help-39

[shrewd rune]

which is true right

[acoustic tangle]

Yep

[shrewd rune]

so for any negative number the ecuation is correct ?

[acoustic tangle]

Yes

[shrewd rune]

now do we include 0 in there

[acoustic tangle]

I think so

[shrewd rune]

0 + 0 = 0

thats right

so for what value of x is the ecuation correct

.

[acoustic tangle]

X>=0

[shrewd rune]

if we have to use negative numbers?

[acoustic tangle]

Oh wait

No

Other way round

X<=0

[shrewd rune]

nice

thats all

[acoustic tangle]

Tysm

[pearl pond]

@acoustic tangle Has your question been resolved?

[pearl pond]

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[acoustic tangle]

How do I do iv

[acoustic tangle]

Finding max height

[sturdy bane]

do yk how to find max/mins from calc 2

[acoustic tangle]

Er

Kinda

[sturdy bane]

what property is true about a max or min

[acoustic tangle]

Rate of change is 0

[sturdy bane]

yea

[amber wind]

Hey hey

what have you tried

[sturdy bane]

so check where the derivative is 0

[acoustic tangle]

I got

-20/6-h tho

Maybe I differentiated wrong idk

[sturdy bane]

oh im tripping ngl

[sturdy bane]

as the tree grows what happens to the change in height

[acoustic tangle]

Tends to 0

[sturdy bane]

yea

the derivative gets smaller as the tree grows

[sturdy bane]

and once its 0 it can no longer grow

[acoustic tangle]

Yeah

[sturdy bane]

at what height does the derivative become 0

[acoustic tangle]

6

[sturdy bane]

ignore t here just look at the top equation

yea

[sturdy bane]

so this is the answer

and you can verify that the change in height is > 0 up till then

[acoustic tangle]

How

[sturdy bane]

which part dont you see about the answer being 6

[acoustic tangle]

No I’m asking abt this

[sturdy bane]

while the height is less than 6, is dh/dt positive or negative

[acoustic tangle]

Positive

[sturdy bane]

yep

so while the tree is shorter than 6 it will be growing towards 6

[acoustic tangle]

Thx

[pearl pond]

@acoustic tangle Has your question been resolved?

[acoustic tangle]

Also

Im not sure how to put everything to the power of e

[pearl pond]

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[bronze glen]

is there an error in this

[bronze glen]

acc to our book this is the ans

not sure if its a misprint or i messed up

but js confirming

[candid coral]

you didn't mention the domain issue x >= 0 and x != 1
the og expression has: sqrt(x) - 1 in the denominator, so at x = 1 it is undefined

it would actually be: dy/dx = 0, for x >= 0 and x != 1

[bronze glen]

the q i wrote is the only info given

[bronze glen]

still tho

our book's ans is that

thing

i wrote above

[delicate temple]

Idk bro tis tuff

Have some ez doubts man

[odd kelp]

hey bbg

[bronze glen]

hi

[rotund nymph]

,w (arccos((sqrt x - 1)/(sqrt x + 1))+arcsin((sqrt x - 1)/(sqrt x + 1)))'

[bronze glen]

yo wha

[jolly parrot]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by boredamogi

[bronze glen]

oh

so our book is wrong

[odd kelp]

hmmm

[bronze glen]

printing error perhaps

[rotund nymph]

,w simplify arccos((sqrt x - 1)/(sqrt x + 1))+arcsin((sqrt x - 1)/(sqrt x + 1))

[rotund nymph]

what did your book say?

[bronze glen]

soln is that thing i wrote above

this

its an mcq

so prob printing error

[rotund nymph]

oh

[bronze glen]

i have another q

[jolly parrot]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by boredamogi

[bronze glen]

ye its an identity

cos^-1 (x) + sin^-1 (x) = pi/2

[rotund nymph]

ok

[bronze glen]

so from an initial glance

f(x) = sin (x)

but what after that

[tall vault]

You dont need f(x).
Just apply the chain rule

[bronze glen]

oh

chain rule?

how tho

we dont have the function

now that i do i can use

[tall vault]

Take g(x) as the inside function

[bronze glen]

oh

yea

got it

im dumb

f'(g(x)) x g'(x)

meaning (A)

[tall vault]

Yep

[bronze glen]

ight

tysm

.close

[pearl pond]

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[pearl pond]

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[rare field]

for (d), is there any interpretation other than "green area = funny number"

i feel like im missing something

[pearl pond]

@rare field Has your question been resolved?

[edgy wren]

It's not the graph of the inverse
it's just integrating from the left of f(x) from f(a) to f(b)

If you graph the inverse value of the integral is under the graph of f^-1(x) from the values of f(a) to f(b)

and from the graph we can deduce the fact you are trying to show in c

[rare field]

ohhhhh i see now

thamks for the help

.close

[pearl pond]

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[pearl pond]

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[stoic imp]

examples of region type 1,2,3,4 please

[dense jasper]

\begin{enumerate}[label=\Roman*.]
\item The solid bounded by $z=x^2+y^2$ and $z=4$
\item The solid bounded by $y=x^2+z^2$ and $y=4$
\item The solid bounded by $x=y^2+z^2$ and $x=4$
\item The solid ball $x^2+y^2+z^2 \leq 1$
\end{enumerate}

[jolly parrot]

Civil Service Pigeon

[stoic imp]

examples with graph explanation

[odd kelp]

hmm

[stoic imp]

graph explanations and thorough examples please

[odd kelp]

is there a translation available

for english....

[odd kelp]

alr thanks...

[stoic imp]

.close

[pearl pond]

Channel closed

Closed by @stoic imp

Use .reopen if this was a mistake.

