help-27

[vernal monolith]

The plane obtained by dot product

[empty flame]

its simple if you have P Q and R then just take the plane that $\vec{PQ}$ and $\vec{QR}$ make then you get a plane containing all 3 points

[woven radish]

calculus is fun

[empty flame]

the dot product isnt giving a plane

oh wait i understand what you mean

[vernal monolith]

It should only consist Q and R

[empty flame]

the dot product is 0 then use the formula that uses coordinates and you get an equation of the form $ax+by+cz+d=0$ which will be the equation of the plane containing Q and R

[woven radish]

calculus is fun

[vernal monolith]

And not P since its a point on a vector that is perpendicular to the point

[empty flame]

P will not be in this plane it wiil be outside it

[vernal monolith]

Arjun was asking to obtain the plane that consists all three by using this plane and the normal vector

[empty flame]

what did arjunn mean by middle point

[vernal monolith]

Oh that is from his first question, you can ignore that

The plane contained middle point of given two points

So after finding equation of plane, he had to use middle point to find constant on the plane equation

Since a point on the plane must satisfy it

[empty flame]

what is a middle point

middle point of what

do you mean the midpoint of the segment joining the 2 given points

[vernal monolith]

These 2 points

They are irrevelant now

Ignore

[empty flame]

ok so middle point is the midpoint of the segment

what i understood from above is that he wanted to know the eq of plane containing Q and R

@wooden axle

and read the 2 lines after this

is this what you are asking about ?

[vernal monolith]

Wouldnt that make it worse since you need to find a plane that intersects 2 parallel planes

[empty flame]

why would you need that

wait a sec forget what i said above i was assuming that we know 3 non collinear points in the plane and we want to find its equation

bruh how did i assume that in my mind

[wooden axle]

.close

[devout snow]

Channel closed

Closed by @wooden axle

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
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[empty flame]

wait

[empty flame]

@wooden axle

[wooden axle]

.reopen

[empty flame]

you have a normal vector of the plane

because you are assuming that PQ is perpendicular to the plane is that right

[wooden axle]

Right

[empty flame]

and you have 2 points which are Q and R which are contained in the plane

in this case the equation of the plane containing Q and R is $ax+by+cz+d=0$ where a,b,c are the components of the normal vector which is PQ in our case

[woven radish]

calculus is fun

[empty flame]

and $d=-ax_0-by_0-cz_0$ where $(x_0,y_0,z_0)$are the coordinates of the point in the plane which is R

[woven radish]

calculus is fun

[empty flame]

so the dot product solves the issue

here you cant use cross product

because you are only given 2 points in the plane

if you know three non collinear points in the plane then you can use cross product

[empty flame]

and in this case its not necessary to know any point outside of the plane

[wooden axle]

I am not able to see why we cannot

[empty flame]

ok i will tell you

if you have 2 points how many different vectors can you do

[wooden axle]

One

[empty flame]

ok what if you have 3 points

[wooden axle]

2

[empty flame]

ok good now the cross product requires how many vectors ?

[wooden axle]

2 vectors

[empty flame]

ok so if you have 2 points you cant

because you cant do cross product in the first place

[wooden axle]

Yes we can't

[empty flame]

so you should know at least 3 noncollinear points to be able to do cross product

[wooden axle]

True

[empty flame]

if you have just Q and R that not enough thats why i was sayng that you cnt use cross product

[wooden axle]

Yes right

My next question

If we are given a point a,b,c on the plane

And line from origin is perpendicular

What will be the equation of the plane

[empty flame]

perpendicular to the plane ?

[wooden axle]

So here one vector is i guess a-0,b-0,c-0

X-a,y-b,z-c

Second vector right

[empty flame]

does the line pass through P(a,b,c)?

[wooden axle]

Yes

[empty flame]

so the line is perpendicular to the plane at P

[wooden axle]

Yes

So dot product is 0

Am i right?

[vernal monolith]

Yes

[empty flame]

dot product between which vectors

[vernal monolith]

There are infinitely many planes which satisfy this though

[wooden axle]

These vectors

So here we get
(x-a)a+(y-b)b+(z-c)c=0

[empty flame]

no the equation is missing

[wooden axle]

ax+by+cz=a^2+b^2+c^2

What is the equation missing

My answer is this

[empty flame]

also your naming got you confused

[wooden axle]

Whatt??

[empty flame]

you named the components of P as a,b,c

[wooden axle]

Yes

[empty flame]

oh wait nvm

[wooden axle]

What?

[vernal monolith]

Its not missing anything though?

You just didnt give enough properties to make that plane unique

So there are infinitely many

[wooden axle]

What?

[empty flame]

wait i will tell you the mistake you did

[amber marlin]

can you guys help me with this

[vernal monolith]

?gethelp

#❓how-to-get-help

[empty flame]

please occupy another channel check #❓how-to-get-help

[amber marlin]

ohk

[empty flame]

what are a,b,c in the general equation of the plane $ax+by+cz+d=0$?

[woven radish]

calculus is fun

[empty flame]

@wooden axle

[wooden axle]

Normal of the plane

[empty flame]

correct

[wooden axle]

But here it is on the plane points

[empty flame]

wdym by on the plane points

the normal always intersects the plane in a point

[wooden axle]

Yes

In the question it is a,b,c where the line intersect

[wooden axle]

Tell me

[wooden axle]

This is my answer

[empty flame]

ok but the mistake you have appeared because you named the components of P as (a,b,c) also

you gave different components the same name

you should've named the coordinates of P as (e,f,g) for example

ax+by+cz=ae+bf+cg

[vernal monolith]

If components of intersection point are (a,b,c) this should hold true still though

[empty flame]

but the question you asked was answered before

[vernal monolith]

Because this vectors comes from the origin

[empty flame]

how will the normal vector have the same components as P

the normal vector doesnt come from the origin of the coordinate axes

[vernal monolith]

So we get the vector (a-0,b-0,c-0) = (a,b,c)

[empty flame]

it starts from P

[vernal monolith]

That is how i evaluated the norm vector?

