P-1: The screw eye is subjected to two forces, Ԧ𝐹1 and Ԧ𝐹2. Determine themagnitude and direction of the resultant force.

P-2: Resolve the horizontal 600-N force in the figure into componentsacting along the u and v axes and determine the magnitudes of thesecomponents. 600-N

1

P-3: The vector 𝑉 lies in the plane defined by the intersecting lines LA and LB. Its magnitude is 400

units. Suppose that you want to resolve 𝑉 into vector components parallel to LA and LB. Determine the

magnitudes of the vector components.

80°

60°

V

LA

LB

P-4: Determine the projections Pa and Pb of 𝑉 onto the lines LA and LB.

2

P-5: Express forces as the vector form in terms of the unit vectors Ԧ𝑖 and Ԧ𝑗 .

(a) (b) (c)

Fx

Fy

jiF

jiF

jFiFF yx

39.32102.383

40sin50040cos500

Fx

Fy

Fx

Fy

40o

30o

jiF

jiF

jFiFF yx

2004.346

30sin40030cos400

jiF

F

F

jFiFF

y

x

yx

28.4

512

52.5

512

122.5

13

22

13

22

P-6: An elastic rubber band is attached to points A and B.Determine its length and its direction measured from A toward B.

P-8: The force acts on the hook. Express it as aCartesian vector.

P-7: Express the force as a Cartesian vector.

4

P-9: Express the forces shown in the figures as a Cartesian vector.

5

F=50 N

P-10: Determine the magnitude of the resultant 𝑉=𝑉1+𝑉2 and the angle q which 𝑉 makes with the

positive x–axis.

V1 = 9 units

V2 = 7 units

30°

y

x

4

3

6

7

P-11: The frame shown is subjected to a horizontal force Ԧ𝐹 = 300Ԧ𝑗. Determine the magnitude of thecomponents of this force parallel and perpendicular to member AB.

Ԧ𝐹 = 300Ԧ𝑗

P-12: Ԧ𝐴=3Ԧ𝑖 +2Ԧ𝑗 5𝑘 and 𝐵 =2Ԧ𝑖 7Ԧ𝑗 6𝑘. Determine;

?AB

?BA

?AB

?BA

?BA

?BA

7

Consider the straight lines OA and OB.

a) Determine the components of a unit vector that is

perpendicular to both OA and OB.

b) What is the minimum distance from point A to the line OB?

A (10, 2, 3) m

y

xq

z

B (6, 6, 3) m

O

P-13:

9

a) Determine the components of a unit vector that is perpendicular to both OA and OB.

A (10, 2, 3) m

q

B (6, 6, 3) m

O

b) What is the minimum distance from point A to the line OB?

OBr /

OAr /

kjirkjir OAOB

3210366 //

md

drrrr

rd

OBOA

r

OBOA

OA

71.9

36636.87sin

sin

9

222

////

/

q

q

Vector normal to both OA and OB is

kjir

ijikjkr

kjikjirrr OBOA

724812

18186123060

3663210//

Unit vector normal to both OA and OB is

kjinkji

r

rn

824.0549.0137.0724812

724812

36.87

222

10

Consider the triangle ABC.

a) What is the surface area of ABC?

b) Determine the unit vector of the outer normal of surface ABC.

c) What is the angle between AC and AB?

A (12, 0, 0) m

y

x

z

B (0, 16, 0) m

C (0, 0, 5) m

P-14:

11

a)What is the surface area of ABC?

A (12, 0, 0) m

y

x

z

B (0, 16, 0) m

C (0, 0, 5) mB

A

C

c) What is the angle between AC and AB?

kiACjiAB

5121612

AC

AB

ikjACAB

kijiACAB

8019260

5121612

2222

24.1082

1926080mABCs

b) Determine the unit vector of the outer normal of surface ABC.

kjikji

ACAB

ACABn

887.0277.0369.0

48.216

1926080

368.56cos1320144

cos51216125121612

cos

2222

kiji

ACABACAB

12

P-15: Determine the magnitude and direction angles of theresultant force acting at A .