[odd kelp]

query resolved?

yk you can always experiment on desmos for graphing....

[stoic imp]

nobody is helping

[odd kelp]

I was....

[stoic imp]

when?

[odd kelp]

when I asked for the translation?

and have patience, it isnt our duty to help someone here

we have lives as well, atleast wait for a few hours

and yk you can ping @helpers after 15mins

[stoic imp]

what

[odd kelp]

so that gets you more reach

[stoic imp]

a few hours?

[odd kelp]

yea well its 4 am here for me...

[odd kelp]

yep

[stoic imp]

what about it

[odd kelp]

Timezones

[stoic imp]

go to sleep if you are tired dawg

[odd kelp]

see renato, all im saying is

have patience

rest is upto you

someone will eventually help you out

[stoic imp]

nobody helped

[odd kelp]

just keep the channel open and someone will

and ping the helpers after 15mins of posting your question

if no one came to help

rn, 2mins remain

[stoic imp]

whats your point

[odd kelp]

my point is to have patience with your questions

dont be so restless

wait a bit

wait for an hour or two

[stoic imp]

I waited for an hour

[odd kelp]

you did not....

[stoic imp]

or half an hour srry

[odd kelp]

you opened this channel 15mins back

[stoic imp]

well I only asked for examples

I wasnt asking for some complicated logic problem

[odd kelp]

doesnt matter, a question is a question....

[stoic imp]

whats your point

[odd kelp]

my point is, your query is a query

jst cuz its simpler compared to others

doesnt mean you will get 'priority'

[stoic imp]

I was never saying it was not

[odd kelp]

can we take this to dms if you want to...
because this is a math help server and I cant be engaging in conversations in a 'help' channel

[spiral pivot]

Please keep things on topic

[odd kelp]

yep was jst gonna say that

[odd kelp]

anyways this channel has been closed....

[summer imp]

And.. you were given examples Renato, to which you just responded you were seeking thorough examples without mentioning (1) why you considered those weren't, and (2) whether you had tried to graph them in the first place.
What about those examples do you not find satisfactory?

[stoic imp]

.reopen

[pearl pond]

✅ Original question: #help-39 message

[odd kelp]

exactly....

[stoic imp]

I wasnt asking for an AI to just throw me a bunch of examples but to explain like a human dawg

is that too much to ask in 2026?

[odd kelp]

sigh

[summer imp]

Those weren't AI generated though. They're just simple examples of the different types of regions.

[odd kelp]

that AI was used to just put words in a neat manner
the real explination was done by a human....

is this ragebait....

[stoic imp]

whether they were ai generated or not, they just basically thrown them without any explanation whatsoever

[odd kelp]

and did you ask for further explination?

[stoic imp]

i did

[summer imp]

Type I just means bounded above and below by the graph of functions f(x,y) and g(x,y).
Type II just means bounded by the graphs of functions f(x,z) and g(x,z).
Type III just means bounded by the graphs of functions f(y,z) and g(y,z).
Can you tell in each of the examples which functions are bounding the region?

[odd kelp]

ykw lemme make this clear
im working on ur query
be patient
give me some time...

[odd kelp]

he/she needs a graphical explination

[stoic imp]

can you give graphical aid to understand

[odd kelp]

Im sorry
I got some urgent work to do rn....

[odd kelp]

ping the helpers, your query has remain unsolved for more than 15 mins

[summer imp]

This is for type I regions. You can change the bounding functions to your liking although it might get weird. I gave you an example of two such regions which are bounded by the same graphs, but which have different regions D as per your definition. The intuition is just that for type 1, you can stretch extrude this region D in the xy-plane into 3D until it hits the graphs above and below.

[stoic imp]

what am I looking at

[summer imp]

You can toggle the bounding functions on and off

[summer imp]

To see that they indeed bound the regions.

The point is to imagine that you take your initial region in the xy plane and you extend it out until it hits the bounds, kind of like those toys which imprint your hand.

So if your D is a disk in the xy plane, you extend it out to a solid cylinder in 3D and carve it out by the graphs of f and g,

If D is a triangle in the xy plane, you extend it out to a solid triangular prism in 3D and carve it out by the graphs of f and g.

And then what changes between the types of regions is just the plane in which you initially place your region D in to extend it out.

[stoic imp]

my hand is the region in R3?

[summer imp]

I mean it's like a mold. You extend each of the little pegs out until it hits the graph of the function.

[stoic imp]

pegs?

[summer imp]

Like the little pins in the image.

[stoic imp]

can you make a drawing

what does carve even mean

[summer imp]

This is the approach Rafilou was explaining earlier essentially.
You take any point $(x,y)$ in your region $D$ and extend it outwards until it hits the bounds. This looks like ${(x,y)} \times (f(x,y), g(x,y))$ so it creates a line perpendicular to your region between the points $(x,y, f(x,y))$ and $(x,y,g(x,y))$. Then you take the union of all those lines for each $(x,y) \in D$.

[stoic imp]

i am not from the usa and my english sucks

[jolly parrot]

Azyrashacorki

[stoic imp]

seems hard

is there any easier route

[summer imp]

Maybe just working with examples that appear after this definition and just with integrals involving those in general will help.

The idea is no different than for the 2D region types though

Like the definition is the definition, Those things you're being told are ways of digesting what it means for a region to be type 1, type2 or type 3.
At the end of the day, all it means like I said earlier is that your region is bounded above and below by the graphs of some 2-variable functions.

[stoic imp]

we dont need to introduce integrals

[stoic imp]

there is not a general sense of ABOVE in R3 thou, correct

you mean in x,y, or z

[summer imp]

No indeed. The choice of your independent axis will dictate whether we mean above/below, left/right, in/out.

