up benson problem ch02

7/25/2019 UP Benson Problem CH02

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University Physics

Harris Benson

CH02Vectors

Exercise 2-1(I) The magnitude of the vectors A and B shown in Fig. 2.24 are 3mA = and

2mB = . Find graphically: (a) +A B (b) A B .

2-24 0 03cos30 3sin 30 2.60 1.5= + = +A i j i j

0 02cos 45 2sin 45 1.41 1.41= = B i j i j

(a) 4.01 0.09+ = +A B i j

(b) 1.19 2.91 = +A B i j

2-24-1

2-24-1

Exercise 2-2(I) The magnitude of the vectors C and D shown in Fig. 2.25 are 4 mC = and

2.5mD = . Find graphically: (a) +C D (b) C D .

2-25


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0 04sin 60 4cos60 3.46 2= = C i j i j 2.5=D j

(a) 3.46 0.5+ = +C D i j

(b) 3.46 0.5 = C D i j

Exercise 2-3(I) For the three vectors shown in Fig. 2.26, take 1.5mA = , 2mB = , and 1mC = .

Find graphically: (a) + +A B C (b) A B C .

2-26 0 01.5cos 20 1.5sin 20 1.41 0.51= + = +A i j i j

0 02cos 20 2sin 20 1.88 0.68= = B i j i j 0 0cos60 sin 60 0.5 0.87= = C i j i j

(a) 2.79 1.05+ + = A B C i j

(b) 0.03 2.06 = +A B C i j

Exercise 2-4(I) Consider the three vectors shown in Fig. 2.26. Find the vector D which when

added to + A B C , produces a null vector.

3.79 0.7+ = +A B C i j

( ) 0+ + =D A B C

3.79 0.7= = D C A B i j

Exercise 2-5(I) Three vectors have equal magnitude of 10m. Draw a vector diagram to illustrate how

the magnitude of their resultant can be: (a) 0; (b) 10m; (c) 20m; (d) 30m.


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Exercise 2-6(II) Two vectors have equal magnitudes of 2m. Find graphically the angle between them

if the magnitude of their resultant is (a) 3m; (b) 1m. In each case use the law of cosines

to confirm your answer.

(a) 083(b) 0151

Exercise 2-7(I) The resultant of two vectors A and B is 40m due north. If A is 30m in the

direction0

30 S of W, find B graphically.

60 m at 25 E of N

In the following problems take the x axis to point east, the y axis to point north, and, if

needed, the z axis upward. An unknown vector should be expressed in terms of its

components:x y

V V= +V i j . The magnitude and direction can be found from the

components.

Exercise 2-8(I) A person undergoes a displacement of 4m in the direction 040 W of N followed by

a displacement of 3m at 020 S of W. Find the magnitude and direction of the resultant

displacement.

Exercise 2-9(I) Three vectors are specified as follows: A is 5m at 045 N of E, B is 7m at 060

E of S, and C is 4m at 030 W of S. Find the magnitude and direction of their sum.


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Exercise 2-10(I) A person walks 5m south and then 12m west. What is the net displacement?

= 12 5 i j

= 2 212 5 13m+ = 1 05tan 22.612

=

Exercise 2-11(I) An insect walks 50cm in a straight line along a wall. If its horizontal displacement is

25cm, what is its vertical displacement?

2 250 25 43.3m =

Exercise 2-12(II) An airplane is flown in the direction 030 W of N. If the magnitude of the westerly

component of the displacement is 100km, how far north does it travel?

0100 173mtan30y = =

Exercise 2-13(I) Four vectors each of magnitude 2m, are shown in Fig. 2.27. (a) Express each in unit

vector notation. (b) Express their sum in unit vector notation. (c) What is the magnitude

and direction of the sum?

2-27(a)

(b)


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(c)

(d)

Exercise 2-14(I) Each of the vectors in Fig. 2.28 has a magnitude of 4m. (a) Express each in unit

vector notation. (b) Express their sum in unit vector notation. (c) What is the magnitude

and direction of their sum?

2-28

Exercise 2-15(I) Given two vectors, 2 3 m= +A i j k and 2 m= + B i j k , find: (a) R = A + B ;

(b) R ; (c) R.

