SCALAR

A physical quantity that is completely characterized by a

real number (or by its numerical value) is called a scalar.

In other words, a scalar possesses only a magnitude.

Mass, density, volume, temperature, time, energy, area,

speed and length are examples to scalar quantities.

VECTOR

Several quantities that occur in mechanics require a description in terms of

their direction as well as the numerical value of their magnitude. Such

quantities behave as vectors. Therefore, vectors possess both magnitude

and direction; and they obey the parallelogram law of addition. Force,

moment, displacement, velocity, acceleration, impulse and momentum are

vector quantities.

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

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. In

other words, a force may be applied at any point on its given line

of action without altering the resultant external effects on the

rigid body on which it acts.

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.

PROPERTIES OF VECTORS

Addition of Vectors is done according to the parallelogram law

of vector addition.

UVVU

MVUMVU or WVU

W

U

V

Subtraction of Vectors is done according to the parallelogram law.

Multiplication of a Scalar and a Vector

VaUaVUa UbUaUba

UabUba UaUa

ZVUVU

Z

U V

V

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. Therefore,

may be expressed in terms of both its magnitude and direction separately. U (a

scalar) expresses the magnitude and (a dimensionless vector) expresses the

directional sense of .

U

n

U

Vector Components and Resultant Vector Let the sum of

and be . Here, and are named as the components

and is named as the resultant.

U

V

W

U

V

W

sinsinsin

WVU

cos2222 UVVUW

Sine theorem

Cosine theorem

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.

Vector Components in Two Dimensional (Planar) Cartesian Coordinates

unit vector along the x axis, , unit vector along the y axis, i

j

i

jVUiVUjViVjUiUVU jViVV

jUiUU jUU iUU

yyxxyxyxyx

yxyyxx

x

y

yxyx

U

U

UUUUUU

an t

22

Vector Components in Three Dimensional (Spatial) Cartesian Coordinates

kVUjVUiVUVU

kVjViVV

zzyyxx

zyx

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

Position Vector It is the vector that describes the location of one

point with respect to another point.

jyyixxr

yyrxxr

rrr

jrirrrr

ABAB

AByABx

yx

yxyx

B/A

B/AB/A

B/AB/AB/A

B/AB/AB/AB/AB/A

,

22

In two dimensional case

In three dimensional case

kzzjyyixxr

zzryyrxxr

rrrr

krjrirrrrr

ABABAB

ABzAByABx

zyx

zyxzyx

B/A

B/AB/AB/A

B/AB/AB/AB/A

B/AB/AB/AB/AB/AB/AB/A

, ,

222

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

vectors.

VU

VUVUVU

aUVaVU

cos cos

irrelevant istion multiplica oforder

zzyyxx

zyxzyx

VUVUVUVU

kVjViVVkUjUiUU

ikkjjiji

kkjjiiii

, ,

, ,

s,Coordinate Cartesian in vectors unit of terms In

00090cos

1110cos

U

V

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

nUUUnUnU

UU

//

1

//

, coscos

cos

Magnitude of parallel component

Parallel component

Normal (Orthogonal) component

//

//

UUU

nnUU

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

VUVUVU

WUVWVU

sinsin

,

jkiijkkij

jikikjkjijiji

kkjjiiii

, ,

, , , 190sin

0 , 0 , 00sin

s,Coordinate Cartesian in orsunit vect of In terms

-- -

kVUVUjVUVUiVUVUVU

VUkVUiVUjVUkVUjVUi

V

U

j

V

U

i

VVV

UUU

kji

VU

kVjViVkUjUiUVU

xyyxzxxzyzzy

xyyzzxyxxzzy

y

y

x

x

zyx

zyx

zyxzyx

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

WWW

VVV

kji

kUjUiUWVU

kWjWiWW

kVjViVV

kUjUiUU

or