[summer imp]

Well sure you don't need to compute them but the process of translating a 3D region into bounds for an integral is exactly the process of deciding what type of region you have and writing it as such.

[stoic imp]

can you help me understand

[stoic imp]

wdym?

[summer imp]

It's like with integration over 2D regions.

You have to decide which type it is in order to have bounds for your integral

If you pick the wrong one you likely will need to split your region of integration.

[stoic imp]

whats the difference

[stoic imp]

we dont need to know integration to discuss regions type1234

[summer imp]

I know you don't need it explicitly in the definition. But the whole point of defining those regions in the first place is to integrate over 2D and 3D regions alike.

[stoic imp]

how so?

[summer imp]

Is your section on those regions not followed by examples of integral computation?

[stoic imp]

no

[summer imp]

I told you the difference. At best you'll need to split your region, so more work. At worse you won't even be able to integrate.

[summer imp]

Then a future class maybe? There is just no way this is not introduced as a means to compute double/triple integrals (which you've done before, so I would hope you would've seen those things).

[stoic imp]

we first understand the types of region

then we use integrals and stuff

@summer imp

[summer imp]

Sure, what I said was that the setting up of the bounds involves deciding the type of region and writing it like that.

So you get extra exercises in that regards.

[stoic imp]

wdym?

[summer imp]

But It's really something you get used to with examples because if the definition doesn't make sense to you as it is I don't really know how to explain it better. Perhaps videos online would do a better job at providing visuals.

[summer imp]

You've done it before. You had that trapezoid you were integrating over a while back and you had a choice to either break it into three parts or just one, depending on which type of region you saw it as.

[stoic imp]

which one?

[stoic imp]

do you have any examples?

[summer imp]

Did your class/notes not provide any?

[stoic imp]

no

@summer imp

[errant cedar]

to integrate each variable x,y,or z in a multiple integral over a general area or volume, you will have to get bounds from a function of the currently unitegrated variables to another (if you integrate z you will need a to integrate it from f(x,y) to g(x,y)). If one the bound is a piecewise function or something complicated, the integration will be hard or case by case (splitting the integral). That’s where identifying the types of regions help you make the integral simpler or at all doable by helping you choose the order in which integrate x,y and z.

[stoic imp]

example please

[errant cedar]

Have you done 2D integrals yet in your course?

[stoic imp]

yes ofc

[errant cedar]

i mean like double integral over regions like this, because 3d is kinda the same idea but with an extra variable to integrate iteratively

[errant cedar]

.

[stoic imp]

do you have any examples

[quartz hearth]

I want a friend to talk about math

[stoic imp]

is it possible you share an example so we can work it out and see the differences

[errant cedar]

https://tutorial.math.lamar.edu/classes/calciii/TripleIntegrals.aspx

imo this one would be appropriate to understand the process for doing triple integral if you already went through general doubles integrals

[errant cedar]

maybe try this one i just sent?

[stoic imp]

first octant huh

care to elaborate

[errant cedar]

x,y,z>=0

[stoic imp]

basically, x>0, y>0, z >0

[stoic imp]

yeah

the region under the plane

that part is where I get stuck

[errant cedar]

you have to identify an integration order, in the example they decided to go with z first (innermost integral) so they have to figure out two bound(x,y) for z to integrate over f(x,y,z) = 2x.

[stoic imp]

any order works right

[errant cedar]

Here yes because this volume is of type 4 according to your definition of earlier

i think, (it’s been like more than a years since i did multiple ints)

[stoic imp]

can we start from scratch

the problem is that this plane is not a parametrizable surface

[errant cedar]

here you don’t want to parametrize the plane you want express it as a bound function of x,y that (intuitively) we will be able to integrate to from the xy plane

like my way of thinking about this for the innermost integral is that you ingrate z from a surface to another surface easily describle by functions of x and y (the other 2 vars)

[stoic imp]

ah I see

z = -2x -3y + 6

(x,y,z) = (x,y,-2x - 3y + 6)

r(s,t) = (s,t, -2s-3t+6)

[errant cedar]

yeah that would be a valid way to parametrize the plane but that’s not what a triple integral does or how to start to compute it

[stoic imp]

wdym

we have three integrals, one wrt x, wrt y, wrt z

we need -2x - 3y + 6 > 0

how do i find the bounds tho

[errant cedar]

that’s what i’m trying to explain. I think it’s tricky, honestly from all my time here i never was able to feel i made someone understand this idea purely in a discord format so i will demand you to be indulgent

[stoic imp]

are u here ?

@errant cedar

[errant cedar]

my intuition for this is not the parametrization of a surface like you did there, what i find useful to understand this is to think the triple integral computes a weighted volume (weighed by f(x,y,z) here 2x). Conceptually, it’s the perfect infinitesimal version of summing a bunch of little cubes of volume dV times f(x,y,z) (the weight) inside the volume to integrate (here the kinda pyramid shape we got in this example)

[stoic imp]

right

[errant cedar]

in this discrete analogy, you sum these little cubes by 3 axis in different possible order (the order of integration) but you have to limit the sum by the constraint imposed by the volume (the pyramid)

[stoic imp]

simpler route?

