scalars and vectors

scalars and vectors:

This topic is covered in Physics as well as in maths (in calculus). I'm covering in Physics as most of it's application is in Physics.

Scalars: Things which don't have any sense of direction, are called scalars. As an example of scalar, mass of a thing, is a scalar quantity. It doesn't have any sense of direction, whether it's going north, south, etc, the mass is the same.

Vectors: Things which have direction, are called vectors. As an example of vector, force is a vector quantity. It depends on which direction you apply force, so it has a sense of direction. IF the force is applied in same direction as the movement of the body, then the body will accelerate, while if it's applied in opposite dir, then the body slows down.So, direction of force matters.

Link => https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/1%3A_Vector_Basics

Vector: A vector is rep by an arrow, whose length is the magnitude of the vector, and whose dir is the angle that that vector makes wrt x and y axis (for 2D vector), or wrt x, y and z axis (for 3D vector). Vectors can be moved anywhere in space and will remain same, as long as mag and dir remain same (i.e starting point and ending pt of vector can be anywhere).

Vector is rep by i, j, k for x,y and z dir or by coords (x,y,z) or in matrix form as x, y, z written vertically in 1st column.

 We'll study some operations (algebra) on vector quantities.

  • Addition/subtraction: Addition/subtraction of vectors is done by placing end point of a vector to the start point of next vector and so on, until all vectors are connected. Then you connect the start point of 1st vector to end point of last vector, and that's the sum of all the vectors. For subtraction, you still treat it as addition, except that the direction of vector is reversed for the subtracting vectors.
    • For vector <x1,y1> and <x2,y2>, the sum is <x1+x2,y1+y2>. The magnitude of vector is sq rt of [(x1+x2)^2 + (y1+y2)^2]. Mag may also be calc by using cosine thm, 
  • Multiplication: There's no multiplication of vectors. We call it the product of vectors. Product of vectors may give a scalar or a vector depending on how we define it. There are 2 kinds of product. Link => https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/1%3A_Vector_Basics/1.5%3A_The_Dot_and_Cross_Product 
    • Dot product: It's a scalar, and is determined by multiplying coefficients as shown in link above.
      • Dot product is simply multiplying the length of one vector, with the length of the shadow (or projection) of the other vector on the 1st vector. So, u.v = ||u||.||v cos θ||  =  ||u||.||v|| cos θ. Ot can also be calculated by using formula u.v=u1.v1 + u2.v2 + u3.v3.  
    • Cross product: It's a vector, and is determined by multiplying coefficients as shown in link above. It also has dir (as the product is a vector).
      • Cross product is found by getting determinant of the matrix with alternating +ve and -ve sign. It also has dir which is perpendicular to the plane of the 2 vectors u, v. 

 

 Matrix: Matrices are useful when evaluating dot and cross product of matrices.

 

 

Chemical Reactions

Chemical Reactions:

Khan Academy => https://www.khanacademy.org/science/ap-chemistry-beta/x2eef969c74e0d802:chemical-reactions

Chemical reactions cause chemical changes, while non-chemical reactions cause only physical change. Ex: ice melting to water is physical change, while propane burning is a chemical change as it results in formation of new compounds. Chemical change results in making and breaking of bonds.

Consider the reaction between NaCl (sodium Chloride) and AgNO(silver Nitrate). They are both solids, but they can be dissolved in a solution (usually water), and then they are denoted by "aq" to indicate they are in aqueous solution. This changes how the 2 compounds react.

Salt (NaCl) dissolves in water. We think of crystalline salt as solid NaCl, and these indeed exist as molecule of NaCl. But when dissolved n water, Na and Cl actually dissociate to form Naions and Cl- ions, surrounded by water molecules, with Na+ surrounded by partially -ve Oxygen atoms of water, while Cl- surrounded by partially +ve Hydrogen atoms of water. Similarly Silver Nitrate dissociates into Ag+ and NO3-.

We write eq as follows in Molecular Eq as follows: NaCl (aq) + AgNO3 (aq) => NaNO3 (aq) + AgCl (s) [NOTE: AgCl solid is formed from the above reaction, as it precipitates out of the solution, it doesn't have high solubility]

To better represent that the aq solution actually consists of dissociated ions, we can write eq in complete Ionic Eq as follows:  Na+ (aq) + Cl(aq) +  Ag(aq) + NO3- (aq) => Na+ (aq)  + NO3- (aq) + AgCl (s) 

We can write above eqn in Net ionic eqn as follows: Cl(aq) +  Ag(aq) => AgCl (s) [By cancelling out spectator ions which are the ions which are just watching the reaction, In this case Na+ (aq) and NO3- (aq) are spectator ions]

+ve ions are called cations, as they get attracted to cathode (-ve terminal), while -ve ions are called anions, as they get attracted to anode (+ve terminal),

Balancing number of atoms of both sides of eqn is needed, as no atoms can disappear or appear out of no where.

ex: CH4 (g) + H2O (g) => CO (g) + 3H2 (g) [Here we have to have 3 molecules of H2 gas on RHS to have same number of H atoms on LHS (total of 6 H atoms on both sides of the eqn]

Stoichiometry: Here we solve chemical eqn to find how many grams of compounds react to produce how many grams of final product. This can be done by first balancing the eqn, then finding out molar mass of molecules on the 2 sides, and then covert it to grams. This is pure Maths addition/subtraction.

