#Class 12 Maths Vector Algebra

[quietimtryingtostudy]

me looking at my math notebook blurry eyed: 

i sit by myself talking in my roooomm tryna understand youuuu

[shiva10010]

JEE Main 2025! Prep Your Math Strategy with the Latest Syllabus

The Joint Entrance Examination (JEE) Main is a critical gateway for aspiring engineers in India. While the official syllabus for JEE Main 2025 will likely be released by the National Testing Agency (NTA) in November 2024, you can get a head start on your preparation using the previous year's syllabus as a guide.

This article focuses on the Mathematics section of the JEE Main 2025 exam.

Key Elements of the JEE Main 2025 Maths Syllabus

The JEE Main Maths syllabus covers a wide range of topics from Class 11 and 12 NCERT textbooks. Here's a quick glimpse of the major areas you'll encounter:

Algebra: Sets, relations and functions, complex numbers, matrices and determinants, mathematical induction, binomial theorem, sequence and series.

Calculus: Limit, continuity and differentiability, integral calculus, differential equations.

Geometry: Coordinate geometry, three-dimensional geometry, vector algebra.

Other Quantitative Techniques: Statistics and probability, trigonometry, mathematical reasoning.

Why eSaral is Your Perfect JEE Main 2025 Maths Companion

While the syllabus provides a roadmap, in-depth preparation is essential for success. This is where eSaral steps in:

Complete Syllabus Coverage: eSaral ensures you cover all the essential JEE Main Maths topics.

Detailed Learning Material: Access a treasure trove of resources, including crisp explanations, illustrative examples, and topic-wise revision notes [refer to JEE Main 2025 Maths Revision Notes on eSaral for details].

Catered Learning Styles: Whether you prefer short and focused content or a deep dive into concepts, eSaral offers options to match your learning style.

Practice Makes Perfect: Sharpen your skills with ample practice problems available on the platform.

Additional Tips to Conquer JEE Main 2025 Maths

Focus on NCERT Textbooks: They form the foundation for the entire syllabus.

Regular Practice: Solve problems consistently to build speed, accuracy, and exam temperament.

Mock Tests: Take regular mock tests to simulate the exam environment and identify areas for improvement.

Clear Your Doubts: Don't hesitate to seek clarification from teachers or online forums.

Conclusion

JEE Main 2025 Maths can be aced with the right approach and resources. By leveraging the detailed syllabus and comprehensive learning materials offered by eSaral, you can approach the exam with confidence and a strategic plan. Remember, consistent effort and focused practice are key to achieving your engineering dreams.

[mpboardsolutions-blog]

