Problem 2.1 In Active Example 2.1, suppose that the
vectors Uand Vare reoriented as shown. The vector
Vis vertical. The magnitudes are jUjD8 and jVjD3.
Graphically determine the magnitude of the vector
UC2V.
V
U
45⬚
Problem 2.2 Suppose that the pylon in Example 2.2 is
moved closer to the stadium so that the angle between
the forces FAB and FAC is 50°. Draw a sketch of the
new situation. The magnitudes of the forces are jFABjD
100 kN and jFACjD60 kN. Graphically determine the
magnitude and direction of the sum of the forces exerted
on the pylon by the cables.
Problem 2.3 The magnitude jFAjD80 lb and the
angle ˛D65°. The magnitude jFACFBjD120 lb.
Graphically determine the magnitude of FB.
FB
FA
a
FC

Problem 2.4 The magnitudes jFAjD40 N, jFBjD
50 N, and jFCjD40 N. The angle ˛D50°and ˇD80°.
Graphically determine the magnitude of FACFBCFC.
FB
FA
a
FC

8
Problem 2.5 The magnitudes jFAjDjFBjDjFCjD
100 lb, and the angles ˛D30°. Graphically determine
the value of the angle ˇfor which the magnitude
jFACFBCFCjis a minimum and the minimum value
of jFACFBCFCj.
FB
FA
a
FC

Solution: For a minimum, the vector FCmust point back to the
Problem 2.6 The angle D50°. Graphically determine
the magnitude of the vector rAC.60 mm 150 mm
AC
B
rAB rBC
rAC
Solution: Draw the vectors accurately and then measure jrACj.
Problem 2.7 The vectors FAand FBrepresent
the forces exerted on the pulley by the belt.
Solution: Draw the vectors accurately and then measure jFAC
FBj.
Problem 2.8 The sum of the forces FACFBC
FCD0. The magnitude jFAjD100 N and the angle ˛D
FC
Solution: Draw the vectors so that they add to zero.
Problem 2.9 The sum of the forces FACFBC
FCD0. The magnitudes jFAjD100 N and jFBjD
80 N. Graphically determine the magnitude jFCjand the
angle ˛.30
FB
FA
FC
a
10
Problem 2.10 The forces acting on the sailplane are
represented by three vectors. The lift Land drag D
are perpendicular. The magnitude of the weight Wis
500 lb. The sum of the forces WCLCDD0. Graph-
ically determine the magnitudes of the lift and drag.
W
D
L
25⬚
Problem 2.11 A spherical storage tank is suspended
from cables. The tank is subjected to three forces, the
Solution: Draw the vectors so that they add to zero. Then measure
the unknown magnitudes.
Problem 2.12 The rope ABC exerts forces FBA and
FBC of equal magnitude on the block at B. The
FBA
B
A
Solution: Draw the vectors accurately and then measure the
unknown magnitudes.
Problem 2.13 Two snowcats tow an emergency shelter
to a new location near McMurdo Station, Antarctica.
(The top view is shown. The cables are horizontal.)
Top View
FA
Solution: Draw the vectors accurately and then measure the
unknown magnitudes.
Problem 2.14 A surveyor determines that the horizon-
tal distance from Ato Bis 400 m and the horizontal
distance from Ato Cis 600 m. Graphically determine
Solution: Draw the vectors accurately and then measure the
unknown magnitude and angle.
12
Problem 2.15 The vector rextends from point Ato
the midpoint between points Band C. Prove that
rD1
2⊲rAB CrAC⊳.
C
B
rAC
r
rAB
rAB
rDrAB CrBM,and rDrAC CrCM.
Add the two equations and note that rBM CrCM D0, since the two
vectors are equal and opposite in direction.
Thus 2rDrAC CrAB,or rD1
2⊲rAC CrAB⊳
rAC
rAB
r
AB
M
Problem 2.16 By drawing sketches of the vectors,
explain why
UC⊲VCW⊳D⊲UCV⊳CW.