Oh nvm

Shouldnt make much diff

Thats also perpendicular

[empty flame]

which one

[vernal monolith]

Both vectors (a,b,c) and (-a,-b,-c) are perpendicular to the plane

[empty flame]

yes they both are

that doesn't make any difference

[vernal monolith]

Then arjun took an arbitrary point Q = (x,y,z) from the plane

[empty flame]

but why bother yourself working with more signs this will increase the chance of making a calculation mistake

arjunn took arbitrary point Q which also lies in the plane and got the coordinates of $\vec{PQ}$

[woven radish]

calculus is fun

[empty flame]

getting the coordinates of $\vec{PQ}$ doesnt do anything all we need is a known point which is P in our case and a normal vector $\vec{n}$ then we can do the following

[woven radish]

calculus is fun

[vernal monolith]

This should be correct in that case

[empty flame]

no it shouldn't

because (a,b,c) are the coordinates of P

or the coordinates of the normal vector

choose which one do you want

it cant be both at the same time

[vernal monolith]

Wont matter, it will only change the constant the most

[wooden axle]

I should have given but the question took it

[empty flame]

the mistake is in the constant because your calculation literally states that the coorcinates of P are the same as the coordinates of the normal vector which is wrong

so it is still a mistake

[vernal monolith]

You are only considering the normal vector coming from the origin

[empty flame]

choose any other triple of letters which isnt used at all

[vernal monolith]

Going towards the origin*

[empty flame]

no i am considering an arbitrary normal vector starting from P

not necessarily going toward the origin

[vernal monolith]

Consider the vector going to P from origin

Dot product is still 0

[empty flame]

dot product between which vectors

[wooden axle]

Who is saying it is normal corrdinates

[vernal monolith]

Besides even if you take $$\overrightarrow{PO} = (-a,-b,-c)$$
You will still get -a(x-a) -b(y-b) -c(z-c) = a(x-a) + b(y-b) + c(z-c) = 0

[empty flame]

in the equation $ax+by+cz+d=0$ (a,b,c) represent the coordinates of the normal vector n to the plane agree or no

[woven radish]

calculus is fun

Cyrenux

[empty flame]

who told you that $\vec{PO}$ and $\vec{OP}$ are normal vectors

[woven radish]

calculus is fun

[empty flame]

what you are saying is a special case

i am talking in general

oh wait

[wooden axle]

Did you read what I said?

[vernal monolith]

What is point of line that passes through origin intersecting this then

[empty flame]

i forgot that line is passing through origin

[wooden axle]

This is next question

[vernal monolith]

This is just like any other question

But yeah it doesnt say its perpendicular, i dont remember, i just assumed

[empty flame]

bro i am stupid

[wooden axle]

You guys are making things worse

I stated that it is saying perpendicular

[empty flame]

no he said but i forgot

[empty flame]

.

[wooden axle]

I don't why are you guys confusing each other first listen to my statement carefully

[vernal monolith]

Well i listened first but forgot later while explaining

Besides my assumption was correct

Otherwise there is no speciality of this plane lol

[wooden axle]

Yes true

[vernal monolith]

Also @wooden axle you already found equation of the plane

What more do you seek?

[wooden axle]

Nothing right now. But i am sure I'll soon ask

Thank you both of you

[vernal monolith]

I think you'd want to learn how to find intersection of two planes

[wooden axle]

I helped me a lot

[vernal monolith]

Would you not

[wooden axle]

I am reading all the concepts of 3D

[vernal monolith]

Also if you have trouble remembering algebracially, try remembering geometrically

[empty flame]

np yw at anytime

[empty flame]

me too i forgot in the long run

arjunn are you studying 3D geometry to get ready for multivariable calculus ??

[vernal monolith]

This is usually taught in analytic geometry

[wooden axle]

Yes i am doing this

[wooden axle]

Multivarible no

But i need to learn that multivariable functions limit continuity and differentiation

[empty flame]

i am still in high school i learned these just before multivariable calculus they were made as a chapter in james stewart book i studied calc from there

[vernal monolith]

Interesting

So multivariable calc is high school level in your country

[empty flame]

no i am self studying

[vernal monolith]

Speedrun

Are you planning math major or engineering

[wooden axle]

May I know about your study?@vernal monolith

[vernal monolith]

Math major

Finished 2nd year (sorta)

[empty flame]

now after summer vacation i will be last year high school we take integrals but we dont take about integration of trigs and barely do partial fractions at least thats what i heard from someone who was in grade 12 last year

[vernal monolith]

Hmm similiar situation here

[empty flame]

i am planning to do math and physics double major and continue to get a PhD in theoretical physics

i may change plan to a PhD in math idk

but rn it is a PhD in theoretical physics

[vernal monolith]

Math major is deeply based on proofs

[empty flame]

i like math the most then comes physics

[vernal monolith]

Have you worked on them as well

You have nice base for engineering currently

[empty flame]

ik thats why i like it the most because its also general

[empty flame]

i am studying linear algebra rn

@wooden axle whats your major

[vernal monolith]

Also if you dont mind answering are you from turkey or india by any chance

[empty flame]

it can be considered as an intro to proofs is that right

[vernal monolith]

That integration topic kinda hints that you are from one of these countries

[empty flame]

i am from lebanon if you have heard by that name

[vernal monolith]

Nope i havent heard

Nice though

[vernal monolith]

Think of it as a segment, not necessarily an introduction

[empty flame]

where are you from if you dont have a problem in answering

[vernal monolith]

Turkey

[empty flame]

oh nice we are not far from each other

[vernal monolith]

We also have same integral topics here

[wooden axle]

I am graduated b.sc

Without no knowledge

[vernal monolith]

You mean with no knowledge?