[stoic imp]

the problem is that I need to find the integral bounds

[errant cedar]

that’s what i’m getting at. In the innermost integral, you sum the weighted cubes from a surface here the xy plane z=0 to another surface z=6-2x-3y both defined as function of x,y, that’s my intuition for the bounds of the innermost integral

[stoic imp]

why z=0

wasn't z>0

@errant cedar

[stoic imp]

sure.

that would be solid if z >= 0

but we have z > 0

@errant cedar

[stoic imp]

but we have z > 0 so your bounds are incorrect

[errant cedar]

the >= or > does not matter in the bounds as far as i remember only the volume they enclose

[stoic imp]

what would be the geometric explanation of this

[stoic imp]

its z > 0 though

also how is this shit related to regions of type X

@errant cedar we are derailing from the original question dawg

[errant cedar]

as i said this thing is tricky to explain

[stoic imp]

how is region of type X related

[errant cedar]

i want to give you intuition for what a triple integral computes because you asked what is the link between the region types and integration

and imo and as other have already hinted at is that these region types are usually defined in order to help you integrates volumes more easily or at all.

these idea are all interconnected to be able to perform the act of computing a triple integral.

[stoic imp]

alr so?

[stoic imp]

which type of region this is

[errant cedar]

here we use that it’s of type I

[stoic imp]

how so?

[errant cedar]

but it’s really of type 4 (so I II and III) at the same time)

[stoic imp]

but arent all regions in R3 type 4 tho?

[errant cedar]

oh my bad it’s at the same time type 1,2,3

i confused an or for and and

[stoic imp]

what?

[errant cedar]

it’s at the same time type 1, type 2 and type 3, so it’s also type 4

[errant cedar]

as i said i never was able to make someone grok the idea of what a triple int computes, so maybe we can stop there (i need to eat).

[stoic imp]

dawg i just wanted to understand region types

[errant cedar]

maybe i come back later later now i go eat. any helper feel free to take this

[stoic imp]

we could've tried understanding region types first

idk why you guys forced the integration topics

@errant cedar

.close

[pearl pond]

Channel closed

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[pearl pond]

Available help channel!

Send your question here to claim the channel.

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• Show your work, and if possible, explain where you are stuck.
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• 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.

[sterile gazelle]

green

[bronze glen]

green

[sterile gazelle]

[viscid shale]

Does this problem appear in some kind of worksheet?

If so

!original

[pearl pond]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[bronze glen]

that

is

the exact

question

[sterile gazelle]

in green

[bronze glen]

yes

[sterile gazelle]

[viscid shale]

is y_n the nth derivative?

[bronze glen]

yea

btw

i did try but i didnt get any noticable pattern

tbh

[viscid shale]

Do you remember product to sum formulas of trig functions?

[bronze glen]

yea

[viscid shale]

If so, what is sinx * sin3x equal to

[bronze glen]

so i can write y as [cos(3x - x) -] cos(3x + x)]/2

well [cos2x - cos4x]/2

[viscid shale]

Not quite

its not +, its -

[bronze glen]

oh

crap

mb

[viscid shale]

Now, try computing the 1st derivative

[bronze glen]

so now just keep differentiating till a noticable pattern?

[viscid shale]

We will just do the first one and i hope you can see the idea im trying to get to

[bronze glen]

?

[viscid shale]

Yes, to start with, this one should help you notice that a) cant be the correct option

Since for each time you do the derivative, the leading coefficients will get bigger and bigger

[bronze glen]

i didnt understand where does npi/2 come into play

[viscid shale]

Thats the second thing i was going for now

[bronze glen]

oh

[viscid shale]

I hope you can understand what im going for w/ this one

[bronze glen]

yeah

[viscid shale]

Now, do you remember shift identities of trig functions?

[bronze glen]

the coefficient increases with every derivative

so a is eliminated

[bronze glen]

shift?

i prob didnt recognize the term

[viscid shale]

As in f(x-a), shifting the function left or right

theres a notable identity for cos(x + pi/2)

[bronze glen]

ohh

yeah

i know

[viscid shale]

Do you remember this one?

[bronze glen]

-sin x

[viscid shale]

Yes, and going a step forwards too, whats sin(x+pi/2)?

[bronze glen]

cos x

oh

i see where we going w this one

[viscid shale]

neat, i hope you can find the pattern with this one at this point

[bronze glen]

yep

[viscid shale]

It has to do with the fact that each derivative (apart from scaling the coefficients), also shift the previous y_{n-1} by a certain amount

[bronze glen]

tho i think i made an error somewhere

i think i made a sign error here, should be cos4x - cos2x?

[viscid shale]

Also, the way the problem is set up, its really elegible for induction

[bronze glen]

oh this q was apparently user "logarithmic differentiation" section of our book

not sure where we used logs

but ight

also im getting this, the -ve sign is infront of 2

which kinda messes me up

did i do it wrong somewhere

oh

im stupid

tysm for the help

.close

[pearl pond]

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[pearl pond]

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[stuck berry]

any other way to find this other than manually? I tried to do it but it's just error prone and unsatisfying (my solution below)

[night sand]

yep so you can use properties of variance

Var(2X-3Y+8) = Var(2X-3Y) = 4Var(X) + 9Var(Y) - 12Cov(X, Y)

Var(X), Var(Y), and Cov(X, Y) are much simpler integrals to evaluate

you found the marginal distributions, you just need to use those, and you can find E[X] and E[X^2], and similarly for Y

and then E[XY] for covariance

[stuck berry]

ah yeah I was avoiding it since I did cover covariance but my teacher didn't cover this for some reason

[night sand]

well you don't need to write it as covariance if you don't want to
you would get the same thing by expanding E[(2X-3Y)^2] - (E[2X-3Y])^2

you would get it in terms of E[X], E[Y], E[X^2], E[Y^2}, and E[XY]

[stuck berry]

fair enough

.close

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[fiery lance]

hey

[fiery lance]

for part c

i dont get why

its 7 <= k < 9

and not

7 <= k <= 9

[smoky musk]

Yeah, I got 7 <= k <= 9 too.

