SCALARS AND VECTORS

All physical quantities in engineering mechanics are

measured using either scalars or vectors.

Scalar. A scalar is any positive or negative physical

quantity that can be completely specified by its

magnitude. Examples of scalar quantities include

length, mass, time, density, volume, temperature,

energy, area, speed.

VECTOR

A vector is any physical quantity that requires both a magnitude and

a direction for its complete description. Examples of vectors

encountered in statics are force, position, and moment. A vector is

shown graphically by an arrow. The length of the arrow represents the

magnitude of the vector, and the angle q between the vector and a

fixed axis defines the direction of its line of action. The head or tip of

the arrow indicates the sense of direction of the vector

TYPES OF VECTORS

Physical quantities that are vectors fall into one of the three classifications as

free, sliding or fixed.

A free vector is one whose action is not confined to or associated with a

unique line in space. For example if a body is in translational motion, velocity

of any point in the body may be taken as a vector and this vector will describe

equally well the velocity of every point in the body. Hence, we may represent

the velocity of such a body by a free vector.

In statics, couple moment is a free vector.

A sliding vector is one for which a unique line in space must be

maintained along which the quantity acts. When we deal with the external

action of a force on a rigid body, the force may be applied at any point

along its line of action without changing its effect on the body as a whole

and hence, considered as a sliding vector.

A fixed vector is one for which a unique point of application is

specified and therefore the vector occupies a particular position in

space. The action of a force on a deformable body must be specified

by a fixed vector.

Principle of Transmissibility (Taşınabilirlik İlkesi)

The external effect of a force on a rigid body will remain

unchanged if the force is moved to act on its line of action.

Equality and Equivalence of Vectors

Two vectors are equal if they have the same dimensions, magnitudes and directions.

Two vectors are equivalent in a certain capacity if each produces the very same effect

in this capacity.

Addition of Vectors is done according to the parallelogram principle of

vector addition. To illustrate, the two “ component ” vectors 𝐴 and 𝐵 are

added to form a “ resultant ” vector 𝑅.

R

A

B

RBA

Parallelogram law

RBA

R

A

B

Triangle law

Subtraction of Vectors is done according to the parallelogram law.

Multiplication of a Scalar and a Vector

VaUaVUa UbUaUba

UabUba UaUa

BABAR

R

A B

B

Vector Addition of Forces

Experimental evidence has shown that a force is a vectorquantity since it has a specified magnitude, direction, and senseand it adds according to the parallelogram law. Two commonproblems in statics involve either finding the resultant force,knowing its components, or resolving a known force into twocomponents.

Finding the Components of a Force.

Finding a Resultant Force.

If more than two forces are to be added, successiveapplications of the parallelogram law can be carriedout in order to obtain the resultant force.

Addition of Several Forces

Vector Components and Resultant Vector Let the sum of 𝐴 and 𝐵 be 𝑅. Here, 𝐴 and 𝐵

are named as the components and 𝑅 is named as the resultant.

sinsinsin

RBA

cos2222 ABBAR

Sine theorem

Cosine theorem

RBA

R

A

B

R

A

B

q

q

qcos2222 ABBAR

Cosine theorem

Note that

(Magnitude of the resultant force can be determined using the law of cosines, and itsdirection is determined from the law of sines.)

Example-1: The screw eye is subjected to two forces, 𝐹1 and 𝐹2.Determine the magnitude and direction of the resultant force.

Example-2: Resolve the horizontal 600-N force in thefigure into components acting along the u and v axesand determine the magnitudes of these components.

600-N

Example-3: It is required that the resultant force acting on the

eyebolt be directed along the positive x axis and that 𝐹2 have aminimum magnitude. Determine this magnitude, the angle q, andthe corresponding resultant force.

The relationship between a force and its vector components must

not be confused with the relationship between a force and its

perpendicular (orthogonal) projections onto the same axes.

For example, the perpendicular projections of force onto axes a

and b are and , which are parallel to the vector components of

and .

F

aF

bF

1F

2F

F

a

b

//a

//b

1F

2F

F

a

b

a

b

aF

bF

Components: F1 and F2 Projections: Fa and Fb

It is seen that the components of a vector are not necessarily equal to

the projections of the vector onto the same axes. The components and

projections of are equal only when the axes a and b are

perpendicular.

F

F

a

b

//a

//b

1F

2F

F

a

b

a

b

aF

bF

Components: F1 and F2 Projections: Fa and Fb

Example-4: 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

Example-5: Determine the projections Pa and Pb of 𝑉 onto the lines LA and LB.

Unit Vector A unit vector is a free vector having a magnitude of 1 (one) as

eornU

U

U

Un

It describes direction. The most convenient way to describe a vector in a certain

direction is to multiply its magnitude with its unit vector.

nUU

U

1

U

n

and U have the same unit, hence the unit vector is dimensionless. U

CARTESIAN COORDINATES Cartesian Coordinate System is

composed of 90° (orthogonal) axes. It consists of x and y axes in two

dimensional (planar) case, x, y and z axes in three dimensional (spatial)

case. x-y axes are generally taken within the plane of the paper, their

positive directions can be selected arbitrarily; the positive direction of

the z axes must be determined in accordance with the right hand rule.

x

y

z

z

z

x yx

y

Cartesian Unit Vectors In three dimensions, the set of

Cartesian unit vectors, 𝑖 , 𝑗, 𝑘, is used to designate the

directions of the x, y, z axes, respectively.