[empty flame]

what is that bachelor in computer science ?

[vernal monolith]

University entrance exam here is harder than most university exams here though

You have to be a speedrunner basically

[wooden axle]

Bachelor in science

[wooden axle]

May i know you guys country

I'm from India

[vernal monolith]

We typed above

[empty flame]

in our country there is a university call lebanese university

students in this university study on their own literally

[vernal monolith]

Without attending lessons?

[empty flame]

they attend but the professors just throw information and the test comes from another world bro

my brother is studying engineering in english they give sone tests in french

[vernal monolith]

[empty flame]

i am telling the truth

so basically you have to do everything by yourself

[vernal monolith]

Kinda same here but except its for uni exam

[empty flame]

the information you get from attending arent enough to solve the test questions

[vernal monolith]

You have to spend crapton for books

[vernal monolith]

My university is somewhat fair compared to that

[empty flame]

the good news is that you gain knowledge and learn to be self-dependent more than students in other universities

[vernal monolith]

Lectures are enough for exams

But missing a lecture is fatal

But yeah you are forced to do active recalling

Lecture alone isnt enough

[empty flame]

but the bad news is that you spend years just waking up eating studying till night then sleeping and repeating the same procedure over the course of 3-11 years depending on the major and degree

[vernal monolith]

In my country one becomes independant while studying for uni entrance exam

[empty flame]

[vernal monolith]

You have 1 min for each question + 15 mins bonus for circling every answer on sheet

120 questions in 135 mins

Worse part is, you are forced to study lectures that arent your interest

Its forced to be a good all-rounder

[empty flame]

yes in lebanese uni same thing happens

[vernal monolith]

In uni or uni entrance exam?

If in uni then even more 💀💀💀

[empty flame]

moreover if you fail in only 2 subjects you repeat the whole year

[vernal monolith]

Fail 3 then

[empty flame]

first year math studies math bio chem physics idk if anything more

same for first year physics and first bio and every first year science faculty students

[vernal monolith]

Physics is a must, but biology ????

[empty flame]

no 2 or more

[empty flame]

bro i told you its like when you a start a call of duty match you here the voice saying "you are on your own,survive as much as you can"

[wooden axle]

Let me upload today's question again

So here p and middle point make a vector

[empty flame]

ok

[wooden axle]

Which is perpendicular to plane in the question

So we will do dot product

For plane equation

[empty flame]

correct

[wooden axle]

Fine. Continue discuss

[empty flame]

discuss the question or continue the chat above

wait

dot prosuct between which 2 vectors ?

[vernal monolith]

Middle and some other of the two given

[wooden axle]

We need to find the equation of a plane which passes through three given points

[empty flame]

wait a sec

[vernal monolith]

Oh this is different now

[wooden axle]

Vector middle point and p

And middle point and random point vector

[vernal monolith]

First, make sure they are NOT colinear

[wooden axle]

This is a different question

[empty flame]

wait he only wants the equation of plane that contains the midpoint of P and Q he doesnt want the plane that contains all of the 3 points is that right arjunn

[wooden axle]

Right

I will request you to not mix two questions

[empty flame]

let the midpoint be M

[wooden axle]

PM vector and PQ vector where q is random point x,y,z

[vernal monolith]

If it contains P and Q, will it not contain its middle point by default?

[wooden axle]

PM.Pq

[vernal monolith]

Well lets ball it

[empty flame]

$\vec{PM}.\vec{PK}=0$ then work this out in terms of coordinates and we are done

[vernal monolith]

Call the plane ax + by + cz + d = 0

Input the points

Solve linear algebra style

[wooden axle]

PM is perpendicular

To the plane

[empty flame]

yea so the dot product is 0

[wooden axle]

Yes right

That is what i said

Now next question

[wooden axle]

This

[empty flame]

yes you are right other steps are just calculation so next question

[empty flame]

the given isnt enough

you need 3 noncollinear points or you need the normal vector of the plane containing P Q M

[woven radish]

calculus is fun

[empty flame]

i just changed the name of the variable point in the previous question because we already have a point called Q to avoid confusion if you want to refer to this channel later on

[empty flame]

if we have the normal vector to this plane then its same steps as the previous question

[vernal monolith]

Guys these 3 are colinear i think

[empty flame]

if we have 3 noncollinear points but we dont have normal vector then we do the cross product to get a normal vector then the next steps will be the same as previous question

[empty flame]

they are because M is given to be the midpoint of P and Q

[vernal monolith]

Exactly

[empty flame]

thats why i said not enough given

[vernal monolith]

No, its enough given

It

It says find a plane

[wooden axle]

Yes

[vernal monolith]

So do this thing i said above

[empty flame]

oh wait take any point that lies on the plane of the previous question which is not on the line (PQ) and we are done

[vernal monolith]

.

[empty flame]

yes input only one point

[vernal monolith]

2 needed actually

[empty flame]

but make sure that it doesnt lie on the line (PQ)

no we already have P and Q

or P and M

or Q and M

[vernal monolith]

You need to input 2 points on equation of plane

Also there is no need to scout for middle point...