[fiery lance]

bruh

[smoky musk]

Can I look at the key's answer?

[smoky musk]

Probably the key is wrong. If k = 9, then x=9 is still inside the domain of g. Probably ask your teacher.

Or maybe I'm also wrong. So, <@&286206848099549185>

[fiery lance]

yeah

aight

[smoky musk]

Probably the key mistook 6 as still being part of the domain, so they mistakenly excluded k=9.

[fiery lance]

why would that changei t

[fleet ridge]

No actually

Wait nvm

[fiery lance]

fastest switch up

[fleet ridge]

I didnt see x can be equal to k

[smoky musk]

Or that the question is a misprint. The domain is intended to be x > k

[fiery lance]

but then

wouldn't it be 7 < k < 9

?

[smoky musk]

No, if x > k, the answer for c would be 7 <= k < 9. Because at k=7, the function g still has an inverse.

[fiery lance]

yeah

i see

bomboclaat

[fleet ridge]

!done

[pearl pond]

If you are done with this channel, please mark your problem as solved by typing .close

[fiery lance]

.close

[pearl pond]

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[pearl pond]

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[latent glen]

Given parallelogram ABCD with I as the intersection between AC and BD. Choose any point K on AB, draw KM parallel to BD (M is on AD) and KN parallel to AC (N is on BC). Find AK/AB so that the area of KMN is 2/9ths the area of ABCD.

[sterile gazelle]

this looks v interesting

[pearl pond]

@latent glen Has your question been resolved?

[latent glen]

it's from the entrance test in the morning

<@&286206848099549185>

[jaunty lintel]

ptnk?

[latent glen]

you know?

[jaunty lintel]

yes

i took 2024-2025 problems

2 year-ish ago

[jaunty lintel]

2024-2025 entrance exams

if ur looking for sols then its probably already solved by the school itself and posted online

since its what they do each year

[latent glen]

yeah but i kinda want to do it by myself (partly bcs i don't have any idea yet)

[smoky musk]

Do you see any similar triangles?

[latent glen]

i know DMI = INB

[smoky musk]

There are more.

Let's define AK/AB = r. What is the area of AKM relative to the area of ABCD?

[latent glen]

AK^2/2AB^2?

[smoky musk]

?

[latent glen]

we know that triangle ABD = BCD

so S(AMK)/S(ABD) = (AK/AB)^2

[smoky musk]

Yeah, but we defined AK/AB = r, so we can reduce further.

And also, we're expressing it relative to ABCD, not ABD

[latent glen]

but S(ABD) = 1/2 of S(ABCD)

[smoky musk]

You mean S(ABD) = 1/2 of S(ABCD)?

[latent glen]

oh yeah

[smoky musk]

Yeah. Also, we expressed AK/AB as r, because you're going to repeat this ratio very much thorought the proof.

[latent glen]

which gives us 1/2r^2

<@&286206848099549185>

[smoky musk]

No. Thread carefully.

[latent glen]

i keep forgetting the square

[smoky musk]

Also, I would avoid notating something like 1/2r^2. It's pretty ambiguous whether you mean (1/2)r^2 or 1/(2r^2).

For many people, they would read it as the latter. (Including me)

Anyway. We know AKM. Now what about KNB?

[latent glen]

(1/2)(1-r)^2

[smoky musk]

And finally, what about MNCD?

[latent glen]

can we splice it into the smaller triangles?

or is there an easier way?

[smoky musk]

There is an easier way.

Note that MNCD is a trapezium.

And it has the same height as ABCD.

[latent glen]

so (MD+NC)*h/2?

[smoky musk]

And you can find the h based on the area of ABCD

[latent glen]

any hints to figure out h from ABCD?

[smoky musk]

You don't know the formula for the area of a parallelogram?

[latent glen]

wait my brain short circuited

[smoky musk]

Let's take AD as the base here

[latent glen]

AD*h

the area

[smoky musk]

Yup, plug the h back there, and you should get a ratio of MD over AD, and NC over BC

[latent glen]

i'll come back some time later

.close

[pearl pond]

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[pearl pond]

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[real ermine]

CAN I ASK A QUESTION

[hazy quail]

go ahead

[hazy quail]

you are supposed to post question here ofc

!status

[pearl pond]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[real ermine]

Idk anything 💀

[smoky musk]

What a weird graphical transformation

[indigo kayak]

Hi people I want to learn math from zero.do you have any advice for me ? Please

[fleet ridge]

!occupied

[pearl pond]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[smoky musk]

Honestly I don't know what the question is supposed to mean.

Like, do we assume the hot air balloon is bound to the point C and D by ropes? Or does the hot air balloon move upwards without any kind of rope? Or even, if the hot air balloon move horizontally at all?

[hazy quail]

derivate of theta?

[real ermine]

I mean this is Related Rates and Differential Calculus

[fleet ridge]

@real ermine the main challenge in this is to relate x and theta

How would you think you could do that?

[real ermine]

I think it is 2.55 but I got it wrong though

[fleet ridge]

The answer?

In m/s?

[real ermine]

Uh I got it wrong

Well probably 2.55 m/s

[fleet ridge]

What relation did you get between x and theta, and how did u do it?

[real ermine]

The rate of change probably

[fleet ridge]

Could you maybe show your work?

[real ermine]

Wait

[fleet ridge]

If I'm not wrong, the hot air balloon is moving up at a some speed which changes. There are no ropes or anything, theta is just the angle

[fleet ridge]

Take your time

[smoky musk]

Even then, the question gives the dθ/dt (since the unit is rad/s), and not dθ/dx (because the unit for that is rad/m). However, there is nothing about dx/dt anywhere in the question.

