# Chapter one Vector Analysis - PowerPoint PPT Presentation

1 / 20

Chapter one Vector Analysis. A scalar quantity is one which can be described fully by just stating its magnitude . Some examples are Mass time length temperature density speed energy and volume.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Chapter one Vector Analysis

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

#### Presentation Transcript

Chapter one

Vector Analysis

A scalar quantity is one which can be described fully by just stating its magnitude.

Some examples are

Mass

time

length

temperature

density

speed

energy and volume

A vector quantity is one which can only be fully described if its magnitude and direction stated.

Some examples are

displacement

velocity

acceleration

force

momentum

magnetic density and electric intensity.

### Why vectors are important?

• Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has both a magnitude and direction, such as velocity, the magnitude of which is speed. For example, the velocity 5 meters per second upward could be represented by the vector (0,5). Another quantity represented by a vector is force, since it has a magnitude and direction.

### Properties of a Vector

• A vector is a quantity that has both direction and magnitude. Let a vector be denoted by the symbol A.

• The magnitude of A is |A| ≡A

• We can represent vectors as geometric objects using arrows. The length of the arrow corresponds to the magnitude of the vector. The arrow points in the direction of the vector

A

|A| ≡A

O

### Types of Vector

• Null Vector

• Vector with zero magnitude

• Position Vector

• Vector starting from origin

• Free Vector

• Vector starting from anywhere but origin

• Unit Vector

• Vector with magnitude of 1

• Equal Vectors

• Two vectors equal in magnitude and direction

• Opposite Vectors

• Two vectors equal in magnitude but opposite in direction.

• ### Laws of vector algebra

• A+(B+C)=(A+B)+C (Associative law of addition)

• mA=Am (Commutative law of Multiplication)

• m(nA)=(mn)A (Associative law of Multiplication)

• (m+n)A=mA+nA(Distributive law)

• m(A+B)=mA+mB (Distributive law)

• The addition of two vectors yields another vector known as Resultant vector.

• For example if vector A and vector B are added their sum will be equal to (A+B).

### Methods of Addition of Vectors

There are 3 methods for addition of Vectors.

• For vectors precisely along X or Y axis.

• ### Parallelogram law

• Vector P and Q are drawn from same origin.

• Straight lines are drawn parallel to both vectors so as to form a parallelogram.

• The resultant (P+Q) is represented by the diagonal of the parallelogram that passes through the origin.

+

Q

P

Q

+

Q

P

P

### Triangle law (head to tail rule)

• The result of two vectors could be determined by drawing a triangle.

• Vector Q and P are drawn in such a way that the tail of vector Q touches head of vector P.

• The resultant (P+Q) is represented by the third side of triangle from tail of P to head of Q.

+

P

Q

Q

+

P

Q

P

### Subtraction of vectors

• The subtraction of two vectors can be treated as the addition of a negative vector.

(P-Q)=P+(-Q)

• The vector (P-Q) can then be determined by any of the two methods.

-

P

Q

=

P

-Q

P

-Q

OR

P

P-Q

-Q

### Resolution of a vector

• Vector R could be considered to be the resultant of two vectors.

R=A+B

• Here the vectors A and B are known as the components of vectors.

### Resolution of a vector

• It is useful to find the components of a vector R in two mutually perpendicular directions. This process is known as resolving a vector into components.

• The magnitude of the two components can be written in the form Rcosy and Rsiny

R

Rsin0

0

Rcos0

Rsind

R

d

Rcos d

### Triangle of forces

• If three forces acting on a point can be represented in magnitude and direction by three sides of a triangle taken in order, then the three forces are in equilibrium.

• The converse is also true:

• Three forces acting on a point are in equilibrium, they can be represented in magnitude and direction by sides of a triangle taken in order.

### Triangle of forces

Q

• By the triangle of vectors, the resultant of P and Q is represented in magnitude and direction by side of OC of the triangle OAC.If third force R is equal in magnitude to (P+Q) but in opposite direction, then the point O is in equilibrium and also R could be represented by the side CO of the triangle.

C

A

P

Q

P

P+Q

o

O

R

R

Q

C

A

P

R

O

### Polygon of Forces

• When more than three coplanar forces act on a point , the resultant (or vector sum) of forces can be found by drawing a polygon of forces.

### Polygon of Forces

S

• If forces acting on a point can be represented in magnitude and direction by sides of a polygon taken in order ,then the forces are in equilibrium.

T

o

P

Q

R

D

R

S

E

C

T

Q

P

A

B

### Resultant of a number of forces

• If point O is being acted upon by a number of coplanar forces such as A,B,C,D,E do not form a closed polygon then the forces are not in equilibrium

• The resultant in magnitude and direction is represented by R.

B

D

A

C

o

E

D

C

E

B

R

A