[empty flame]

why 2 points

[vernal monolith]

If it containts two points it will also contain its middle point

Contains*

[empty flame]

you know d

[vernal monolith]

No need to

[empty flame]

you just have to plug x,y,z

of a single point

and we are done

[vernal monolith]

D changes every time surely

Depending on what you give to free variable

[empty flame]

no d stays the same because you will keep the same point in the plane

what i mean is that you need a point to find the equation of the plane

we dont need 2 here

[vernal monolith]

Plane you pick changes, d will also change depending on your free variable

[empty flame]

the plane changes so the whole equation will change

[vernal monolith]

Two points are needed, otherwise you can have one plane that doesnt contain the other point

[empty flame]

but here we have that the new plane contains the old plane

but there is a problem

there is a mistake in the given of the previous question

[vernal monolith]

@wooden axle are you there?

[empty flame]

because $\vec{PM}$ isnt normal to the plane

[woven radish]

calculus is fun

[wooden axle]

I am here but busy

You know that i am not fluent in English

[empty flame]

$\overrightarrow{PM}=<-1,1,0>$

[woven radish]

calculus is fun

[vernal monolith]

Lets say that plane that includes our points P and Q is: ax+by+cz +d =0
Since it includes P and Q it also includes its middle point

[empty flame]

$\overrightarrow{MR}=<2,-4,1>$

[woven radish]

calculus is fun

[vernal monolith]

Since our plane includes these points P and Q, they should satisfy equation of plane, so input them to the plane

You can also input the middle point for more insight but wont help

[empty flame]

so $\overrightarrow{PM}.\overrightarrow{MR}=-2-4+0=-6$ different than $0$

[vernal monolith]

From there you can freely give one of the following three variables a value you desire: a,b,c

[woven radish]

calculus is fun

[vernal monolith]

Try the thing i said and then show your work

Its the cleanest way to do it

Barely any analytic geometry

[devout snow]

@empty flame Has your question been resolved?

[vernal monolith]

@wooden axle have you done any work?

[wooden axle]

Nope

As I have no clue about the problem

Even I didn't understand what you guys did

[vernal monolith]

Well

Is it fair for you if i tell you to assume equation of any plane is ax+by+cz+d=0
@wooden axle

[wooden axle]

Yes

Let me give you three points first

[vernal monolith]

Alright then

[wooden axle]

(0,0,0) (1,2,1) (2,1,1)

Let's go

First tell me what are we gonna do

[vernal monolith]

Since they are in the plane, they should satisfy it, correct?

Like how a point on line satisfies equation of the line

[wooden axle]

True

D=0 by first point

[vernal monolith]

First try (0,0,0)

Yes

Now (1,2,1)

[wooden axle]

a+2b+c=0
2a+b+c=0

[vernal monolith]

Yup

Its just solving equations now

[empty flame]

you got these values by substituting x=0,1,2 and y=0,1,2 in the equation of the original plane ??

[vernal monolith]

Also z too

[wooden axle]

0,1,2?

[empty flame]

oh ok

[wooden axle]

Got it

[vernal monolith]

Ignore that, you just typed in the points

[empty flame]

like that you get the equation of the original plane

[vernal monolith]

To the plane

[wooden axle]

Next

[empty flame]

i am thinking abt it and i am realizing that its the same equation

[vernal monolith]

Solve system of equations

[empty flame]

for both planes

am i right ??

[vernal monolith]

Wdym by original plane

[empty flame]

i mean the plane this plane

the plane that arjunn is working with now is the one containing even P and Q

is that right

[vernal monolith]

There isnt an 'original plane' for that there are infinitely many

Calling 1 original is like giving it a privilege

[empty flame]

i am talking about the one in the picture specifically

[vernal monolith]

Thats just 1 of many infinite planes

[empty flame]

i called it original because we worked with it before this and now we are relating this plane to that

[vernal monolith]

And equation of it is NOT given

[empty flame]

this plane has the same equation as the one before it

[vernal monolith]

Its a coincidence

[empty flame]

its just that the new plane we are taking into consideration is an extension

the question we are solving is wrong

[wooden axle]

What should i do now?

[vernal monolith]

What did you find?

[wooden axle]

This

No idea about solve

[vernal monolith]

Row operation

[wooden axle]

No idea

How to do it?

[vernal monolith]

Or you can use substituion

c= -a -2b from first equation then plug this c to second equation

You should know this from high school though

Or even middle school

[wooden axle]

I want this

Matrix solution

[vernal monolith]

Its hard to type in text

[empty flame]

but the problem is that with substitution is that he will still have 1 equation and 2 unkowns which cant be solved like that

[vernal monolith]

https://youtu.be/eDb6iugi6Uk

[wooden axle]

U have paper

[vernal monolith]

@wooden axle watch this

[wooden axle]

Hard to understand

[vernal monolith]

I think i will need you search what a row means

And a column

[wooden axle]

I know matrices

[vernal monolith]

You eliminate the first variable from the left which is x in our case

And we can do that by multiplying first row by -2 and adding it to 2nd row

2x -2x = 0

[wooden axle]

I don't understand english

Can you write in on a paper I'll understand mathematics written equations

I am exhausted right now. 🤦

[empty flame]

np you can go rest and when you refill energy just open another channel

or i can keep this channel open

@vernal monolith cant he just plug (2,2,1)??

[wooden axle]

We have two equations only

[wooden axle]

Where you got this?

[empty flame]

where did you get other 2 points

that was my question from before

[wooden axle]

Whole 3 points are given in question

[empty flame]

arent these the points

[wooden axle]

Nope

[empty flame]

you didnt give any 3 points other than these

[empty flame]

here

[wooden axle]

Just read

[empty flame]

ok

[wooden axle]

@vernal monolith

[empty flame]

you can do it in another way

make any 2 vectors out of these points

[wooden axle]

I don't know

[empty flame]

ok i will tell you how

[wooden axle]

What will i do with vectors?

