Vector Analysis Lecture Notes 2018, Lecture notes of Mathematics

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VECTOR ANALYSIS

Introduction: scalar and vector

quantities

• Physical quantities can be divided into two

main groups, scalar quantities and vector

quantities.

• A scalar quantity is one that is defined

completely by a single number with

appropriate units, e.g. length , area ,

volume , mass , time , etc. Once the units

are stated, the quantity is denoted

entirely by its size or magnitude.

Introduction: scalar and vector

quantities

• Field – a function that specifies a

particularly quantity everywhere in a region;

may be a scalar field or a vector field

Example:

Scalar field – temperature distribution in a

building, sound intensity in a theater,

electric potential in a region, etc.

Vector field – gravitational force on a body

in space, velocity of raindrops in the

atmosphere, etc.

Vector Representation

• A vector quantity can be represented

graphically by a line, drawn so that:

(a). the length of the line denotes the

magnitude of the quantity, according to

some stated vector scale.

(b). the direction of the line denotes the

direction in which the vector quantity acts.

The sense of the direction is indicated by

an arrowhead.

Vector Representation

• can be represented by a vector

quantity with the same magnitude but

opposite in direction

BA

Vector Representation

Two Equal Vectors

If a = b , then

(a) a = b (equal magnitude)

(b) the direction of a = direction of b , i.e.

the two vectors are parallel.

Vector Representation

Types of Vectors

(a) A position vector occur when the

point A is fixed.

(b) A line vector is such that it can slide

along its line of action, e.g. a mechanical

force acting on a body.

(c) A free vector is not restricted in any way.

It is completely defined by its magnitude

and direction and can be drawn as any one

of a set of equal length parallel lines.

AB

Vector Representation

• Unit vector along - is a vector

whose magnitude is unity (1) and its

direction is along.

magnitude

vector

B

B

a unit vector

B

B

a

B

B

2 2 2

x y z

x x y y z z

B

B B B

B a B a B a

a

 

 



Basic Laws of Vector Algebra

Law Addition Multiplication

Commutative

Associative

Distributive

A  B  B  A

K A  AK

( A  B ) C  A ( B  C )

K ( n A )  ( Kn ) A

K ( A  B )  K A  KB

Cartesian, or Rectangular,

Coordinate System.

• three coordinate axes (known as x,y, and

z) mutually at right angles to each other.

Cartesian, or Rectangular,

Coordinate System.

• A reserve symbol a for a unit vector is use

to identify the direction of the unit vector by

an appropriate subscript. Thus a x, a y, and

a z are the unit vectors in the Cartesian

coordinate system. They are directed along

the x, y, and z axes.

Cartesian, or Rectangular,

Coordinate System.

Dot Product

• The dot product of two vectors A and B is

defined as the product of the magnitude of

A and magnitude of B and the cosine of

the angle between the vectors

where: is the smaller angle between

the vectors.

AB

A  B  A B cos 

AB

Dot Product

Properties:

1. Commutative:

2. Distributive:

3. Scaling:

7. Alternatively:

8. If the dot product of the two vectors is zero, then

the two vectors are orthogonal to each other.

A  B  B  A

A ( B  C )  A  B  A  C )

K ( A  B )  ( KA ) B  A ( KB )

2

A  A  A

      0 x y y z z x

a a a a a a

      1 x x y y z z

a a a a a a

x x y y z z

A  B  A B  A B  A B