MP Board Class 12th Maths Book Solutions in English Medium

MP Board Class 12th Maths Chapter 1 Relations and Functions

Chapter 1 Relations and Functions Ex 1.1

Chapter 1 Relations and Functions Ex 1.2

Chapter 1 Relations and Functions Ex 1.3

Chapter 1 Relations and Functions Ex 1.4

Chapter 1 Relations and Functions Miscellaneous Exercise

MP Board Class 12th Maths Chapter 2 Inverse Trigonometric Functions

Chapter 2 Inverse Trigonometric Functions Ex 2.1

Chapter 2 Inverse Trigonometric Functions Ex 2.2

Chapter 2 Inverse Trigonometric Functions Miscellaneous Exercise

MP Board Class 12th Maths Chapter 3 Matrices

Chapter 3 Matrices Ex 3.1

Chapter 3 Matrices Ex 3.2

Chapter 3 Matrices Ex 3.3

Chapter 3 Matrices Ex 3.4

Chapter 3 Matrices Miscellaneous Exercise

MP Board Class 12th Maths Chapter 4 Determinants

Chapter 4 Determinants Ex 4.1

Chapter 4 Determinants Ex 4.2

Chapter 4 Determinants Ex 4.3

Chapter 4 Determinants Ex 4.4

Chapter 4 Determinants Ex 4.5

Chapter 4 Determinants Ex 4.6

Chapter 4 Determinants Miscellaneous Exercise

MP Board Class 12th Maths Chapter 5 Continuity and Differentiability

Chapter 5 Continuity and Differentiability Ex 5.1

Chapter 5 Continuity and Differentiability Ex 5.2

Chapter 5 Continuity and Differentiability Ex 5.3

Chapter 5 Continuity and Differentiability Ex 5.4

Chapter 5 Continuity and Differentiability Ex 5.5

Chapter 5 Continuity and Differentiability Ex 5.6

Chapter 5 Continuity and Differentiability Ex 5.7

Chapter 5 Continuity and Differentiability Ex 5.8

Chapter 5 Continuity and Differentiability Miscellaneous Exercise

MP Board Class 12th Maths Chapter 6 Application of Derivatives

Chapter 6 Application of Derivatives Ex 6.1

Chapter 6 Application of Derivatives Ex 6.2

Chapter 6 Application of Derivatives Ex 6.3

Chapter 6 Application of Derivatives Ex 6.4

Chapter 6 Application of Derivatives Ex 6.5

Chapter 6 Application of Derivatives Miscellaneous Exercise

MP Board Class 12th Maths Chapter 7 Integrals

Chapter 7 Integrals Ex 7.1

Chapter 7 Integrals Ex 7.2

Chapter 7 Integrals Ex 7.3

Chapter 7 Integrals Ex 7.4

Chapter 7 Integrals Ex 7.5

Chapter 7 Integrals Ex 7.6

Chapter 7 Integrals Ex 7.7

Chapter 7 Integrals Ex 7.8

Chapter 7 Integrals Ex 7.9

Chapter 7 Integrals Ex 7.10

Chapter 7 Integrals Ex 7.11

Chapter 7 Integrals Miscellaneous Exercise

MP Board Class 12th Maths Chapter 8 Application of Integrals

Chapter 8 Application of Integrals Ex 8.1

Chapter 8 Application of Integrals Ex 8.2

Chapter 8 Application of Integrals Miscellaneous Exercise

MP Board Class 12th Maths Chapter 9 Differential Equations

Chapter 9 Differential Equations Ex 9.1

Chapter 9 Differential Equations Ex 9.2

Chapter 9 Differential Equations Ex 9.3

Chapter 9 Differential Equations Ex 9.4

Chapter 9 Differential Equations Ex 9.5

Chapter 9 Differential Equations Ex 9.6

Chapter 9 Differential Equations Miscellaneous Exercise

MP Board Class 12th Maths Chapter 10 Vector Algebra

Chapter 10 Vector Algebra Ex 10.1

Chapter 10 Vector Algebra Ex 10.2

Chapter 10 Vector Algebra Ex 10.3

Chapter 10 Vector Algebra Ex 10.4

Chapter 10 Vector Algebra Miscellaneous Exercise

MP Board Class 12th Maths Chapter 11 Three Dimensional Geometry

Chapter 11 Three Dimensional Geometry Ex 11.1

Chapter 11 Three Dimensional Geometry Ex 11.2

Chapter 11 Three Dimensional Geometry Ex 11.3

Chapter 11 Three Dimensional Geometry Miscellaneous Exercise

MP Board Class 12th Maths Chapter 12 Linear Programming

Chapter 12 Linear Programming Ex 12.1

Chapter 12 Linear Programming Ex 12.2

Chapter 12 Linear Programming Miscellaneous Exercise

MP Board Class 12th Maths Chapter 13 Probability

Chapter 13 Probability Ex 13.1

Chapter 13 Probability Ex 13.2

Chapter 13 Probability Ex 13.3

Chapter 13 Probability Ex 13.4

Chapter 13 Probability Ex 13.5

Chapter 13 Probability Miscellaneous Exercise

[ncertmcq]

Vector Algebra Class 12 Notes Maths Chapter 10

[meritbatch]

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra

[efastforward123]

E Fast Forward

JEE Main Syllabus: Physics, Chemistry and Maths

JEE Main is one of the most prestigious engineering entrance exams in India. The JEE Main exam serves as a gateway for admission to the top engineering colleges in India, including the Indian Institutes of Technology (IITs) and the National Institutes of Technology (NITs). The exam is conducted by the National Testing Agency (NTA) twice a year, and the syllabus for JEE Main is vast and includes topics from class 11 and 12.