Wait

[empty flame]

you will do cross product

this will give you normal vector

[vernal monolith]

You get 3b + c = 0 from row operation

You also have a + 2b + c = 0

Plug c=-3c to get a=b

[wooden axle]

Your operation is wrong

[vernal monolith]

You have 1 free variable since you had 3 variables but 2 equations

[vernal monolith]

Which part

[wooden axle]

You said your operation is 2r1-r2

[vernal monolith]

Oh yeah

LOL

It still gives correct result because that equals to 0

I should have implied -R2-> R2

Then it would have been correct

[wooden axle]

Still unable to find values of a,b,c

And it is taking too much time

a=b
C=-3b

@vernal monolith

[empty flame]

you can do it in another way

i dont acually know gaussian elemination yet to help in this

[wooden axle]

Nope

I tried but elimination gave the same results

[spiral mural]

Row echelon form?

[wooden axle]

Yes it is row echelon form

[empty flame]

my way goes like this : you have 3 points : O(0,0,0),P(1,2,1) , Q(2,1,1) you can make 2 vectors out of these 3 points say $\overrightarrow{OP}$ and $\overrightarrow{PQ}$

[spiral mural]

Whats the original q

By the looks of it show the point of plane intersection?

[woven radish]

calculus is fun

[wooden axle]

I+2j+k

And i-j second

[empty flame]

ok now get the cross product of these vectors

[wooden axle]

I am leaving

Wait

[empty flame]

what

[vernal monolith]

@wooden axle

Pick your free c variable

Like c=k

Actually let c=3k

[wooden axle]

x+y-3z=0

I got this

[vernal monolith]

Huh? How

[empty flame]

wrong

[vernal monolith]

That is wrong yeah

[wooden axle]

Edited

[vernal monolith]

Now its true

[empty flame]

2+1-1=2

oh yea

[vernal monolith]

Since c=3k what is a=b=?

Knowing c=-3b

[empty flame]

how did you get it

[vernal monolith]

Also i didnt notice but its still false

That implies a=b=c=0 which destroys our equation of plane

[wooden axle]

I got this by two vectors cross product

[vernal monolith]

Oh you found equation of plane?

[wooden axle]

Yup

[empty flame]

its correct equation of plane

[vernal monolith]

Wait im dumb LOL its correct

Its not in terms of a,b and c

Pardon me for my blindness

[wooden axle]

Nope. Now my focus is to use your method

[vernal monolith]

Yeah its correct

[empty flame]

you should choose easier and faster way to solve

[wooden axle]

If three points are given we just do cross product

[vernal monolith]

Use this

[empty flame]

three noncollinear points are given you can just take any 2 vectors made by these points then cross product and plug everything in equation of plane using dot product

[vernal monolith]

@empty flame equation solving way is also fast for this

[wooden axle]

a+2b+c=0
3b+c=0

[vernal monolith]

You already did this

[empty flame]

if 1 point iand normal then just plug everything using dot product

[vernal monolith]

You are goiny backwards

Dont go backwards

Go forward

Give k a free value

And question is done

A free value like 1 or 2

[vernal monolith]

Let c=3k

you can pick k as an integer for easiness

[wooden axle]

c=-3k

[vernal monolith]

Now assign an integer value to k by your own liking

Any value will hold true

[wooden axle]

I am saying again it is not giving any value

It will give round and round

[vernal monolith]

Let k=2

[wooden axle]

a=b

[vernal monolith]

And tell me

[wooden axle]

C=-3a/-3b

[vernal monolith]

Im telling you to assign k a value YOURSELF

You DONT find

You MAKE

You DECIDE

[wooden axle]

I am giving 0

a=0

B=0

Then c=0

[vernal monolith]

Giving 0 gives 0=0

[wooden axle]

If i give 1

[vernal monolith]

And destroys equation of plane

[wooden axle]

a=1

B=1

[vernal monolith]

Any but 0 works, my bad

[wooden axle]

C=-3

[vernal monolith]

plug those into original plane now

[wooden axle]

1+2-3=0

0=0

[empty flame]

no he means plug into $ax+by+cz=0$

[vernal monolith]

Original plane equation is ax +by +cz=0

[woven radish]

calculus is fun

[empty flame]

because we got d=0 from before

[wooden axle]

Why were i putting the value in the 2 equations except original😪

[vernal monolith]

Well also you dont have to assign k a value, i lied

[wooden axle]

For x and a?

Leaving again

I don't know what are you guys doing

As i am not seeing any equations, I am not able to understand

[empty flame]

you just found values of a,b,c using cyren method right

[wooden axle]

Can you not write except talking in words?

[vernal monolith]

Since you let c=-3k and a=b=k
You can type those into equation of plane too and get xk+yk-3zk=0
And divide both sides by k to get x+y-3z=0

[wooden axle]

We are taking too much time on just one equation solving

[wooden axle]

Got it

[empty flame]

you can do the way of 2 vectors then cross product

you did it and got the result

[wooden axle]

I did it you know how fast i did

[empty flame]

or do the gaussian elemination and get result

[empty flame]

yea i saw that

anyway you choose the way that you find the most suitable for you and the one which you are good with to avoid mistakes

next question

[vernal monolith]

Or use both to validate answer

4% battery

[empty flame]

cyren's battery is dying take it to operation room to perform the surgery

[vernal monolith]

Also cramers rule really helps

If it was a system with 3 equations, using it would have been the most optimal

[empty flame]

yes i wanted to say cramer's but to use it he should have same number of equations and unkowns

because it involves determinants so the work will be with square matrices

[wooden axle]

You used row echelon form for non square matrix

[vernal monolith]

Forgot about that lol

[empty flame]

gaussian elemination can be used for non square matrices

what's your next question arjunn

or you want to get some rest now

[wooden axle]

I want to do more question

But need to rest for few minutes

[empty flame]

ok i will go for some mins too

[vernal monolith]

Which

[empty flame]

if i am late then i went to eat lunch

[wooden axle]

Here

[devout snow]

@empty flame Has your question been resolved?