(a) (2 3 ) ( 2 ) m= + + + = R i j k i j k i j

(b) 2 21 ( 1) 2 1.414 mR = + = =

(c)1 [ ( )] (0.707 0.707 ) m2

mR

= = = R

R i j i j

Exercise 2-16(I) Given two vectors, 4 3 m= + C i j k and 2 3 5 m= +D i j k , find: (a) S = C - D ;

(b) S; (c) S .

(a) (4 3 ) (2 3 5 ) 2 4 8 m= + + = + S i j k i j k i j k

(b) 2 2 22 4 ( 8) 84 9.17 mS = + + = =

(c) 0.218 0.436 0.873S

= = + S

S i j k

Exercise 2-17


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(II) The vector A has a magnitude of 6m and vector B has a magnitude of 4m. What

is the angle between them if the magnitude of their resultant is (a) the maximum

possible; (b) the minimum possible; (c) 3m; and (d) 8m. Do each part graphically and

by components. (Let A lie along thexaxis.)

(a) 0 (b) 0180

(c) 2 2(6 4cos ) (4sin ) 9 + + =

cos 0.896 =

0153 =

(d) 2 2(6 4cos ) (4sin ) 64 + + =

cos 0.25 =

0

75.5 =

Exercise 2-18(I) The resultant R of two displacements is 10m at 037 W of N. If the second

displacement was 6m at 053 N of E, what was the first?

Exercise 2-19(I) In a yacht race the boats sail around three buoys as shown in Fig. 2.29. What is the

displacement from the last buoy to the starting point? Express your answer (a) in unit

vector notation, and (b) as a magnitude and sirection.

2-29

Exercise 2-20


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such that = +C A B , but whose direction is perpendicular to +A B .

Exercise 2-25(I) Given two vector, 2= + A i j k and 3 2= + +B i j k , find the unit vector in the

direction of 2 3= S B A .

Exercise 2-26

(I) Given two vector, 5 2= +A i j , and 2 3= B i j , find (a) A B+ ; (b) +A B ; (c)

A B ; (d) A B .

Exercise 2-27(I) If vector 6 2 3 m= +A i j k , find (a) a vector in the same direction as A with

magnitude 2A ; (b) a unit vector in the direction of A ; (c) a vector opposite to A

with a magnitude of 4m.

Exercise 2-28(I) Given two vectors, 2 3= +A i j k and 4 5= + B i j k , find a third vector, C ,

such that 2 03

+ =C

A B .

2 03

+ =C

A B

3 6 3(2 3 ) 6( 4 5 )= + = + + + C A B i j k i j k

30 15 33= + i j k


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Exercise 2-29(II) Show that if the sum of three vectors is zero, they must all lie in the same plane.

Does the same restriction apply to the zero sum of four vectors?

Exercise 2-30

(I) The vectors A and B have the following components: 2mx

A = , 3.5myA = ,

1.5mxB = , and 2.5myB = . Find the magnitude and direction of 3 2= C A B .

Exercise 2-31(I) Find the components of the following vectors: (a) P of length 5m directed at 0150

counterclockwise from the +x axis; (b) Q of length 3.6m directed at

0

120 clockwisefrom the +y axis.

Exercise 2-32(I) A body moves from a position with coordinates (3m, 2m) to ( 4m,4m) . Find its

displacement (a) in unit vector notation, and (b) in terms of its magnitude and direction.

Exercise 2-33(I) Given that vector A is 5m at 030 N of E, find vector B such that their sum is

directed along the negativexaxis and has a magnitude of 3m.


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Exercise 2-34(II) The hour hand of a clock is 6cm along. Take 12 noon to lie along the y axis and 3

p.m. to lie along the x axis. Find the displacement (in unit vector notation) of the tip

between each of the following times: (a) 1 p.m. to 4 p.m.; (b) 2 p.m. to 9:30 p.m.

Exercise 2-35(II) Figure 2.31 shows the directions of three vectors whose magnitudes in arbitrary

units are 20W = , 10F = , and 30T = . The x and y axes are titled as shown. Find: (a)

the components of the vectors; (b) their sum in unit vectornotation.