Mathematics

Trigonometry is an integral part of JEE Main maths and includes various subtopics such as trigonometric ratios, trigonometric identities, and trigonometric equations. Coordinate geometry involves understanding the relationship between geometrical figures and the coordinate plane. Vectors are utilized to describe the direction and magnitude of a particle. Understanding the concepts of all these topics is crucial to score well in the JEE Main maths section. One of the most common mistakes students tend to commit in the JEE Main maths section is that they tend to waste a lot of time on simple questions. To avoid this, students should strategically plan their exam and focus on solving the difficult questions first.

Physics

Optics is another crucial topic in JEE Main physics and involves the study of light and its behavior. Students need to have a good understanding of concepts such as reflection, refraction, lenses, and mirrors. Modern physics is also an essential topic in JEE Main and includes subtopics such as wave-particle duality, photoelectric effect, and nuclear physics.

Chemistry

Chemistry requires a lot of practice, and it is recommended to attempt as many questions from previous year’s papers as possible. One way to prepare for JEE Main chemistry is to focus on practicing chemical reactions and reaction mechanisms. Students must have a thorough understanding of the periodic table and chemical bonding to score well in JEE Main chemistry.

Exam Pattern and Marking Scheme

The JEE Main exam is conducted both online and offline and consists of three sections, including physics, chemistry, and mathematics. The physics and chemistry sections consist of 30 questions each, while the maths section consists of 25 questions. The JEE Main exam is for a total of 360 marks, and each question carries four marks. However, for every wrong answer, one mark is deducted, and for any unanswered question, no marks are deducted.

It is essential to read the question paper carefully and plan the exam strategy accordingly. It is always advisable to attempt the questions in which students are confident, and they must avoid negative marking.

NCERT Books

NCERT books are an essential part of JEE Main preparation. More than 60% of the JEE Main syllabus is covered in NCERT books. NCERT books are beneficial in providing a strong foundation for students by developing their concepts from the basics. Students should start with NCERT books before moving on to other reference books.

Students must focus on the key topics of NCERT books, which are crucial for JEE Main. These topics include calculus, algebra, mechanics, optics, and chemical reactions. Students should read the NCERT books carefully, focus on important sections, and take notes for future revisions.

Online Resources and Mock Tests

There are many online resources that students can use to prepare for JEE Main. Students can access online sample papers, previous year’s JEE Main question papers, and tutorials to help them prepare for the exam. There are many websites and apps that provide video lectures, practice questions and solutions for JEE Main.

Download Android App

Mock tests are important for JEE Main preparation as they help in identifying the strengths and weaknesses of students. By taking mock tests, students can gauge their level of preparation and get a feel for the actual exam. Many websites provide free mock tests that mimic the JEE Main exam pattern, which students should take to improve their scores.

Time Management for JEE Main

During the exam, students should allocate their time wisely and attempt only those questions they are confident about. They should avoid getting stuck on a particular question and move forward if they are unable to solve it. Students must keep track of time during the exam and must allocate sufficient time for revising their answers.

Revision Techniques

Revision is important for JEE Main as it helps in consolidating knowledge and improves memory retention. Revision techniques such as active recall, spaced repetition, and mind maps can help students in retaining information more effectively.

Students should create revision notes while studying for JEE Main and revise them regularly. They should focus on understanding the concepts and not just memorizing them. They can use flashcards to revise key concepts or formulas, which will help them in retaining the information for longer.

JEE Main is a challenging exam, and it requires dedication, focus, and hard work. To score well in JEE Main, students must have a clear understanding of the syllabus and develop a sound strategy for preparing for the exam. Students must focus on the essential topics, practice regularly, and take mock tests to enhance their chances of success. Effective time management, revision, and online resources can help students in achieving their goal of cracking JEE Main.

[marvellouspinecone]

Ok so it's been approximately forever since you wrote this amazing lecture to me, i had time enough to graduate and i finally have some headspace to start untangling this mystery.