[vernal monolith]

It seems like a square matrix though?

[wooden axle]

I am back

[vernal monolith]

Wb

[devout snow]

@empty flame Has your question been resolved?

[noble niche]

Hi

What is where question?

[devout snow]

@empty flame Has your question been resolved?

[empty flame]

Arjunn What's next question

[vernal monolith]

@wooden axle

[wooden axle]

Hello

There is no question but i want to understand some formulas

[vernal monolith]

Sure, do ask

[wooden axle]

in the equation of plane what is D represent

distance?

[vernal monolith]

Addition of xo yo and zo

So assume normal vector <a,b,c> and plane points P=(xo,yo,zo) and Q=(x,y,z)

[wooden axle]

what is o

[vernal monolith]

Construct vector PQ and notice its perpendicular to normal vector

[wooden axle]

origin?

[vernal monolith]

nah i mean $x_0$

[woven radish]

Cyrenux

[vernal monolith]

But i typed like that

$x_0$ and $x_1$ etc

[woven radish]

Cyrenux

[vernal monolith]

Its just used to show a point on the plane

$$P=(x_0,y_0,z_0)$$ is a certain point on plane while $$Q=(x,y,z)$$ is a arbitrary point

[wooden axle]

is this right way to explain what is D

[woven radish]

Cyrenux

[wooden axle]

I guess you are making it more difficiult than it is

[empty flame]

$\overrightarrow{PQ}.\overrightarrow{n}=0$

[vernal monolith]

Dont you mean +d ?

[woven radish]

calculus is fun

[wooden axle]

d in the plane equation

[empty flame]

Is that right arjunn

Because n is normal of the plane

[wooden axle]

and what is PQ

[empty flame]

i will tell you

[vernal monolith]

I just explained...

[empty flame]

P is a point in the plane of known coordinates and Q is a variable point on the plane

let $P(x_0,y_0,z_0)$ and $Q(x,y,z)$

[woven radish]

calculus is fun

[empty flame]

let $\overrightarrow{n}=<a,b,c>$ be the normal vector to the plane containing P and Q

[woven radish]

calculus is fun

[wooden axle]

you are just making the plane equation

[vernal monolith]

Yes?

[wooden axle]

okay do fast

[vernal monolith]

Do it for us

[empty flame]

$\overrightarrow{PQ}=<x_Q-x_P,y_Q-y_P,z_Q-z_P>$ is that right

[woven radish]

calculus is fun

[wooden axle]

i see

right

[empty flame]

ok now $\overrightarrow{PQ}.\overrightarrow{n}=a(x-x_0)+b(y-y_0)+c(z-z_0)=0$

[woven radish]

calculus is fun

[vernal monolith]

@wooden axle expand

[empty flame]

now expand and collect the _0 terms together

[wooden axle]

ax+by+cz=ax0+by0+cz0

[wooden axle]

yes i started before saying

[vernal monolith]

Now you should be able to see what d is

[wooden axle]

so D is the dot product of a point and normal

[empty flame]

thats true you can keep everything before the = sign this will lead to $ax+by+cz-ax_0-by_0-cz_0=0$

[woven radish]

calculus is fun

[empty flame]

now to make it look better let $d=-x_0-y_0-z_0$

[woven radish]

calculus is fun

[vernal monolith]

I didnt know you could dot product a vector with a point?

[empty flame]

we didnt dot product a vector with a point

where did we do that

[vernal monolith]

Here

[wooden axle]

here

[vernal monolith]

Its 2 vectors

[empty flame]

i dont see a point here these are 2 vectors

[wooden axle]

I meant yes vector

[empty flame]

oh ok

[wooden axle]

not alone P

[vernal monolith]

To construct a vector you need 2 points

[wooden axle]

PQ with n

yes we need

[vernal monolith]

Clean explanation @empty flame

[empty flame]

here we grouped these terms together because they are all constants so instead of having all 3 constants we substituted the sum of these constants by only one constant which is d

[empty flame]

tysm your explanation is awesome too its all clear

[vernal monolith]

Do you also know how to parametrize two points? @wooden axle

[wooden axle]

do you guys know any short?

[vernal monolith]

Yeah

[wooden axle]

like in which form

polar or normal?

[vernal monolith]

Both

[wooden axle]

xcos+ysin

[vernal monolith]

Do you know what a linear combination is?

[wooden axle]

t=xi+yj+zk

[steel sage]

Clearly 2, 3

[vernal monolith]

Not exactly what i wanted, i will show you later

[wooden axle]

sure just after the question

[empty flame]

yes because all triples consist of 0s and 1s so its easy to check whether any of the elements is a linear combination of the other 2

[wooden axle]

tell me how to check?

square and add?