2-31

Exercise 2-36(I) The instruction to a treasure hunt state: Start at the oak tree and walk 5m in a straight

line at 030 W of N. Turn to your right through 045 and walk 4m. Dig a hole 2m deep.

Where is the treasure relative to the base of the tree?

Exercise 2-37

(II) Given the vector 2 3 m= +A i j , find the vector B of length 5m that is

perpendicular to A and lies in the following planes: (a) thexzplane; (b) thexyplane.

Exercise 2-38(I) A helicopter rises 100m from its pad and travels a horizontal distance of 200m at

025 S of W. What is its displacement?


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181 84.5 100 m +i j k

Exercise 2-39(I) What is the angle between the vectors 2= A i j and 2 3= +B i j ?

2 2 2 2

( 2 ) (2 3 ) 4cos

(5)(13)( 1 ( 2) )( 2 3 )AB

+ = = =

+ +

A B i j i j

1 04cos 12064

= =

Exercise 2-40(I) Given the vectors 2 3= + A i j k and 5 2= + B i j k , find: (a) A B , and (b)

( ) ( )+ A B A B

(a) ( 2 3 ) (5 2 ) 10 2 3 5 = + + = + + = A B i j k i j k (b) 2 2( ) ( ) A B+ = A B A B

2 2 2 2 2 2

[( 2) 1 ( 3) ] [5 2 ( 1) ] 14 30 16= + + + + = =

Exercise 2-41(I) The dot product of two vectors, whose magnitudes are 3m and 5m, is 24m . What

is the angle between them?

24m 4

cos(3 m)(5 m) 15AB

= = =A B

1 04cos 105.515

= =

Exercise 2-42

(I) The components of two vectors are 2.4x

A = , 1.2y

A = , 4.0z

A = , and 3.6x

B = ,

1.8y

B = , 2.6z

B = . Find the angle between them.


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2.4 1.2 4= +A i j k 2 2 2(2.4) ( 1.2) (4) 4.82A = + + =

3.6 1.8 2.6= +

B i j k

2 2 2

( 3.6) (1.8) ( 2.6) 4.79B= + + =

21.2cos 0.918

(4.82)(4.79)AB

= = =

A B

0157 =

Exercise 2-43(I) The vectors A and B are in the xy plane where A is 3.2m at 045 to the +x

axis, and B is 2.4m at0

290 to +x axis. Find A B .

0 2(3.2)(2.4)cos115 3.25m = = A B

Exercise 2-44(II) The Vector A and B in Fig 2.32 define two sides of a parallelogram. (a) Express the

diagonals in terms of A and B . (b) Show that the diagonals are perpendicular if

A B=

.

2-32(a) +A B

A B or B A

(b) ( )( ) 0+ =A B A B A B=

Exercise 2-45(II) Show that the angles , , and between a vector A and thex,y, andzaxes,

respectively, are given by cosA

=A i

, cosA

=A j

, cosA

=A k

. If

3 2= + +A i j k , find the angle between A and each axis.

3 2= + +

A i j k

2 2 2

3 2 1 14= + + =

A


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3cos

14 = 036.7 =

2cos

14 = 057.7 =

1cos

14 = 074.5 =

Exercise 2-46(II) Given the three vector 4= A i j , 3=B i , and 2= C j evaluate the following

expressions if they are allowed mathematically: (a) ( ) +C A B ; (b) ( ) C A B ; (c)

C+ A B ; (d) ( )C A B ; (e) ( )C A B .

Exercise 2-47(II) What is the component of the vector 2 m= +A i j k in the direction of the vector

3 4 m= +B i k ?

Exercise 2-48(I) Give two vectors, 2 4= + A i j k and 3 5= +B i j k , find A B .

( 2 4 ) (3 5 ) = + +A B i j k i j k

1 2 43 1 5

i j k

=

2 4 1 4 1 2

1 5 3 5 3 1

= +

i j k

(10 4) (5 12) ( 1 6)= + + i j k

6 17 7= i j k

Exercise 2-49


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(I) (a) Show, for arbitrary vectors A and B , that ( ) 0 =A A B . (b) How could you

have arrived at this conclusion without any computation?