Vector Components in Two Dimensional (Planar) Cartesian Coordinates

jUiUU

jUU

iUU

yx

yy

xx

x

y

yx

U

U

UUU

qtan

22

x

y

U

i

j

yU

xU

q

yx UUU

Vector Components in Three Dimensional (Spatial) Cartesian Coordinates

unit vector along the x axis, ,

unit vector along the y axis, ,

unit vector along the y axis, ,

ji

k

222

zyx

zyx

UUUU

kUjUiUU

x

y

z

U

i

j

k

zU

yU

xU

In three dimensional case

kzzjyyixxr ABABABB/A

A (xA, yA, zA)

B (xB, yB, zB)

x

y

z

B/Ar

i

jk

Position Vector: It is the vector that describes the location of one point with respect

to another point.

In two dimensional case

jyyixxr ABABB/A

y

i

j

A (xA, yA)

B (xB, yB)

B/Ar

x

Example-6: An elastic rubber band is attached to points Aand B. Determine its length and its direction measuredfrom A toward B.

* When the direction angles of a force vector are given;

The angles, the line of action of a force makes with the x, y and z axes are named as

direction angles. The cosines of these angles are called direction cosines; they

specify the line of action of a vector with respect to coordinate axes.

In this case, direction angles are qx, qy and qz.

Direction cosines are cos qx, cos qy and cos qz.

cos qx = l cos qy = m cos qz = n

222

cos

cos

cos

zyx

zz

yy

xx

FFFF

FF

FF

FF

q

q

q

kjiFF

kFjFiFF

kFjFiFF

zyx

zyx

zyx

qqq

qqq

coscoscos

coscoscos

11coscoscos

1

coscoscos

222222

2

222

2

22222

nml

F

FFF

F

FFFFF

knjmiln

kjin

zyx

zyx

zyx

F

zyxF

qqq

qqq

kjiFF zyx

qqq coscoscos

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

2

12

2

12

2

12

121212

zzyyxx

kzzjyyixxF

AB

ABFF

nFF F

* When coordinates of two points along the line of action of a force are given;

Example-8: The force acts on the hook. Express itas a Cartesian vector.

* When two angles describing the line of action of a force is given;

qq

qq

sincosFsinFF

coscosFcosFF

sinFF

cosFF

xyy

xyx

z

xy

First resolve F into horizontal and vertical components.

Then resolve the horizontal component Fxy into x- and y-components.

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

F=50 N

Addition of Cartesian Vectors

jVUiVUjViVjUiUVU

jViVVjUiUU

yyxxyxyx

yxyx

kVjViVV zyx

In two dimensional case

In three dimensional case

kUjUiUU zyx

kVUjVUiVUVU zzyyxx

Example-10: Two forces act on the hook shown.

Specify the magnitude of 𝐹2 and its coordinate

direction angles so that the resultant force 𝐹R

acts along the positive y axis and has amagnitude of 800 N.

Dot (Scalar) Product A scalar quantity is obtained from the dot product of two vectors.

VU

VUcos cosVUVU

aUV irrelevant is tionmultiplica of order

aVU

zzyyxx

zyxzyx

VUVUVUVU

kVjViVV kUjUiUU

ik ,kj ,cosjiji

kk ,jj ,cosiiii

00090

1110

U

V

In terms of unit vectors in Cartesian Coordinates;

//

//

UUU

nnUU

Normal and Parallel Components of a Vector with respect to a Line

cos // qUU Magnitude of parallel component:

Normal (Orthogonal) component:

n

U

U

//U

q

Parallel component:

nUUUnUnU

//

1

, coscos qq

Example-11: The frame shown is subjected to a horizontal force 𝐹 = 300 𝑗. Determinethe magnitude of the components of this force parallel and perpendicular to member AB.

𝐹 = 300 𝑗

Cross (Vector) Product: The multiplication of two vectors in cross product

results in a vector. This multiplication vector is normal to the plane containing the

other two vectors. Its direction is determined by the right hand rule. Its magnitude

equals the area of the parallelogram that the vectors span. The order of

multiplication is important.

YUVUYVU

VaUVUaVUa

VU

VUsin sinVUVU

WUV , WVU

qqU

q

V

U

V

W

Wq

jk i ,ijk , kij

jik ,ikj , kji

sinjiji

kk ,jj ,siniiii

190

0000

In terms of unit vectors in Cartesian Coordinates;

kVUVUjVUVUiVUVUVU

VUkVUiVUj VUkVUjVUi

V

U

j

V

U

i

VVV

UUU

kji

VU

kVjViVkUjUiUVU

xyyxzxxzyzzy

xyyzzxyxxzzy

y

y

x

x

zyx

zyx

zyxzyx

x

y

z

i

j

k

i

j

k

i

j

k

+ +

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

Example-12:

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

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

Example-13:

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

Mixed Triple Product: It is used when taking the moment of a force about a line.

zyx

zyx

zyx

zyx

zyxzyx

zyx

zyx

zyx

WWW

VVV

UUU

WVU

or

WWW

VVV

kji

kUjUiUWVU

kWjWiWW

kVjViVV

kUjUiUU