[vernal monolith]

Let v1 v2 and v3 be vectors

[wooden axle]

ok

[vernal monolith]

If they can be expressed as $$a_1 v_1 + a_2 v_2 + a_3v_3=0 $$ where at least one of $a_1$, $a_2$, $a_3$ ISN'T zero then they are linearly depended

An easy example is:
$$(1,1,2) + (0,1,1) = (1,2,3) $$
These 3 vectors are linearly depended because 3rd vector can expressed as sum of other two

[woven radish]

Cyrenux

[vernal monolith]

So any form of $$av_1 + bv_2 = v_3$$ where $a\neq 0$ or $b\neq 0$

[woven radish]

Cyrenux

[empty flame]

you can go further to say anything of the form $$av_1+bv_2=cv_3$$ for $a\neq 0$ or $b\neq 0$ or $c\neq 0$

[vernal monolith]

Another example:
2(1,0,0) + 3(0,1,1) = (2,3,3)
Because of this (1,0,0), (0,1,1) and (2,3,3) are linearly depended

[woven radish]

calculus is fun

[vernal monolith]

@wooden axle

[empty flame]

also the order of vectors can be changed so it can be in the form of $av_1+cv_3=bv_2$ or $bv_2+cv_3=av_1$

[woven radish]

calculus is fun

[wooden axle]

i am here

i unedrstand the equation you are telling

but how to solve the question

[vernal monolith]

Add the two vectors try to see if you get the other

[wooden axle]

there is no 2

wait

[vernal monolith]

If you get the other vector: they are linearly depended

[wooden axle]

should I need to add them 6 times

i meant 1 to 2 2 to 3 3 to 1

or just in order 1 and 2

[vernal monolith]

Random order

No particular order

[wooden axle]

damn

[vernal monolith]

Randomly add two vectors of your liking

[wooden axle]

it will take 5 minutes

[vernal monolith]

anyw

[wooden axle]

and i do not think it is good idea

[vernal monolith]

Nah

It takes like 5 secs lol

If

If you do in your head

[vernal monolith]

For no 1: v3- v2 = v1

Which makes them linearly depended

[wooden axle]

so any pair which fails in addition

linearly independent

[vernal monolith]

They dont actually fail but their coefficient is 0 instead of another integer

[wooden axle]

2 and 3

[vernal monolith]

Because 0v1 + 0v2 = 0v3 always lol

0 coefficient is always a solution

If its the only solution then linearly independant

If there are other solutions then its linearly dependant

[vernal monolith]

These 3 vectors are elementary vectors

for no 2 i mean

e1, e2, e3

[wooden axle]

i want to solve it without chinese

[wooden axle]

yes

they are special

[vernal monolith]

Elementary vectors are independant by default lol

[wooden axle]

attack

[vernal monolith]

But to show that they are independent you can type them as a matrix

a1v1 + a2v2 +a3v3= 0

Plug v1 v2 and v3

Find a1 a2 and a3 where at least one isnt 0

[wooden axle]

Long method

[vernal monolith]

Yes

The shortcut i do is to show that its linearly depended

Dont use it to show its independant lol

[vernal monolith]

You should see that x= 5k+3 and x= 7l +5

I made an error on my previous message, fixed now

5k+3 = 7l+5

5k= 7l+2

Notice that k>l

Find 7l+2 where its a multiple of 5

k=6 and l=4 satisfies it

To find next pair

Add 7 to k value

And add 5 to next l value

k is associated with 5, and l is associated with 7 so you add them up with opposite numbers

[wooden axle]

Wait

What I do next?

How many pairs should I find?

@vernal monolith

[vernal monolith]

There are Infinitely many

[wooden axle]

Yes

[vernal monolith]

In this order:
k l
6 4
13 9
20 14
27 19
... ...

[wooden axle]

Thanks

[wooden axle]

let us talk about perpendicular feet

[devout snow]

@empty flame Has your question been resolved?

[rain edge]

can i get help?

[restive river]

yes

[wispy cargo]

Can anyone solve these please

[sturdy atlas]

Yes i can solve

[sturdy atlas]

Yes sure

[deft jay]

ub

uh

thats way to complex

[devout snow]

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[queen garden]

Hi I’m currently curious on how to do these three limits?

The answers are already there but I’m wondering how to get there

They’re probably pretty simple and I’m missing something

Cause I thought coming in from the left of x->3- would’ve been two

[devout snow]

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[winter torrent]

you deleted your first post; that will close the channel and is irrecoverable. you'll need to make a new one

[queen garden]

Ok

[winter torrent]

not sure why you think lim x-> 3- would be 2

[devout snow]

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[velvet bay]

Hii

[valid iron]

hey!!

you got a question?

[velvet bay]

Super easy questions

[scarlet sequoia]

,rccw

[velvet bay]

I just need to figure out the way to do these

[velvet bay]

I just can’t ever get the correct answer for some reason :/

Im always doing smth wrong

This question.

Came in my exam

.close

[devout snow]

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[unkempt quiver]

How do I do this?

If f(x) = |2x + 3| - |x - 4|

Redefine f(x) piecewise

[toxic sandal]

Do you know, how |x| is defined?

[devout snow]

@unkempt quiver Has your question been resolved?

[unkempt quiver]

Uh

Wdym defined??

[weak cove]

|x| is a function

it behaves differently when x is negative as compared to when x is positive

it is a piece-wise functio

[unkempt quiver]

Yes

[weak cove]

by definition

but what is this definition?

[unkempt quiver]

I know what y = |x| looks like

It's reflected across the y axis

[weak cove]

knowing what it looks like could help you figure out how it is defined, but once you tell us how it is defined then you will be able to continue with your original question

[unkempt quiver]

I know how it's defined

Any neg x values will have the same y value as their pos x value counterparts

@weak cove

But idk what u mean by "defined"

Is this correct?

[weak cove]

write it as a piecewise function

that is what I mean by defined

write |x| as a piecewise function (this is the definition of |x|)

[unkempt quiver]

@weak cove are u still here

Idk how to do it

Where does it change??

Sorry

You can ping me

[weak cove]

google definition of |x| function

you will find it

[devout snow]

@unkempt quiver Has your question been resolved?