Yes, i'm only familiar with high school level of maths, however bc of how our education is structured, somewhere around grade 9 my knowledge intake of maths became increasingly more sloppy and uneven bc the teachers were putting more resources into kids who were going to pass high level state exam in math, amd i was only basic level (and on the other hand all my efforts went into English and literature exams). So i immediately forgot practically all of the.most basic algebra after graduating high school bc it wasn't my specialty anymore. I feel very much like a five year old in this sphere, so you don't have to be afraid of coming off as condescending.

I'm assembling all the questions in the notesapp and some bits here are much older than others.

So first: i don't think this question is important for my understanding of the Gauss sum formula bc it's pretty intuitive anyways, but i'm just curious: what does k mean there? What does it stand for and how it influences the calculations?

Second: with affine transformations you lost me at matrices. Up until that point i was following it, but then i got confused by the lowercase l and the numbers in the matrix like 11, 21, 31 - i don't understand where they came from. And there must be some pattern that is important to further steps, bc without understanding this the rest is starting to look a bit arbitrary to me.

Actually no, i think the issue started with the vector length - i don't understand where the number comes from. It seems like the vector length (the n in R^n) equal how many numbers there are in a column inside the brackets, but i understand it's kind of a dumb assumption on my part. The matrix-vector-multiplication looks sooo pretty tho, i can easily imagine incorporating it in the visual texture in the background if i fail to wrap my head around it and come up with better ideas for it.

The symbolism part makes perfect sense to me, but i'm still curious about the math process.

Third: don't ask me how, but i think i understand on a subconscious level why you can represent a sphere by a vector from centre to radius (if my understanding of vectors is correct. I'm basing it more on my brief hysics phase in grade nine tho, so it might be completely wrong at this level of geometry). Also correct me if i'm wrong, but sharp brackets are supposes to mean that we are more interested in the quantity of a number than it being negative or positive, right? I don't know the English term for it i think in my classes we called it something like modules? It was five years ago don't @ me.

The whole lie geometry thing makes sense to me like metaphysically. Like someone would claim to inderstand God on some level, even if He is ultimately unknowable. Idk it just looks right. The vibes resonate with me. The mention of gears gave me Ideas tm and i think they can be a really cool interactive element.

Those were my math remarks and questions, i'd be super happy if you want to elaborate on any of that!

Moving on to the creative ideas, get children and sensitive people away from the screens, local linguist will be commiting math blasphemy

Now the Gauss sum formula got me thinking of the actual memory equation. I want something like "the sum of Time Lord's memory equals infinity, if and only if they keep moving forward and changing". Maybe starting as something like this?

(I used 12 up there, bc by the end of Twelve's life his memory is the sum of his 12 regenerations, and the number is just symbolic here, does that make sense?)

Unfortunately here i really feel the lack of my understanding of vectors and transformations. I feel like they will fit really well here in place of the question mark, since they are about change and movement. I hope you can dumb that bit down even more <3

Also i know that this equation doesn't make sense bc 12(12+1)/2 ≠ ∞, but since i don't speak math i'm more free to make this language both my canvas and my collage pieces, plz don't get mad🥺 I feel like the "iff" sign serves as a kind of indulgence that says "i know this is absolute nonsense but it works in these secific circumstances". And the impossible things working in very specific circumstances is a very Doctor Who thing, don't you thinkthink?

Hi, i'm here with my blank page!

So it's not entirely blank, it's whole spread and i already did something on the left side (it's a movie ticket from when i went to see Twice Upon A Time in theater, with some packaging taped on top because i've noticed that the writing started rubbing off over the years, and the background is the inbetween bits from some stickers for some visual interest, because as a junkjournaller i don't throw anything away lol)

The prompt on top says "if a Time Lord is a sum of their memories, write down the equation" And there is this fun grid pattern with some numbers in the corner and another number in the other corner (hi Twelve). I feels like it wants something maths-related, but i'm hopeless in maths. I know that you like maths. Maybe you can help me, like with a smart math trick, and/or something related to Twelve, maybe a quote that might be fitting or just a nudge in a certain direction. It doesn't have to be an equation as the prompt says, but the main themes i'd like to reflect are memory, maths (science as poetry perhaps?) and Twelve's story/character to tie in with the ticket.