Oxidation Reduction (redox reaction):  An oxidation–reduction is a reaction that involves the transfer of electrons between chemical species (the atoms, ions, or molecules involved in the reaction). During a redox reaction, some species undergo oxidation, or the loss of electrons, while others undergo reduction, or the gain of electrons.

Oxidation is process of something combining with oxygen. Since oxygen has 6 electrons in outermost shell (n=2), it needs 2 more electrons to complete it's shell, which requires those electrons coming from the other compound, so oxidation is defined as loss of electrons (more +ve charge, gain of charge). Oxygen itself gains electrons, and that is defined as reduction (more -ve charge, loss of charge).

For example, consider the reaction between iron and oxygen to form rust:

4Fe(s) + 3O2 (g) => 2Fe2O3 (s) (rusting of iron)

Fe has 2 electrons in outermost 4s shell, so it loses 2 electrons to for Fe2+. But more stable form is Fe3+ where it loses an extra electron from 3d shell to form half filled 3d shell (from 3d6 to 3d5 ). So Fe2O3 is a chain like structure with 2 oxygen on the end attaching to 2 Fe atoms, while oxygen in the center forms 1 bond with each Fe atom, forming O2- for all 3 O atoms, and Fe loses 3 electrons for each Fe atom to form Fe3+. Fe is oxidized, while O is reduced.

An atom’s oxidation number (or oxidation state) is the imaginary charge that the atom would have if all of the bonds to the atom were completely ionic. Oxidation involves an increase in oxidation number, while reduction involves a decrease in oxidation number.

The sum of the oxidation numbers for all atoms in a neutral compound is equal to zero, while the sum for all atoms in a polyatomic ion is equal to the charge on the ion. Look in KA for examples on Oxidation number.

Half reaction method: To balance a redox equation using the half-reaction method, the equation is first divided into two half-reactions, one representing oxidation and one representing reduction, and then both are balanced independently and then combined in the end.

 


 

Dissociation of Water molecules: 

When we talk of water, we treat it as having all it's molecules present as H2O. In reality, some of it's molecules dissociate into H+ and OH- . The H+ combines with a lone H2O molecule to form hydronium ion, H3O+.  About 6 in every 100 million (6 in 108) water molecules undergo the following reaction: 

  H2O (l) +  H2O (l) <=>  H3O+ (aq) + OH(aq) Here l refers to liquid while aq refers to aqueous soln. The extra proton forms a covalent bond with 1 of the lone pair electron in oxygen. The reaction is bidir.

Dissociation of water => https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introduction_to_General_Chemistry_(Malik)/06%3A_Acids_and_bases/6.05%3A_Dissociation_of_water

This process is called the autoionization of water and occurs in every sample of water, whether it is pure or part of a solution. This ionization of water gives rise to conductivity of water, albeit very low.

Conductivity of water: 

Let's take water as a solution in our standard cell to measure it's conductivity: electrodes of area A=1 cm2A=1 cm^2, gap d=1 cmd=1 cm, applied voltage V=1 VV=1 V, measured current I=5.5×10−8 AI=5.5×10^-8A (the 55 nA). It's extremely low current. Using R=V/I, we get R=1.8*10^7 ohms. So, resistivity ρ = R*L/A = 1.8*10^7 Ω-cm. So, resistivity is very high (almost insulator) as metals have conductivity in range of 10^-7 Ω-cm.

The ionic conductivity of water arises from conductivity of  H+ and OH-. We saw in "passive elements" section, that for any conducting medium, σ = -q*n*µ = n*q2*t/m, where n=number of electrons crossing per unit area, q=charge and µ= electron mobility. If there are other carriers besides electrons, then they all have to be added, i.e in semiC, we have holes (p), so σ = - (q*n*µn + q*n*µp) . For solutions, we have similar mobility concept for ions, as the free ions in a solution are what become carriers (as the ions are the ones that move in response to electric field). 

Electrolyte (ion) conductivity in a solution = ∑ (zi * F) * ui * C= ∑  λi * Ci , where λi= (zi * F) * ui = molar conductivity of ions per mole (charge_per_ion*Faraday*mobility_of_ion) and Ci= ion concentration (mole/cm^3). 