[fleet ridge]

Thats what you need to find... im pretty sure the chain rule will be used

[smoky musk]

Nevermind, yeah. There is a way.

[fleet ridge]

Uhhh... @real ermine u there?

[real ermine]

Yeah

My handwriting is unpleasant today

Anything wrong?

[fleet ridge]

I got to go for around 20 minutes. @smoky musk could you try helping?

[real ermine]

Uhhh… Anyone online?

[fleet ridge]

Ping <@&286206848099549185> . Ill be back in a while

[real ermine]

I didn’t mean to send that

[smoky musk]

Hmm, I didn't see anything wrong?

[real ermine]

Oh

[pearl pond]

@real ermine Has your question been resolved?

[real ermine]

Pretty much I will quit for now

[pearl pond]

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[junior anchor]

What's with ts about checking risk of an axiom by it's ability to prove the existence of large cardinals

[pearl pond]

@junior anchor Has your question been resolved?

[junior anchor]

Like why is it that consistency of ZF + AD <=> consistency of ZFC + woodin cardinals

[rough forge]

@waxen agate

[cinder flower]

[junior anchor]

Hiiiiii

The amount of melody emojis increase by the day

[cinder flower]

[waxen agate]

relative consistency strength

we cannot usually prove 'this new axiom is consistent' outright so set theorists, bless them, ask what other theories it is equiconsistent with

you could say if an axiom implies, or can be transformed into, a model with large cardinals, then it is at least that strong, and if large cardinals can also produce a model of that axiom, then the two have the same consistency strength

AD obviously contradicts full choice so what we're really saying is if ZF + AD has a model, then there is enough structure to get a model of ZFC with infinitely many Woodins

[junior anchor]

(My net was down the messages just appeared)

[waxen agate]

and conversely if such Woodins are consistent they can produce a module of AD

wait this blud is arohi

[junior anchor]

Okay I see

[junior anchor]

Module?

[waxen agate]

model*

[junior anchor]

Okay so

[cinder flower]

just a module of AD 🤲🏻

[junior anchor]

Since the models satisfying ZF + AD will be a subset of models satisfying ZF, we are proving that the same subset can satisfy ZFC + existence of woodin cardinals?

[waxen agate]

not the same models

[junior anchor]

Oh wait

If model exists for ZF + ad

Then another exists for ZFC + woodin

And vice versa

?

[waxen agate]

yes

[junior anchor]

So uh how do we go about

Proving ts sort of result

Like can u give me an example with another axiom

[waxen agate]

'another exists' here means a related model can be constructed or shown to exist, not necessarily inside the original model in a simple way

[junior anchor]

Not sure what "not necessarily... way" means

That part

[junior anchor]

Bwo

[waxen agate]

I guess Con(ZF) <=> Con(ZFC + GCH)

what does this say

[junior anchor]

Yea so how do we prove such a equiconsistency result

[waxen agate]

roughly that the second model does not have to be the same model, and it does not have to be a simple submodel sitting inside the first one

[junior anchor]

Yea I got that, it'll be very unlikely for that to be the case

Because then that model 'll have to satisfy all the axioms, ZFC + ad + woodin. (And ZFC is known inconsistent with ad)

[waxen agate]

The general pattern to prove Con(T) <=> Con(S). You assume there is a model M |= T. Define some related structure N. Prove N is actually a model. Verify every axiom of S holds in N. Therefore, if T has a model, then S has a model. Then you do the reverse.

very roughly

[junior anchor]

Hmm

[waxen agate]

for this suppose M |= ZF define the hierarchy L_0 = emptyset, L_{a+1} = Def(L_a), L_lambda = bigcup_{a<lambda} L_a for limit lambda, L = bigcup_{a \in Ord}L_a.. at each state you take only subsets definable from earlier material

then Gödel proves L^M |= ZFC + GCH

The reverse direction is trivial

any model of ZFC + GCH is automatically a model of ZF

[waxen agate]

the full proof is a bit long but the idea is prove L^M |= ZF then L^M |= V = L then V = L => AC then V = L => GCH so L^M |= ZFC + GCH

[junior anchor]

What's L^M here

[waxen agate]

the constructible universe as computed inside M

diamond$ds L^M = bigcup_{alpha in mrm{Ord}^M}L_\alpha^M$

the class of sets that M thinks are constructabke

[jolly parrot]

Oléagineux Distilliànus VIVII

[junior anchor]

Interesting... How long can I keep the channel open? I'm still reading stuff, would you possibly be free after an hour ?

[waxen agate]

50/50

[junior anchor]

Okay np. Thanks for the help so far.

[pearl pond]

@junior anchor Has your question been resolved?

[pearl pond]

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[rough forge]

yapmatics

[pearl pond]

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[sterile gazelle]

hi renato

[stoic imp]

hi hyzae good afternoon

[sterile gazelle]

[stoic imp]

I drew the region

what i dont understand is that part that is saying C is traversed by going through (1,0), (0,0) , (0,2)

@summer imp

[summer imp]

C is the part of the ellipse in the first quadrant along with a line going from (0,2) to (0,0).
They just mean that you traverse C from (1,0) to (0,0) along those two curves.

[stoic imp]

yeah so counterclockwise

[summer imp]

Yes

[stoic imp]

thats what they mean

prolly.

any ideas, no spoilers please

just hints

im pretty sure this shit screams GREEN

[viscid shale]

When they give you a list of points traversed its usually so you can correctly orient the curve

[summer imp]

They have some weird notation going on for the actual integral they want you to compute.

