This document provides a summary of vectors and their properties. It defines vectors as physical quantities having magnitude and direction. It describes methods for representing, adding, subtracting and resolving vectors. It also defines dot products and cross products of vectors, and explains how to calculate them for vectors represented in Cartesian form. The document aims to concisely summarize important vector concepts and formulas.
SUMMARY OF CHAPTER:-
Definition of Gravitation
Acceleration Due to Gravity
Variation Of “G” With Respect to Height And Depth
Escape Velocity
Orbital Velocity
Gravitational Potential
Time period of a Satellite
Height of Satellite
Binding Energy
Various Types of Satellite
Kepler’s Law of Planetary motion
A vector has magnitude and direction. There is an algebra and geometry of vectors which makes addition, subtraction, and scaling well-defined.
The scalar or dot product of vectors measures the angle between them, in a way. It's useful to show if two vectors are perpendicular or parallel.
SUMMARY OF CHAPTER:-
Definition of Gravitation
Acceleration Due to Gravity
Variation Of “G” With Respect to Height And Depth
Escape Velocity
Orbital Velocity
Gravitational Potential
Time period of a Satellite
Height of Satellite
Binding Energy
Various Types of Satellite
Kepler’s Law of Planetary motion
A vector has magnitude and direction. There is an algebra and geometry of vectors which makes addition, subtraction, and scaling well-defined.
The scalar or dot product of vectors measures the angle between them, in a way. It's useful to show if two vectors are perpendicular or parallel.
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Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
2. I'd like to express my greatest gratitude to the people who
have helped & supported me throughout my project. I’ m
grateful to my Physics Teacher Mr. Chhotelal Gupta for his
continuous support for the project, from initial advice &
encouragement to this day. Special thanks of mine goes to
my colleagues who helped me in completing the project by
giving interesting ideas, thoughts & made this project easy
and accurate.
____________
Shivam Rathi
4. Content
1. Introduction
2. Representation of Vectors
3. Addition and Subtraction of Vectors
4. Resolution of vector
(i ) Rectangular Component
(ii) 3-D resolution of vector
5. Unit Vector
6. Multiplication of Vector
( i ) Dot Product
(ii) Cross Product
5. Introduction
Scalar Quantities
Physical quantities having magnitude alone are known as
Scalar quantities.
Examples:- Mass, Time, Distance etc.
VectorQuantities
Physical quantities having both magnitude and direction
and also follow vector rule of addition are known as vector
quantities.
Examples:- Displacement, Momentum ,Force etc.
Tensor Quantities
Physical Quantities which are neither vectors nor scalars
are known as tensor quantities.
Examples :- Moment of inertia, Stress, etc.
6. Note:- Some quantities like area, length, angular velocity,
etc. are treated as both scalars as well as vectors.
Representation of a vector
Vectors are represented by alphabets (both small and
capital) with an arrow at its top.
Examples:-𝑎⃗ ,𝐴⃗ etc
Magnitude of vector is represented as a or | 𝑎⃗|.
Graphically a vector is represented as an arrow, and
head indicating direction of vector.
Example :-
head(indicating direction)
𝑎⃗
tail of vector
7. Addition of vectors
Graphical Law
According to this law if two vectors are represented in
magnitude and direction by two consecutive sides of a triangle
taken in same order then the 3rd side of triangle taken in opposite
order gives the resultant of two vectors.
Example:-
𝑅⃗⃗ = 𝑎⃗ + 𝑏⃗⃗
𝑅⃗⃗ 𝑏⃗⃗
𝑎⃗
Note:-
Same order of Vectors- Head of one vector matches with tail
of other vector.
Example:- 𝑎⃗ 𝑏⃗⃗
8. Opposite order of Vectors- Two vectors are said to be in
opposite order if either tail matches with tail or head
matches with head of other vector.
Example:- 𝑎⃗ 𝑏⃗⃗
Parallelogram Law
If two vectors are represented in magnitudeand direction by two
adjacent sideof parallelogram intersecting at point then the resultant is
obtained by the diagonal of the parallelogram passing through the same
point.
Polygon Law
It states that if a no. of vectors are represented in magnitude and
directionby sides of a polygon taken in same order then the resultant is
obtained by closing sideof polygon taken in oppositeorder.