Solution: Additive associativity for vectors is usually given as an
of U. The result is the vector UC⊲VCW⊳.
V
V
U+[V+W]
its magnitude jFj?
Strategy: The magnitude of a vector in terms of its
components is given by Eq. (2.8).
Problem 2.18 An engineer estimating the components
of a force FDF
xiCF
yjacting on a bridge abutment
Solution:
Problem 2.19 A support is subjected to a force FD
F
xiC80j(N). If the support will safely support a force
Solution: Use the definition of magnitude in Eq. (2.8) and reduce
algebraically.
Problem 2.20 If FAD600i800j(kip) and FBD
Solution: Take the scalar multiple of FB, add the components of
Problem 2.21 The forces acting on the sailplane are its
weight WD500j⊲lb⊳, the drag DD200iC100j(lb)
and the lift L. The sum of the forces WCLCDD0.
Determine the components and the magnitude of L.
y
L
14
Problem 2.22 Two perpendicular vectors Uand Vlie
in the x–yplane. The vector UD6i8jand jVjD20.
What are the components of V? (Notice that this problem
has two answers.)
Solution: The two possible values of Vare shown in the sketch.
The strategy is to (a) determine the unit vector associated with U,
y
Problem 2.23 Afish exerts a 10-lb force on the line
that is represented by the vector F. Express Fin terms
Solution: We can use similar triangles to determine the
components of F.
Problem 2.24 A man exerts a 60-lb force Fto push a
crate onto a truck. (a) Express Fin terms of components
x
Solution:
Problem 2.25 The missile’s engine exerts a 260-kN
force F. (a) Express Fin terms of components using the
coordinate system shown. (b) The mass of the missile
is 8800 kg. Determine the magnitude of the sum of the
forces exerted by the engine and the missile’s weight.
y
F
3
4
Solution:
(a) We can use similar triangles to determine the components of F.
Problem 2.26 For the truss shown, express the
position vector rAD from point Ato point Din terms of
Solution: Coordinates A(1.8, 0.7) m, D(0, 0.4) m
16
Problem 2.27 The points A,B,… are the joints of
the hexagonal structural element. Let rAB be the position
vector from joint Ato joint B,rAC the position vector
Solution: Use the xy coordinate system shown and find the
locations of Cand Fin those coordinates. The coordinates of the
points in this system are the scalar components of the vectors rAC and
Problem 2.28 For the hexagonal structural element in
Problem 2.27, determine the components of the vector
Solution: rAB rBC.
The angle between BC and the x-axis is 60°.
Problem 2.29 The coordinates of point Aare (1.8,
3.0) ft. The ycoordinate of point Bis 0.6 ft. The vector
rAB has the same direction as the unit vector eAB D
0.616i0.788j. What are the components of rAB?
y
A
rAB
Problem 2.30 (a) Express the position vector from
point Aof the front-end loader to point Bin terms of
components.
(b) Express the position vector from point Bto point C
in terms of components.
(c) Use the results of (a) and (b) to determine the
distance from point Ato point C.
45 in
98 in
50 in
55 in
35 in
A
50 in
y
x
C
B
Solution: The coordinates are A(50, 35); B(98, 50); C(45, 55).
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Problem 2.31 In Active Example 2.3, the cable AB
exerts a 900-N force on the top of the tower. Suppose
that the attachment point Bis moved in the horizontal
direction farther from the tower, and assume that the
magnitude of the force Fthe cable exerts on the top
of the tower is proportional to the length of the cable.
(a) What is the distance from the tower to point B
if the magnitude of the force is 1000 N? (b) Express
the 1000-N force Fin terms of components using the
coordinate system shown.
A
B
80 m
40 m
A
y
Solution: In the new problem assume that point Bis located a
distance daway from the base. The lengths in the original problem
(b) The force F is then
Problem 2.32 Determine the position vector rAB in
terms of its components if (a) D30°, (b) D225°.