[unkempt quiver]

Yes but I literally don't know how to write it as a piecewise

Sorry

[unkempt quiver]

How do they know to do that

[robust dust]

that's a definition

you don't "know" it, it's just what the function actually is

when we write |x|, we mean {x if x>=0, -x if x<0}

[devout snow]

@unkempt quiver Has your question been resolved?

[unkempt quiver]

Ok

I just have no ide how to write my question as a piece wise

[mystic scarab]

Use the definition of the absolute value, but instead of x you have to input x - 4 and 2x + 3

[devout snow]

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[uncut lintel]

I need help

Checking my soln

Soln:
Draw a line through one of the outermost points of the n points. Rotate the line anticlockwise... Till it touches another point. The line then 'snaps' to that point and then starts to rotate anticlockwise w.r.t that point. Do this till we touch all the points. If at some point we touch a line already drawn then we just ignore it

[eager lodge]

I don't think it guarantees the line touches all points

[uncut lintel]

Why

[eager lodge]

let's consider a triangle

[uncut lintel]

Nah wait

I get what you ar trying to say

[eager lodge]

ok

[uncut lintel]

I also said

If the line at any point meets an already drawn line

Then It ignores that

And then continues to rotate

[eager lodge]

oh

[uncut lintel]

And snaps to the next point

So it would make a sort of spiral or sth

No?

[eager lodge]

hmm i think it works then

[uncut lintel]

Yey thanks

U awesome

.close

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[devout snow]

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[granite arch]

ok so i would start by letting u = (x^2-4)

du = 2x dx

we can cancel out the x

when x = sqrt5, u = 1 when x = sqrt8, u = 4

for integrate ln u / sqrtu id do ibp

[devout snow]

@remote vigil Has your question been resolved?

[granite arch]

i got 2ln(4)-2

feel free to ping if you have any questions

[devout snow]

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[untold crow]

Hellooo!
So I am currently going through Calculus during my Electronics Bachelor's Degree.
This:

[untold crow]

$\bar{w}w = |w|^2$

[woven radish]

HqppyFeet

[untold crow]

I am struggling to understand this equation... like what does it mean?
This is about complex numbers.
I learned about real and imaginary numbers.
I learned a bit about modulus and argument, but quite foggy there.
Addition, Subtraction, multiplication, and division is similar to traditional ways of solving using real numbers.

[past compass]

Hi there

Do you know what the bar means

[sand dove]

Do you know about the formula for the module | | ?

[pseudo basin]

the product of a complex number with its own conjugate equals the square of its magnitude/modulus/absolute value/distance-from-zero/whatever else you wanna call it.

is that what you were looking for?

[untold crow]

Oooh wait, this is similar to (a+b)(a-b). You get a^2 + b^2, which is just w^2

[pseudo basin]

⚠️

[untold crow]

Took me a while hehe, thank you all ❤️

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[zenith fable]

anyone able to help me understand how to convert 1/3 to ternary decimal format?

[zenith fable]

I understand its (0.022222)_3

but i dont know how that was gotten

[robust dust]

you can basically do long division, but "wrap around" your digits at 3 instead of 10

[zenith fable]

do you know of a worked example of this online?

[robust dust]

looking for a video

[zenith fable]

this is kinda like changing bases of larger numbers right?

[robust dust]

yeah

I have a worked example of a binary one

[zenith fable]

i understand pretty much everything about this cantor set proof except for this conversion 😛

[robust dust]

but it's the same idea for ternary

[zenith fable]

AHH okay thanks dude

thank you so much

[robust dust]

lol this question was about cantor-bernstein

different theorem from cantor but still

[zenith fable]

he seemed to like to split sets up

😛

bright philospher to realize all this

i find this entire thing very cool how the measure has been taking out, but the set isnt empty

bordering beautiful imo

[robust dust]

yeah, topology and measure theory is very nice

[zenith fable]

Iver never had a topology class, so no idea. Im just bored and self teaching this

[robust dust]

you can construct a whole load of weird sets like this either of those two fields

if you have the chance, I'd recommend sitting in one of those classes

[zenith fable]

I want too, well see if i get into a masters or not

its a feild? thats an algebraic thing right

[robust dust]

I meant field as in "area of mathematics"

but yes, a field is also an algebraic structure

[zenith fable]

AHH okay good, i dont know that stuff. i chose this since i thought my ignorance in algebra wouldnt be too big of a deal

and i find measure theory interesting

[robust dust]

measure theory is more analysis than algebra

but there are places for both

[zenith fable]

Hah, well someday ill have the time to learn it maybe 🙂

have a good day brother, thanks for the help!

[robust dust]

you too, take care

[zenith fable]

.close

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[lavish cypress]

is there a way to graph these type of questions

[devout snow]

Please don't occupy multiple help channels.

[pseudo basin]

no and it is not necessary

!onechannel

[devout snow]

Please stick to your channel.

[jovial juniper]

We only have one variable. If there were two of them, one might think)

[devout snow]

@lavish cypress Has your question been resolved?

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Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[restive river]

If we consider the $+$ as a function defined as [+: \R^n\longrightarrow \R \quad (a_1,a_2, \hdots a_n) \mapsto a_1 + a_2 + a_3 + \hdots+ a_n]
Can we define a function composition similar to $\map f +$

[placid rover]

grrr

[robust dust]

as in, f is a function R to whatever?

[robust dust]

infix notation mixing with function notation is

[placid rover]

heck is f(+) bro

[robust dust]

but yes, you can compose those functions

[supple knot]

Composition of functions is indeed a binary operation

[robust dust]

though you probably shouldn't write it like that because it's super unclear

[supple knot]

Theorem 2.2.1
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/02%3A_Introduction_to_Groups/2.02%3A_Binary_Operation

[placid rover]

i dunt get it, what yall on abou