Any thoughts? No pressure to come up with anything big or deep tho, even just a hint might be helpful to figure it out

Firstly, thank you so much again for giving me this chance to talk about maths ❤ Also, I feel so honoured I get to contribute and spark ideas (hopefully?) for your beautiful, beautiful art project!!

Also, very sorry, but this will be very maths heavy, obviously. I'll try to put many pictures so you don't really have to understand anything; you can just go with the aesthetics.

That said, if you really do want to understand something, I will probably not put enough info here to do so. So, just ask, I can go in small steps and explain it from the start. Like, I love doing that. My mum regularily has to indulge me and let me explain higher mathematics until she (kinda) gets it.

But, I mean, most of these things I'm about to mention, you learn about after at least two semesters of intensive mathematics studies at uni, so...they're not that easy to understand.

Also, I will assume, that you only know high school maths, I hope that is alright? I don't actually know your level, please forgive me if you feel like I am treating you like a 5 yo. (I will basically explain this like I would for my family).

I'm gonna put this under the cut again :)

Sum Formula by Gauss

This is probably the simplest bit of maths that I am going to mention, so let's put it first.

'A timelord is a sum of their memories' obviously lets you think that you'll have to include a sum, somewhere. In mathematics, we normally use the greek letter Σ (capital sigma) to show that we are doing summation. All the things you want to sum are put after the letter and the start and finish are marked on top. Then

would be the sum of all (natural) numers 1 to n. If n=4, the above would read 1+2+3+4, f.ex.

Here, we want to sum over memories. So, obviously, we could find some sort of symbol or artistic interpretation of 'memory' and put it behind the sum, where the k is.

Or, what I would propose, we concentrate the 'memory' into the 'name' or rather number of the different regenerations. Because then, if the memories of the timelord 12 are the timelords Doctor 1 to 11, which would give us the above sum with n=11:

Here, I am just pretending, 'Timelord' is a function on natural numbers, defined by the sum on the right.

(Sidenote: A function is like a rule that takes some input and gives you an output, f(x)=x-1, for example would be a commonly known one or f(x)=x^2, or f(x)=sin(x). We often see them called f or g or something in school and make them work with numbers but technically they can work on any set and have any name and do anything. Quite cool :) )

The proper definition of the function would have to come first and could look something like this:

Which just means Timelord is a function that takes a natural number and gives you a result in the Universe. Which does not mean much mathematically, except for that you can plug only natural numbers into the function and get an element from the unknown set 'Universe'. Actually, the result would be another natural number, of course.

Now, there is a well-known formula for this specific sum, namely

that is all natural numbers 1 to n summed together gives n times n+1 divided by two. So, 1+2+3=3*4/2=6 for example. So you could definitely include that with like a little extra equation.

Also, you could obviously change this whole thing up a bit, because for now, we have not included the war doctor at all. Or the fugitive Doctor. Or any other forgotten Doctor or spin-off Doctor or or or - although admittedly, the Doctor IS missing those memories, so...?

But, there is one more neat thing with this formula, and that is that there exists a nice geometric visualisation:

So, here the idea is, that we want to add the numbers 1 to n and those are the green squares from top to bottom. See? there is one in the first row, then two in the second and so on and n in the bottom row.

And all together, the green squares are exactly half of the number of the squares in the right rectangle, which are n times (n+1) many.

Or, to see it another way: you could always add up two of the rows to get a full row of green squares. The top row with the last but one row and the second and third row. And then you always either have n+1/2 full rows (, i.e. n*(n+1)/2, like here) or n/2 full rows and one half row ((n/2)*n+n/2=(n/2)*(n+1)).

So, I think there are some themes and images here you could work with :)

Affine Transformations

Okay, so, this one is my favourite because of the WORDS :D (Also because I am taking discrete geometry right now and affine transformations are important there.)

So, first, let's have a little look at LINEAR transformations. Slightly simplified, those are functions between vectorspaces that fulfill certain properties.