(a)(b) ( ) A A B

Exercise 2-50(I) Vectors A and B are in the xy plane with 3.6mA = at 025 counterclockwise

from the +x axis, and 4.4mB = at 0160 counterclockwise from +x axis . Find

A B .

0 0

3.6 cos 25 3.6 sin 25= +A i j 0 04.4cos160 4.4sin160= +B i j

A B

Exercise 2-51

(II) Show that the area of a parallelogram, as in Fig. 2.32, is A B .

= ( )( sin )B A =

= A B

Exercise 2-52(II) Given three vectors, 2 5= A i j , 4=B j , and 3=C i , evaluate the following

expressions if they are mathematically allowed: (a) ( )C A B ; (b) ( ) C A B ; (c)

( ) C A B ; (d) ( ) C A B ; (e) + C A B .

(a)(b)

(c)

(d)

(e)


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Exercise 2-53(II) Vector A has a magnitude of 4m and lies in the xy plane directed at 045

counterclockwise from the +x axis, whereas B has a magnitude of 3m and lies in the

yzplane directed at 030 clockwise from the +z axis; see Fig. 2.33. Find A B .

2-33 0 04cos 45 4sin 45 2 2 2 2 2.83 2.83= + = + = +A i j i j i j

0 0 3 3 33sin 30 3cos30 1.5 2.62 2

= + = + = +B j k j k j k

7.36 7.36 4.25 = +A B i j k

Exercise 2-54(II) Find a vector of length 5m that is perpendicular to both vectors 3 2 4 m= +A i j k and 4 3 m= B i j k .

2 2 2

14 19

14 19 1

+ = =

+ +

A B i j k n

A B

1=n

70 95 5

5 2.96 4.02 0.21558 558 558= + = + n i j k i j k

Exercise 2-55(I) Vector A is 5m directed at 035 above the +x axis, both in the xy plane. Find

A B . Draw a vector diagram.


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Exercise 2-56(I) Vector A is 5m directed at 030 W of N and vector B is 3m directed at 020 s of

W. Find +A B . Draw a vector diagram.

Exercise 2-57(I) Vector A is 4m directed at 035 above the +x axis and B is 2.5m directed at 020

above the x axis, both in thexyplane. Find B A . Draw a vector diagram.

Exercise 2-58(I) Given the three vector 2m=A at 028 N of E, 1.5m=B at 025 W of N, and

2.5m=C due E, find 2= + D A B C . Draw a vector diagram.

Exercise 2-59(i) A person starts a two-stage journey at the origin and first walks 6m in the direction

050 N of E. The final position is 3.5m from the origin in the direction 035 N of W.

What was the second stage? Draw a vector diagram.

Exercise 2-60

(I) For the vector A we know 2mx

A = , 1.5my

A = and 4mA = . What isz

A ?

2 2 2x y zA A A= + +A

2 2 2 2

4 ( 2) (1.5) ( )zA= + +


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2 2 2(4) ( 2) (1.5) 9.75 3.12mz

A = = =

Exercise 2-61

(I) The vector 2 3 m= +A i j and the sum +A B is 4m long directed at 0120 to the

+xaxis in thexyplane. What is B ?

Exercise 2-62(I) Vector A is 4m along the +xaxis and +A B is 6m along the +yaxis. Determine

B .

Exercise 2-63

(I) Three vectors in thexyplane have equal magnitude and their sum is zero. If one is2 mi , express the other two in unit vector notation.

Exercise 2-64(II) The vector A is 2m directed at 030 N of E and B is directed at 050 N of W.

The sum +A B is 2.6m long. Find graphically (a) B , and (b) the direction of +A B .

(A compass is needed.)

Exercise 2-65(II) Vectors A and B , both in the xyplane, have the same magnitude. Vector A is

directed at 030 above the +x axis and B isperpendicular to A , where


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2.12m+ =A B . (a) Find A and B . Determine +A B given that yB is (b) positive,

or (c) negative.