I don't like that you're integrating a vector field w.r.t. arclength?

[viscid shale]

Its the green equivalent to stokes

since ds implies an oriented curve segment

on a plane*

[summer imp]

Sure but you'd usually have $\int_C \vec{F} \cdot \dd{\vec{r}}$

[jolly parrot]

Azyrashacorki

[viscid shale]

But yea, this is made for green's theorem

[summer imp]

And ds (unbolded) I've seen mostly used for scalar line integrals

Anyway

[stoic imp]

,w partial wrt x of x^2 + x + 2y

[jolly parrot]

Results provided by WolframAlpha

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[viscid shale]

Gang its a polynomial 🥀

[stoic imp]

,w partial wrt y of 2xy + 4x + y

[jolly parrot]

Results provided by WolframAlpha

Click here to refine your query online
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Requested by renatoxbz

[stoic imp]

wut, I am getting 0 for the integral

[viscid shale]

Remember that your integral for greens have to be a positively (counterclockwise) oriented closed curve

our curve is not closed

[stoic imp]

our curve is closed

[viscid shale]

Try to graph the curve

[stoic imp]

we need to add the segment that goes from (0,0) to (1,0) for making this shit closed

[viscid shale]

Well, yes, but you also have to substract to preserve equality to what they asked you for

[stoic imp]

yes ofc

so let C3 = C1 u C2 u CL
where CL = {(x,y) in R2 | y = 0, 0<=x<=1}

now the line integral that traverses C of the vector field F is equal to the surface integral of C3 (pdx Q - pdy P) dx dy - line integral of CL

wait chat am I actually getting good at this shit?

,, \int_C F ds = \iint_{D} \left(\pdv{Q(x,y)}{x} - \pdv{P(x,y)}{y}\right) \ dx \ dy - \int_{CL} F ds

[viscid shale]

Btw, just for reference, when you wanna compute the double integral

use _D as prefix

since its a domain enclosed by curve C3

[stoic imp]

wdym?

[viscid shale]

C3 is the curve around D

D is the domain enclosed by C3

So instead of doing $\iint_{C_3} \cdots dA$. you can do $\iint_D \cdots dA$

[jolly parrot]

Renato

[viscid shale]

Since its not over the curve youre integrating with, its the area inside it.

[stoic imp]

yeah but idk what should be inside by my line integral of CL

[viscid shale]

Well, you already said and know that the curl integral in the middle = 0

Because the components are equal to each other

[stoic imp]

whats your point

[viscid shale]

,, \iint_D \left(\pd x Q - \pd y P\right) dA = 0

[stoic imp]

yeah, so?

[viscid shale]

Now, you can follow with the line integral by just doing the usual work

,,\iint_C F\cdot dr = \int_a^b F(r) \cdot r' dt

[stoic imp]

,align \int_C F ds &= \iint_{D} \left(\pdv{Q(x,y)}{x} - \pdv{P(x,y)}{y}\right) \ dx \ dy - \int_{CL} F ds
\ \int_C F ds &= - \int_{CL} F ds

[jolly parrot]

Renato

[stoic imp]

wut

[viscid shale]

its just a line integral over a vector field

using parametrized curves

[stoic imp]

wut

[rough forge]

hello

wut's the matter

[viscid shale]

Can you write CL in terms of a parametrized curve?

[stoic imp]

someone help for christ sake

[rough forge]

like Dexter isn't already

[stoic imp]

CL is a line segment

I can do it

but we are integrating over the vector field ye?

[viscid shale]

Go ahead

I want to believe you know how to do line integrals over a vector field

Cause if not i have 0 clue what the hell we are doing using greens theorem

[stoic imp]

the problem is you are multiplying by the derivative of r

[viscid shale]

ah

,, \int_C \vec F(x,y) \cdot d\vec r = \int_a^b \vec F(\vec r(t)) \cdot \vec r\ '(t) dt

[viscid shale]

what i wrote previously and this are the same thing
the second one took me a lot more time to write tho

[stoic imp]

why and how

[viscid shale]

Thats how you compute a line integral over a vector field????

[stoic imp]

it depends because when we first did the line integral over a vector field at the start we used greens

because the region was closed after we added the line

[viscid shale]

Because green's is an identity used to simplify them

The fact that you previously used greens doesnt invalidate other techniques

[rough forge]

https://mathinsight.org/line_integral_vector_field_introduction

[stoic imp]

sure

[stoic imp]

dont be so judgemental dawg, im not einstein

CL = {(x,y) in R2 | y = 0, 0<=x<=1}

r(t) = (t, 0)

0<=t<=1

,, \int_{CL} F \ ds = \int_0^1 F(r(t)) \cdot r'(t) dt

[jolly parrot]

Renato

[rough forge]

In a sense if you have a force F and a curve r you are essentially accumulating all the components of the force F along the curve r, in other words you are accumulating how much the force F has an effect on the curve at a point, and that accumulation is what we could call work

[stoic imp]

F(x,y) = (2xy + 4x + y, x^2 + x + 2y)

F(x,0) = (4x, x^2 + x)

F(r(t)) = (4t, t^2 + t)

r'(t) = (1,0)

,align \int_{CL} F \ ds &= \int_0^1 F(r(t)) \cdot r'(t) dt \ &= \int_0^1 4t \ dt \ &= 4 \int_0^1 t dt \ &= 4 \cdot \frac{1}{2} \ &= 2

[jolly parrot]

Renato

[stoic imp]

@rough forge @viscid shale @summer imp

[rough forge]

atp ping everyone

[viscid shale]

Remember to "-"

Since we were originally substracting

[stoic imp]

yeah so the answer is just -2

alr dawg thanks.