Example:-
𝑑⃗ 𝑐⃗
9. 𝑅⃗⃗ 𝑏⃗⃗
𝑎⃗
𝑅⃗⃗=𝑎⃗+𝑏⃗⃗+𝑐⃗+𝑑⃗
Analytical Method
B Let ø is angle b/w 𝑎⃗ & 𝑏⃗⃗ and
𝑎⃗ 𝑅⃗⃗ 𝑎⃗ let | 𝑎|⃗⃗⃗⃗⃗⃗ = a , | 𝑏|⃗⃗⃗⃗⃗⃗ = b and | 𝑅|⃗⃗⃗⃗⃗⃗ =
R
ø ø A
O 𝑏⃗⃗ C
From vertex B drop a on OA(extended)
so , cos ø = CD/BC & sin ø = BA/BC
CD = BC cos ø & BA = BCsin ø
so , R2 = (BA)2 + (OA)2
R2 = b2 sin2
ø +(OC +CA)2
R2 = b2 sin2
ø + a2+b2 cos2
ø + 2abcos ø
R = √ 𝑎2 + 𝑏2 + 2𝑎𝑏𝑐𝑜𝑠ø
Let 𝑅⃗⃗ makes an angle α with 𝑏⃗⃗
10. SUbtraction of vectors
Negative Vector
Negativevector of a given vector is a vector of same magnitude in
oppositedirection.
𝑎⃗
- 𝑎⃗
Subtraction of 𝑏⃗ from 𝑎⃗ is nothing but additionof 𝑎⃗ +(− 𝑏⃗⃗ ) .
𝑎⃗ − 𝑏⃗⃗ = 𝑎⃗ + (− 𝑏⃗⃗ )
Graphical Law
11. Example:-
𝑅⃗⃗ = 𝑎⃗ - 𝑏⃗⃗
𝑏⃗⃗
𝑎⃗
− 𝑏⃗⃗
𝑅⃗⃗
Analytical Method
B Let ø is angle b/w 𝑎⃗ & 𝑏⃗⃗ and
𝑎⃗ 𝑅⃗⃗ 𝑎⃗ let | 𝑎|⃗⃗⃗⃗⃗⃗ = a , |−𝑏|⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = b and | 𝑅|⃗⃗⃗⃗⃗⃗ =
R
ø ø A
O - 𝑏⃗⃗ C
From vertex B drop a on OA(extended)
so , cos ø = CD/BC & sin ø = BA/BC
CD = BC cos ø & BA = BCsin ø
so , R2 = (BA)2 + (OA)2
R2 = b2 sin2
ø +(OC +CA)2
R2 = b2 sin2
ø + a2+b2 cos2
ø - 2abcos ø
R = √ 𝑎2 + 𝑏2 − 2𝑎𝑏𝑐𝑜𝑠ø
Let 𝑅⃗⃗ makes an angle α with 𝑏⃗⃗
12. Resolution Of Vectors
The process of splitting a vector into two or more
vectors along different directions is called “resolution
of vectors”.
The splitted vectors are called “components of given
vector”.
A vector can have ‘infinite’ components.
13. Resolution of vectors is reverse of addition of vector.
𝑐⃗ 𝑐⃗ 𝑏⃗⃗
𝑎⃗
Vector 𝑐⃗ is resolved to 𝑎⃗ and 𝑏⃗⃗
Rectangular component
If the components of a vector are mutually perpendicular ,the
components are called rectangular components of the given vector.
Resolution in 2-Dimensions
Consider 𝑂𝐴⃗⃗⃗⃗⃗⃗ vector equal to 𝐴⃗ and it makes angle ø with X-
axis .Project 𝐴⃗ along X & Y axis. Let rectangular components of
𝐴⃗ be Ax and Ay respectively.
14. Y-axis
𝐴⃗
Ax = Acos ø
Asinø Ay= Asin ø
Tan ø = Ay/Ax
ø Ax
2+Ay
2=A2
O Acos ø X-axis
(A vector can have maximum 2 rectangular component in a plane & maximum 3 in space)
3-d Resolution of vector
Consider a vector 𝑂𝐴⃗⃗⃗⃗⃗⃗ when projected along space making α,β
(α+β≠90) & γangles with X,Y & Z axis respectively.
