Scalars and Vectors

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 forced 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.

To sum up, the equality of two vectors is

determined by the vectors themselves, but the

equivalence between two vectors is determined

by the situation at hand.

PROPERTIES OF VECTORS

Addition of Vectors is done according to the parallelogram

law of vector addition.

Subtraction of Vectors is done according to the parallelogram law.

Multiplication of a Scalar and a Vector

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

U

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

Cartesian Coordinates Cartesian coordinate system is composed of 90°

(orthogonal) axes. It consists of x and y axes in two dimensional (planer) 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 (Planer) Cartesian Coordinates

,

jVUiVUjViVjUiUVUjViVV

jUiUUjUUiUU

yyxxyxyxyx

yxyyxx

Vector Components in Three Dimensional (Spatial) Cartesian Coordinates

kVUjVUiVUVU

kVjViVV

UUU UkUjUiUU

zzyyxx

zyx

zyxzyx

222

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

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 equal to 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