60 mm 150 mm
x
y
A
C
B
rAB
rBC
θ
Solution:
rAB D51.96iC30jmm. And
(b) rAB D⊲60⊳cos⊲225°⊳iC⊲60⊳sin⊲225°⊳jor
rAB D42.4i42.4jmm.
F
BC
F
AB
mm
y
x
CF
A
B
θ
Problem 2.33 In Example 2.4, the coordinates of the
fixed point Aare (17, 1) ft. The driver lowers the bed of
the truck into a new position in which the coordinates
of point Bare (9, 3) ft. The magnitude of the force F
exerted on the bed by the hydraulic cylinder when the
bed is in the new position is 4800 lb. Draw a sketch of
the new situation. Express Fin terms of components.
A
F
x
30⬚
y
B
30⬚A
B
Solution:
D14.04°
20
Problem 2.34 A surveyor measures the location of
point Aand determines that rOA D400iC800j(m). He
wants to determine the location of a point Bso that
jrABjD400 m and jrOA CrABjD1200 m. What are the
cartesian coordinates of point B?
x
y
A
B
O
rAB
rOA
Proposed
roadway
N
Solution: Two possibilities are: The point Blies west of point A,
or point Blies east of point A, as shown. The strategy is to determine
the unknown angles ˛,ˇ, and . The magnitude of OA is
B
y
BA
Problem 2.35 The magnitude of the position vector
rBA from point Bto point Ais 6 m and the magnitude of
Solution: The coordinates are: A⊲xA,y
A⊳,B⊲0,0⊳,C⊲3m,0⊳
Problem 2.36 In Problem 2.35, determine the compo-
nents of a unit vector eCA that points from point Ctoward
point A.
Strategy: Determine the components of rCA and then
divide the vector rCA by its magnitude.
Solution: From the previous problem we have
Problem 2.37 The xand ycoordinates of points A,B,
and Cof the sailboat are shown.
(a) Determine the components of a unit vector that
Solution:
rAB D⊲xBxA⊳iC⊲yByA⊳j
22
Problem 2.38 The length of the bar AB is 0.6 m.
y
xD0.183 m,yD0.356 m
Thus
eAB DrAB
rAB D⊲0.183 m [0.3m]⊳iC⊲0.356 m⊳j
⊲0.183 m C0.3m⊳2C⊲0.356 m⊳2
D⊲0.806iC0.593j⊳
Problem 2.39 Determine the components of a unit
vector that is parallel to the hydraulic actuator BC and
0.6 m Scoop
0.15 m
1 m
y
rBC D0.75iC0.6j⊲m⊳
Problem 2.40 The hydraulic actuator BC in Problem
Solution: From the solution to Problem 2.39,
Problem 2.41 A surveyor finds that the length of the
line OA is 1500 m and the length of line OB is 2000 m.
(a) Determine the components of the position vector
from point Ato point B.
(b) Determine the components of a unit vector that
points from point Atoward point B.
y
60⬚
B
A
Proposed bridge
N
Solution: We need to find the coordinates of points Aand B
rOA D1500 cos 60°iC1500 sin 60°j
24
Problem 2.42 The magnitudes of the forces exerted
by the cables are jT1jD2800 lb, jT2D3200 lb, jT3jD
4000 lb, and jT4jD5000 lb. What is the magnitude of
the total force exerted by the four cables?
x
y
29⬚
9⬚
40⬚
51⬚
T4T3
T2
T1
Solution: The x-component of the total force is
TxDjT1jcos 9°CjT2jcos 29°jT3jcos 40°CjT4jcos 51°
Problem 2.43 The tensions in the four cables are equal:
jT1jDjT2jDjT3jDjT4jDT. Determine the value of
Tso that the four cables exert a total force of 12,500-lb
magnitude on the support.
x
y
29⬚
9⬚
40⬚
51⬚
T4T3
T2
T1
Solution: The x-component of the total force is
26