Let 𝑂𝐴⃗⃗⃗⃗⃗⃗ = 𝑎⃗ & |𝑂𝐴⃗⃗⃗⃗⃗⃗| =a.
Let the rectangular components of 𝑎⃗ be ax ,ay& az.
Thus ax=a cos 𝛼 , ay=a cos 𝛽 , az= a cos 𝛾
15. Y-axis 𝑎⃗
𝛽
𝛾 𝛼
X-axis
Z-axis
Further
ax
2+ay
2+ az
2=a2
so cos 𝛼 2+ cos 𝛽 2 + cos 𝛾2 =1
unit vector
Vector having magnitude as unity are called unit vector . They
are represented as 𝑎̂ (‘cap’or ‘hat’).They are used to indicated
direction .A unit vector may also be defined as vector divide by
its magnitude i.e.
𝑎̂=
𝑎⃗⃗
| 𝑎⃗⃗|
16. Orthogonal Unit Vectors
Three unit vectors(called orthogonal unit vectors) 𝑖̂, 𝑗̂ & 𝑘̂ are
used to indicate X,Y & Z axis respectively.
𝑗̂
𝑘̂ 𝑖̂
Multiplycation of Vectors
1. Dot(or Scalar ) Product :- 𝑎⃗. 𝑏⃗⃗
17. 2.Cross (or Vector) Product:- 𝑎⃗x 𝑏⃗⃗
Dot Product of twovector
Let the two vectors be 𝑎⃗& 𝑏⃗⃗.
𝑎⃗. 𝑏⃗⃗=abcos 𝛼 where 𝛼 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑏/𝑤 𝑎⃗& 𝑏⃗⃗.
18. Ex- W= 𝐹⃗. 𝑠⃗ P= 𝐹⃗. 𝑣⃗
Dot product of vectors given in Cartesian form
𝑎⃗ = ax 𝑖̂+ay 𝑗̂+az 𝑘̂
𝑏⃗⃗ = bx 𝑖̂+by 𝑗̂+bz 𝑘̂
So
𝑎⃗. 𝑏⃗⃗ = ax bx +ay by+ az bz
Note:-(𝑖̂. 𝑖̂ = 𝑖 ∗ 𝑖 cos 0 𝑗̂. 𝑗̂ = 𝑗 ∗ 𝑗 cos 0 𝑘̂. 𝑘̂ = 𝑘 ∗ 𝑘 cos 0
𝑖̂. 𝑗̂ = 𝑖 ∗ 𝑗 cos 90 𝑖̂. 𝑘̂ = 𝑖 ∗ 𝑘 cos9 0 𝑘̂. 𝑗̂ = 𝑗 ∗ 𝑘 cos 90)
Properties of Dot Product
i) Commutative:- 𝑎⃗. 𝑏⃗⃗= 𝑏⃗⃗. 𝑎⃗
ii) Distributive:- 𝑎⃗.( 𝑏⃗⃗ + 𝑐)⃗⃗⃗⃗ = 𝑎⃗. 𝑏⃗⃗+ 𝑎⃗. 𝑐⃗
Cross Product
“Cross -Product”of two vectors is another vector where
magnitude is equal to the product of the magnitude of the
vectors & sin of the smaller angle b/w them.
19. The dir’nof this vector is perpendicular to the plane
containing the given vectors & given by Right Hand Thumb
Rule or Screw Rule.
Let the two vectors be 𝑎⃗& 𝑏⃗⃗. 𝑐⃗ be the cross product of 𝑎⃗ 𝑥𝑏⃗⃗.
|𝑎⃗ 𝑥𝑏⃗⃗|=| 𝑐⃗|=absin 𝛼 where 𝛼 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑏/𝑤 𝑎⃗& 𝑏⃗⃗.
Cross product of vectors given in Cartesian form
𝑎⃗ = ax 𝑖̂+ay 𝑗̂+az 𝑘̂
𝑏⃗⃗ = bx 𝑖̂+by 𝑗̂+bz 𝑘̂