Faraday is defined as total charge (in Coulombs) on 1 mole of electrons = 1.6*10^-19 C * 6.02*10^23 e/mole = 96,500 C. So, (zi*F)= total charge on 1 mole of ions, which is equiv to "q" in our above classic resistivity eqn. This multiplied with mobility and conc, gives total conductivity σ = ∑  q*n*µ, where q=zi*F, n=Ci , µ=ui

Molar conductivity per mole λi= (zi*F)*ui = charge*mobility = C*cm^2/(V-s)/mole = S.cm^2/mol (units obtained by dividing σ (S/cm) by conc (mol/cm^3)).

 

Acid and Base: 

Definition of Acid/Base:

Bronsted and Lowry defined acid and base based on this. They defined acid as anything that is a proton/H+ donor. Coversely base was defined as anything that is a proton/H+ acceptor. 

ex: HCl (aq) + H2O (l) → Cl+ H3O+ Here HCl donated a proton (H+), so it's a acid, and H2O accepted a proton, so it's a base. The product formed from the acid after donating a proton is called "conjugate base" of the acid. Similarly the product formed from the base after accepting a proton is called "conjugate acid" of the base. 

The above eqn are generally rep by arrow in both direction, indicating the forward and backward reaction are both taking place at the same time, and it's a reaction that is in equilibrium.

Acid: donates H+. Product after donating becomes a conjugate base. It's called conjugate base, as it can accept H+

Base: accepts H+ . Product after accepting becomes a conjugate acid. It's called conjugate acid, as it can donate H+. 

Titration: It's a process for determining the conc of a acid/base in a solution. We find the unknown molar conc of a solution (i.e acid) by mixing it with known molar conc of another slution (i.e base) and note where does the resulting solution go neutral.

 


 

 

IT BPO Companies

IT BPO Companies

BPO is Business Process Outsourcing. It involves outsourcing services such as customer service, technical support, and back-office operations to other countries where cost of operation is lower. IT BPO refers specifically to IT service that is outsourced to other companies. These companies are more efficient in providing entire IT services for other companies. IT works out well for other companies too, as they don't want to have their own IT department to take care of billing, paychecks, etc. Total Market size of IT BPO is $450B as of 2026, and growing at 5%-8% a yr (estimates vary widely). I assume that all markets grow at same rate as what US GDP grows by, which is 5%/yr. USA is the largest market by revenue accounting for 30%-40%. India is the largest provider of IT BPO services, as is evident by the fact that most IT companies are Indian Co, or are situated in India. Besides India, other countries as Philippines, Vietnam, Indonesia are also emerging players. 

Global IT spending is $5.5T as of 2025, of which $1.7T is spent on IT services. BPO part of these IT services seems to be only $0.5T.

 

 

Mag 7 Companies

Mag 7 (Magnificent 7)

This is a term referred to top 7 companies in USA, which are in tech and highly successful. They have dominated the total stock market cap, and have provided higher return than S&P500. These are the 7 stocks => Alphabet, Amazon, Apple, Meta, Microsoft, Nvidia, and Tesla. 

ETF: MAGS (Roundhill Magnificent Seven ETF) is the top ETF with 0.29% expense ratio and $3B in AUM to track Mag 7. It was launched on 04/11/23 and initially traded as ticker "BIGT". It has provided 100% return in 3 yrs since it's launch, excluding dividend.

We discuss the 7 stocks to see how worthy of investment are they:

Apple: 

 

 

Crypto Currency

Crypto Currency (CC):

First of all, crypto currency is just bunch of numbers. They re no different any other currency that the government issues. In case of Crypto currency, this currency is issued by a distributed source controlled by the people, so government has no authority to print or destroy it. To make this currency work, merchants have to be willing to accept this currency, just like they accept dollars or other government issued currency. When these crypto currency were issued in 2000's, people were highly skeptical if any businesses would accept them. But over, big players started accepting them. Thus they got solidified as real currency, and their values started going up in value. Bunch of crypto currency came into existence, each claiming to be better than the other. With so many different types of Crypto currency, it's hard to say which of these will survive long term.

Basics of CC:

CC are encrypted numbers (called as digital currency). The seller/buyer are anonymous, and each CC needs to be mined (i.e numbers generated) using a long series of mathmatical solutions that takes a lot of compute power. Each CC has a limit of how many can be mined till 

Major CC:

As of 2024, combined market cap of all CC is ~$2T. Top 5 cryptocurrency comprise 80% of the market. They are all traded on stock exchanges just like stocks.

  • Bitcoin (BTC) = $1.2T or about 55% of market. This is the most popular CC, and was the first one that started the CC business.
  • Ethereum (ETH) => $0.3T or about 15% of market. 
  • Others: Tether USDt (USDT) @$115B, BNB (BNB) @80B and Solana (SOL) @60B comprise next 12%

There's also an ETF for crypto, so that you can invest in CC, w/o directly buying the CC. This is lot better, since all crypto exchanges charge some fees for buy/sell of crypto, and also add a wide spread, which eats into your profits immediately.