13

P-16: The door is held opened by two chains. Forces in AB and CD are FAB=300 N and FCD=250 N, respectively.

a) Write these forces in vector form.

b) Determine the resultant of these forces and the direction angles of the resultant force.

c) Determine the projection of force FAB on line CD.

14

P-17: Cables extend from A to B and from A to C. The cable

AC exerts a 1000 N force vector Ԧ𝐹 at A.

a) What is the angle between the cables AB and AC?

b) Determine the vector component of Ԧ𝐹 parallel to the

cable AB and write the vector expression of this

component. Also determine the normal component of Ԧ𝐹

to the cable AB.

F

A (0, 7, 0) m

y

x

zB (0, 0, 10) m

C (14, 0, 14) m

15

P-18: Observers on Earth at points A and B measure direction cosines of position vectors to the

space shuttle at C as

For Ԧ𝑟𝐶/𝐴: cos qx = 0.360, cos qy = 0.480 and cos qz <90°.

For Ԧ𝑟𝐶/𝐵: cos qx = –0.515, cos qy = – 0.606 and cos qz <90°.

Determine the x, y, z coordinates of the space shuttle located at point C and also the shortest distance

from point C to direction AB.

A

B

C

x

y

z

(520, 640, 0) km 16

P-19: Consider force Ԧ𝐹 = 12Ԧ𝑖 + 24Ԧ𝑗 + 24𝑘 𝑁. Resolve Ԧ𝐹

into two components, one parallel to line AB and one

perpendicular to line AB.

A(10, 2, 14) Position vector

F

kjir

kjir

AB

AB

884

146261014

/

/

B(14, 6, 6)

Unit vector

n

kjikji

r

rn

AB

AB

3

2

3

2

3

1

884

884

12

222/

/

Parallel component

kjikjinFF

NkjikjinFF

3

8

3

8

3

4

3

2

3

2

3

14

43

224

3

224

3

112

3

2

3

2

3

1242412

////

//

(in vector form)

Normal component

kjikjikjiFFF

67.2633.2167.103

8

3

8

3

4242412//

//F

F

17

P-20: A house with 98000 kg mass is built on a steep slope defined by points A, B, and C.

To help assess the possibility of slope failure (mud slide), it is necessary to

a) Determine the components of the weight in directions normal and parallel (tangent) to

the slope.

b) Determine the component vectors of the weight in directions normal and parallel

(tangent) to the slope.

c) Determine the smallest distance from point O to the slope.

(g=9.81 m/s2)

18

a-b) Determine the components of the weight in directions normal and parallel (tangent) to the slope.

W

Wn

Wt

W: Weight of house

Wn: Normal component of the weight to slope

Wt : Parallel component of the weight to slope

kjiN

jikjACABN

21600720010800

18012060180

jiAC

kjAB

180120

60180

kjikji

N

Nn

7

6

7

2

7

3

21600720010800

21600720010800

25200

222

kkW

96138081.998000

AB

AC

B

A

C

NkjiknWWn 8240407

6

7

2

7

3961380

kjikjinWW nn

706320235440353160

7

6

7

2

7

3824040

(Vector normal to the slope)

(Unit vector normal to the slope)

(its magnitude)

(in vector form)

19

c) Determine the smallest distance from point O to the slope.

W

Wn

Wt

kjiW

kjikWWW

t

nt

255060235440353160

706320235440353160961380

(its magnitude)

(in vector form)

NW

W

t

t

4.495186

255060235440353160 222

O

B

C

n

mkjiinOCd 43.517

6

7

2

7

3120

mkjijnOAd

mkjiknOBd

43.517

6

7

2

7

3180

43.517

6

7

2

7

360

CThe smallest distance can also be determined as follows

20