What is a vectorspace? Some mathematical thing (Sidenote: also a very nice word, vectorSPACE :) ) where vectors live in. Vectors for our purposes look like numbers pressed between brackets, so for example:

would be three vectors. All from different vectorspaces because they have different lengths or as we like to say: DIMENSIONS :)

Vectors can also lie sideways like v=(1 , 2, 0, -3), but I am not a fan of that. And technically the difference is not big and unimportant for our purposes.

Okay, now the vectorspace we are interested in is called R^n, which is vectors of length n. So in the above picture, v is in R^2, w in R^4 and u is 'the origin' of R^3.

Then a linear transformation is a function from R^n to itself (it could go to another vectorspace, too but let's not overcomplicate that) which we write as:

And the linear transformation fulfills the rules:

(1) means, that if I look at the output of a function of the sum of two vectors, I get the same as if I take the sum of the output of both vectors on their own. So, L(3)=L(1)+L(2), for example.

(2) means if I look at the output of the function on the multiple of a vector, then that gives the multiple of the output of the function on the vector. So, L(2)=2*L(1), for example.

If the function fulfills this, we call it linear.

We often write linear functions as MATRICES. So, L could look like this:

which we often write more generically as

And then applying this function to a vector is a matrix-vector-multiplication, which again works with sums :D

Okay- this last bit is probably to much, sorry for that. The takeaway: We 'multiply' matrices and vectors with specific rules and then that fulfills rules (1) and (2) and gives us a LINEAR transformation.

If the matrix is INVERTIBLE - don't worry about what that implies - then we call the linear function BIJECTIVE.

Now, what is an affine transformation? It is a linear transformation with a fixed vector added and looks like this:

Look familiar? Yes, it's exactly the same as before, just with the vector w added! This actually comes from a linear transformation made simpler, where the linear transformation looks like this:

If you were to calculate this you would see that Lv with the little tilde thing really resulted in the thing in the brackets. Point is, if we forget the 1s, we get the affine transformation from before but like this we have one linear transformation again.

And now I am coming back to Doctor Who:

Affine transformations describe rotations, translations, scaling and like slight bending of the space.

And if the underlying linear trafo (gonna start using our neat little shortening from uni) is bijective, INVERTIBLE, you can retract it to where you started from.

And that sounds like memory? Like regeneration? We could assume L is an invertible transformation - regeneration or ourselves becoming different people throughout our lives? - so we could technically go back to who we were or that is still part of us and we can follow it back to the start, aslong as we retain the memory. BUT there is the affine part, too, that little vector w added. What if that was time? The part of the transformation that makes the trafo not linear anymore (at least unless we go to a higher plane of existence in R^(n+1)). What if that vector somehow prevented us from actually going back?

(That is not how it works in maths, I don't think, you could just subtract it, but let's imagine we can't)

Wouldn't that be some beautiful metaphor of changing and remembering and becoming something new? While moving through space and time? (You could technically also split that vector up into a space and a time vector if you liked :) )

Maybe you can do something with that?

Lie Geometry

My favourite part, because Geometry, but now we are entering the part of maths where I have to look stuff up as well. So, I'll try to keep it short and just give you some concepts and images.

Just, let me say that I love this because

Rule 1: The Doctor lies

This is all about spheres in ORIENTED CONTACT

Circles are 1-dim spheres, so CIRCULAR GALLIFREYAN?

Also they look funny, a little like hedgehogs:

So, ask me if you really want to know why but I'm not going to explain it here and I would have to look that up, too, but apparently we can represent 'normal' spheres (Möbius spheres to be precise) by their center c and radius r as the vectors

Don't worry about why the brackets suddenly are sharp, there is a meaning but it's not too important right now. It's still a vector. Then, c is a vector, too - giving us the n-dimensional point of the circles center - so this actually has more than 3 entries. (c,c) is the standard scalar product, just imagine this as the length of the vector c or how far away from the origin we are.

Takeaway here: Spheres are defined by their center and radius. Now that doesn't sound so bad, does it?

