

Hint 1. How to approach the problem

The total force on the post is zero: {\vec F_1}+{\vec F_2}+{\vec F_3}=0.

This can be written as {\vec F_3}=- \left ({\vec F_1}+{\vec F_2}\right ).

Add the x components of \texttip{\vec{F}_{\rm 1}}{F_1_vec} and \texttip{\vec{F}_{\rm 2}}{F_2_vec} to find the

negative of the x component of \texttip{\vec{F}_{\rm 3}}{F_3_vec}. Similarly, add the y components of

\texttip{\vec{F}_{\rm 1}}{F_1_vec} and \texttip{\vec{F}_{\rm 2}}{F_2_vec} to find the negative of the y

component of \texttip{\vec{F}_{\rm 3}}{F_3_vec}. Once you know \texttip{\vec{F}_{\rm 3}}{F_3_vec}, use the

formula

R=\sqrt {R_x 2^ +R_y 2^}

from the introduction.



Hint 2. Determine the vector components of the force exerted by the second person

It is important to use the correct angle and trigonometric function when calculating the component of avector. If the direction of the vector (with magnitude \texttip{A}{A}) is described by the angle

\texttip{\theta }{theta} that it makes with the +x axis, then its components are

A_x = A \cos \theta \qquad A_y = A \sin \theta.

However, the angle given in this problem is not measured from the +x axis. You should use geometry tocalculate an angle from the +x axis before using the above formulas.

Calculations using the angle measured from the +x axis will also give the correct sign (positive or negative)for the components.

Consider the force vector \texttip{\vec{F}_{\rm 2}}{F_2_vec} = 400{\rm N} , 60.0 {^\circ} west of north.

Calculate the x and y components of the force, \texttip{F_{\rm 2x}}{F_2x} and \texttip{F_{\rm 2y}}{F_2y},

using the conventional choice of coordinates, with the +x axis as east and the +y axis as north.

Express the x and y components of \texttip{\vec{F}_{\rm 2}}{F_2_vec} in newtons, separated by a

comma, to three significant figures.

ANSWER:

ANSWER:

Correct

Part C

Determine the direction in which the third person is pulling.

Express the angle in degrees ( 0 \leq \theta < 360^{\circ}) that \texttip{\vec{F}_{\rm 3}}{F_3_vec} makes,

counterclockwise, with the +x axis. Express your answer to three significant figures.

ANSWER:

Correct

EVALUATE your answer

Part D

\texttip{F_{\rm 2x}}{F_2x}, \texttip{F_{\rm 2y}}{F_2y} = -346,200 {\rm N}

F_3 = 781 {\rm N}

\texttip{\theta }{theta} = 296 \^circ



By adding vectors using components, you have been able to determine an unknown force, both magnitude anddirection, to three significant figures:

\texttip{\vec{F}_{\rm 3}}{F_3_vec} = 781 \rm N, 63.7 {^\circ} south of east.

You can check your answer by comparing it to the diagram in Part A above.

Some other students calculated \texttip{\vec{F}_{\rm 3}}{F_3_vec}, not as correctly as you, and came up with a set

of different values. Pretend that you do not know the actual value of \texttip{\vec{F}_{\rm 3}}{F_3_vec}, and by

considering the vector addition diagram in Part A, decide which of the following values could possibly be correct,and which are definitely incorrect.

Drag the appropriate values to their respective bins.

ANSWER:

Correct

Score Summary:

Your score on this assignment is 98.9%.You received 78.13 out of a possible total of 79 points.

ch 01 vector addition and vector components

Documents

vector addition

vector componentshttp

displacement vector

vector componentsdue

vector texttip

magnitude equal

correctpart efind

correctpart cwhich