Exercise 2-66(II) Vector A is 2m directed at 030 N of E, the sum +A B is directed at 020 W of

N, and 3.5mB = . Graphically find (a) +A B , (b) the direction of B . (A component

is needed.)

Exercise 2-67(I) Two vectors are given by 3 2 m= +P i j k and 2 4 m= +Q i j k . Fins: (a)

+P Q ; (b) +P Q ; (c) P Q+ .

(a) (3 2 ) ( 2 4 ) 4 3 6 m+ = + + + = +P Q i j k i j k i j k

(b) 2 2 24 ( 3) 6 61 7.81m+ = + + = =P Q

(c) 2 2 2 2 2 23 ( 1) 2 1 ( 2) 4P Q+ = + + + + +

14 21 3.74 4.58 8.32m= + = + =

Exercise 2-68(I) Two vector are 2 3 m= P i j and 2 m= +Q i j . (a) What is Q P ? (b) What is

the distance between the tips of the vectors?

(a) ( 2 ) (2 3 ) 3 5 m = + = +Q P i j i j i j

(b) 2 23 5 5.83mQ P = + =


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Exercise 2-69(I) Given the vectors 4 2 m= +A i j k and 2 3 m= + B i j k , find: (a) A B (b)

A B (c) A B .

(a) (4 2 ) ( 2 3 ) 6 4 3 m = + + = +A B i j k i j k i j k

(b) 2 2 26 ( 4) 3 7.81m = + + =A B

(c) 2 2 2 2 2 24 ( 1) 2 ( 2) 3 ( 1)A B = + + + +

21 14 4.583 3.742 0.841m= = =

Exercise 2-70(I) Given (3 2 ) m= +R i j k , what is the vector that is twice as long in the same

direction?

2 2(3 2 ) (6 2 4 ) m= + = +R i j k i j k

Problem 2-1(I) Find a vector of length 5m in thexyplane that is perpendicular to 3 6 2= + A i j k

m.(Hint: Consider the dot product.)

( , ,0)x yb b=B xy

0 (3,6, 2) ( , ,0) 3 6x y x yb b b b = = = +A B

3 6 0x yb b+ = 2x yb b=

(2 )c= B i j ( 2 )c +i j

2 25 2 1B c= = + 5c =

2 5 5 4.46 2.23= = B i j i j 2 5 5 4.46 2.23 + = +i j i j

Problem 2-2


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2.35. Find the angle between: (a) the body diagonal shown and the z axis; (b) two face

diagonal on adjacent faces; (c) a face diagonal and a body diagonal that share a corner.

(Assign vectors to each line and express each in unit vector notation.)

2-35(a) ( )L= + +D i j k

3D L=

( ) 1cos

3 3

L

D L

+ + = = =

D k i j k k 1 0

1cos 54.7

3 = =

(b) ( )L= +A i j , ( )L= +B i k

2L =A B 1

cos2AB

= =A B

1 01

cos 602

= =

(c)2

cos6AD

= =A D

1 02

cos 35.36

= =

Problem 2-6

(I) Personnel at an airport control tower track a UFO. At 11:20 a.m it is located at ahorizontal distance of 2km in the direction 030 N of E at an altitude of 1200m. At

11:15 a.m. the location is 1km at 045 S of E at an altitude of 800m; see Fig. 2.36.

What was the displacement of the UFO?

2-36

0 02cos30 2sin 30 1.2 1.73 1.2 km= + + = + +A i j k i j k

0 0cos 45 sin 45 0.8 0.707 0.707 0.8 km= + = +B i j k i j k

1.02 1.71 0.4 km = B A i j k


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2 2 2cos cos cos 1 + + =

cos xr r

= =r i

cos yr

= cos zr

=

2 2 22 2 2

2

( )cos cos cos 1

x y z

r

+ ++ + = =

Problem 2-11(II) A three-dimensional vector A has a length of 10m and makes the angle 065 and

0

40 with the x+ and z+ axes, respectively. Find the magnitudes of its Cartesiancomponents.

010 cos 65 4.23mxA = =

010 cos 40 7.66 mzA = =

2 2 2 2 2 210 4.23 7.66 23.43 4.84my x z

A A A A= = = =

96/01/03

up benson problem ch02

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