Now you may have noticed the radius r only appears as r^2. Lie spheres have an ORIENTATION however, that means they can have negative radius. Hence, we have to add an extra info (how long is our radius and does it go inward or outward?) as another coordinate:

If r>0 we will draw the arrows in the first picture outward, otherwise inward. Or the other way round. Honestly, that's just convention, it doesn't really matter. This is now a lie-sphere. They make up the points in the Lie quadric, defined by the Lie-scalar product. So they are all the points y fulfilling:

So we want that weird angly thing (Lie scalar product) on the left that translates to that rather long sum (yes, you could write this as a sum again, too) to sum up to 0.

This is all very difficult and confusing, I know. But pretty formulas?

Anyway, the thing called 'oriented contact' works with that angly thing, too. It means that two spheres meet tangentially - in exactly one point - and they meet such that their arrows point in the same direction at the contact point:

Like this, those are in oriented contact! (Please ignore my inability to draw circles)

The mathematical condition for that is:

Okay, what was all that? So, first the definition for the lie scalar product works slightly differently for two different vectors, but it fits together with the version we saw earlier.

And then I did some math tricks (or rather my professor did). It all just says in the end: For oriented contact, we need that the distance between centers is the same as the difference between radii.

And the y, y' are defined as:

Idk. I am not expecting you to get super into Lie geometry and its theory now. Rather that you might want to draw some circles with arrows?

What is also interesting is that these circles are often seen as gears. Because the radius orientation can represent a wheel that can only turn one way and then the oriented contact means if the other wheel touches on the outside it has to turn the other way and if it touches from the inside they will turn the same direction.

This also has a relevance to the problem of Appollonius if you want to namedrop, but I won't go into that now.

Markov-Chains

You asked for quotes as well (so sorry, I'm not going to explain what Markov Chains actually are, just know: Stochastics) and I really liked this quote characterising Markov Chains in 'normal' language:

In relation to the present the future is independent of the past.

Or something along those lines. I unfortunately don't actually remember the exact quote. But the above was the spirit.

Other symbols

Imo it is always a good idea to randomly write 'Let Ɛ>0.' somewhere or put little arrows like ⇔ or ⇒ wherever. Or put the little lightning bolt contradiction thingy:

(funfact, my ao3 name is 'hvis-og-bare-hvis' because that means 'if-and-only-if' translated, a phrase we use SO MUCH in maths and that is often abbreviated to 'iff' or written as ⇔)

Conclusion

I know this is all very MATHS now and also I didn't really explain anything properly. Except for the first point maybe. Feel free to ask any question about any topic, I would love to elaborate. I just can't promise it won't take weeks again to answer.

But I had a blast and I feel so honoured I get to weigh in on your art book! Thank you so much!! I hope I could help you a little!

[vectoralgebraclass12]

Vector Algebra is a one of the basic topics that is necessary to understand the various concepts of Physics and Mathematics. Vector Algebra is the prerequisite for concepts like Kinematics and basics for students intended to go for engineering studies like IIT JEE Mains, other entrances, boards, and higher studies in Mathematics. It covers all the basic to advance concepts, each NCERT Solutions, R.D Sharma, R.S Aggarwal, Board’s Question Bank, Important Examples and Comprehensive Educational Resources for Vector Algebra Class 12 Maths including Assignments and high-quality video lectures prepared by expert teacher from the boards point of view.

Vector Algebra Class 12 Maths

[ashishkumarletslearn]

Vector Algebra Class 12 Maths IIT JEE Mains

Detailed explanation of vectors and scalars, magnitude and direction of a vector.Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors including basics to advance level concepts, NCERT Solutions, R.D Sharma, R.S Aggarwal, Important Examples from the boards point of view for NCERT Chapter 10 Vector Algebra Class 12 Maths IIT JEE Mains.

Below is the list of High-Quality Video Lectures links for Vector Algebra Class 12 Maths:

Lecture -1 Vector Algebra Class 12 Maths Chapter 10

NCERT Exercise 10.1 Vector Algebra Class 12 Maths

NCERT Exercise 10.2 Vector Algebra Class 12 Maths

How to study Vector Algebra & Three Dimensional Geometry?

[cbse-board]

Detailed explanation of ‘how to represent vectors, types of vectors and ncert solutions for Exercise 10.1 of Chapter 10 Vector Algebra Class 12 Maths.

#VectorAlgebra #Class12Maths

[penguin-ghost]

Week 0 - What I am taking this semester

Uni starts next Monday!

As a Physics and Philosophy major, I’ve enrolled in 5 units of study this semester:

Physics 1A

This is introductory physics course, and I'm taking the advanced stream that will be more fast paced. We will cover 3 modules this semester:

Mechanics

Thermal Physics

Oscillations, Chaos and Waves

Physics is my main interest, and although the introductory class doesn’t have very exciting modern physics, I’m still pretty hyped about learning Physics at a more advanced level with a lot of maths.

Reality, Ethics and Beauty

This is an introductory philosophy class that discusses the central issues in metaphysics, ethics and aesthetics. We will begin with the philosophy of religion, and the reading for Week 1 is on Thomas Aquinas and his five ways. I've never formally studied philosophy, so I am also super hyped about this class too.

Linear Algebra

I think I'm excited for every class hahaha. We are required to watch pre-lecture videos (produced by the professor) to familiarise ourselves with the material, so that harder content and examples can be covered in the actual lectures. I genuinely enjoy this learning format, and I think it's effective for me. I've already watched and made notes on the first set of videos on Vectors in R^n and Vector Space. It doesn't take long to realise that uni maths is a lot more reliant on symbols rather than words, so I will have to get used to reading and interpreting them faster.

Calculus of One Variable

We will begin with complex numbers! This was my favourite maths topic back in Year 12. But I'm not sure how much of a revision it'll be - it's been more than 100 days since I've last touched any maths, let alone maths on the complex plane. And by the look of the unit schedule, the pace for the class is really fast. A little worried.

Introduction to Programming

I have attempted to learn how to code on at least 3 occasions. Never succeeded. It's got to a point where I don't even know what my attitude towards programming is anymore. Who knows, maybe it’ll be fun. It's also quite essential for my physics career so I will try to enjoy it as much as I can.

p.s. All of my classes are remote, as I am currently away from Campus, and I will be doing weekly updates on how I find uni.

[quietimtryingtostudy]

Some questions of vector algebra are quite simple so it gets frustrating when you’re like ‘I KNEW THAT’ when you see the solution to a problem you couldn’t solve

[valpy-blog]

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4

NCERT Solutions for Class 12th Maths Chapter 10 Vector Algebra Ex 10.4

Get Free NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Vector Algebra Exercise 10.4 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of…

[anushacbse]

CBSE Class 12 Maths Revision Notes

Class 12 Maths Notes: Class 12 Maths Notes is designed to help for the students who are going to appear final exams, students can read this CBSE Class 12 Maths Revision Notes before going to exam to cover all chapters and topics in last minute and can easily revise all chapters of class 12 Maths. Important topics like Trigonometry, integrals, equations and three-dimensional geometry has been included in Class 12 Maths Notes. For CBSE Students board exams will be conducted by NCERT. Class 12 Maths Notes has been designed on the basis of the latest NCERT syllabus textbooks. So students can check CBSE Class 12 Maths Quick Revision Notes and can download the solutions in PDF’s format as well. Students can practice the Class 12 Maths Revision Notes to gain more knowledge.

Class 12 Maths consists of 13 Chapter below is the list of chapter Names.

Chapter 1: Relations and Functions Chapter 2: Inverse Trigonometric Functions Chapter 3: Matrices Chapter 4: Determinants Chapter 5: Continuity and Differentiability Chapter 6: Application of Derivatives Chapter 7; Integrals Chapter 8: Application of Integrals Chapter 9: Differential Equations Chapter 10: Vector Algebra Chapter 11: Three Dimensional Geometry Chapter 12: Linear Programming Chapter 13: Probability

Students can download the PDF’s of Class 12 Maths Revision Notes. The Class 12 Maths Notes will also help in understanding the chapters properly as it will contain the important points for revision purpose in short times