.close

[pearl pond]

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[rough forge]

[pearl pond]

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[cobalt torrent]

Can someone explain what they mean by space is moving at 99.99% the speed of light? I recently thought "if event horizon is the point where nothing can escape, then how much force would I be facing at say 10 metres away from it in a really powerful blackhole like ton 618" and the answer was surprisingly like 0.15m/s .
But how is this possible? Why am I acceralating really low at really near to the event horizon and why does it suddenly jump to a lot at event horizon?

[shrewd rune]

wassup

so basically applying the formula g = GM/r^2

the mass ofcourse is enourmous

but the radius of the ton 618 squared is just uncomparable

and since formulas are considered the centre of the mass

youre gonna be so far from the centre :. it doesnt depend on distance between the horizon but the distance to the centre

[shrewd rune]

and the "jump a lot", where did you get that

[pearl pond]

@cobalt torrent Has your question been resolved?

[pearl pond]

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[ripe trout]

The question for number 2 is asking me to Fold and tape things so I can get my Octahedron with tetrahedrons

[ripe trout]

The print out is here
https://media.apexlearning.com/One-Off/201811/01/b6c62ff0-c0d2-4889-9b1a-633294427031.pdf

But I don't really know what to do for this

can anyone help me?

[pearl pond]

@ripe trout Has your question been resolved?

[copper matrix]

cool

[copper matrix]

can you print the paper out

and get scissors

[pearl pond]

@ripe trout Has your question been resolved?

[pearl pond]

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[desert cliff]

https://cdn.discordapp.com/attachments/903486480075354182/1507794166036304186/image0.jpg?ex=6a13323e&is=6a11e0be&hm=c48e40cd2e7959f549efbea943e392db2111b18f7620cf0c33c9bf3ea0deb5dc&

[desert cliff]

!ststus

!status

[pearl pond]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[desert cliff]

the sum of the two matricies are only symmetric if and only if alpha = beta

How

.close

[pearl pond]

Channel closed

Closed by @desert cliff

Use .reopen if this was a mistake.

[pearl pond]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[ruby wing]

can it be translated please

[pearl pond]

@stoic imp Has your question been resolved?

[viscid shale]

#help-36 message

[stoic imp]

can you help with gauss

[viscid shale]

Well, you first have to create some volume that has as one of its boundaries the surface created by the intersection of the cylinder and f(x,y)

[stoic imp]

we dont know what f(x,y) is tho

[viscid shale]

We do this precisely because we cant, but we know that this field is divergence free, so one side of the identity will become 0

[delicate temple]

?

[viscid shale]

Open the link and see what that message links to

[delicate temple]

Ohh..

[stoic imp]

how

[viscid shale]

be a little more specific

[stoic imp]

one side of the id is 0

[viscid shale]

Write down the identity for divergence theorem

[stoic imp]

wdym?

[viscid shale]

The formula for gauss

[stoic imp]

ah

[stoic imp]

@viscid shale

[viscid shale]

,tex What is $\del \cdot \mathbf F$ here?

[stoic imp]

for this problem?

[viscid shale]

yes

[stoic imp]

its just the dot prod between (pdx, pdy, pdz) and our vector field

[viscid shale]

Do it

[stoic imp]

also, we need the normal first

lets not jump to computations first

what is the surface we are integrating with respect to in gauss here

should be the border of a solid

[viscid shale]

Im just explaining why its 0

[viscid shale]

Again, find this

[stoic imp]

,w divergence of (2ye^z, 2xe^z, (x^2 + y^2)^2)

[jolly parrot]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by renatoxbz

[stoic imp]

its 0 dawg

[viscid shale]

plug that divergence into the formula for Gauss's Divergence Theorem

[stoic imp]

but we dont know which surface we are even integrating

[viscid shale]

Id like for you to think why i do not care based on the fact that the divergence is 0

[stoic imp]

Also we want to know the line integral of F traversing C

@viscid shale

[viscid shale]

.

Without still thinking on what the volume is, try to rewrite the equation for Gauss's Divergence Theorem into a meaningful way knowing the divergence is 0

[stoic imp]

I mean we know the surface integral is 0

[viscid shale]

Depends on what surface integral.

[stoic imp]

but in order to know that you gotta know if gauss applies

[stoic imp]

exactly

[viscid shale]

You gotta construct a volume that has the surface corresponding to f(x,y) as one of its boundaries.

You will have other boundaries too

And you can rewrite the whole expression:
$$\iint_S \mathbf F \cdot \bar\eta dA$$
into
$$\iint_{S_1} \mathbf F \cdot \bar\eta dA + \iint_{S_2} \mathbf F \cdot \bar\eta dA +\iint_{S_3} \mathbf F \cdot \bar\eta dA + \cdots $$

[stoic imp]

The problem is that I dont understand the geometrical intuition of the problem dawg

@viscid shale

[viscid shale]

Say you have a non-closed surface

And you join surfaces to it such their union becomes closed

[stoic imp]

how so?

[viscid shale]

This is a construction, you just add surfaces of your liking

Now, the volume enclosed (which we call V) is ellegible for using Gauss's Divergence Theorem

Which allows you to write this

Substract the integrals of the surfaces you added from both sides

And you get an expression that allows you to find the flux through the original surface.

You generally want to use "extra surfaces" that are easy to integrate with.

[stoic imp]

the problem is I dont follow

because we want the line integral over a region C

[viscid shale]

what line integral???

[stoic imp]

dawg see the attached picture

[viscid shale]

Mark the point in the picture in which youre asked for a line integral

[viscid shale]

Apart from the fact that you couldnt be seriously expecting me to be able to read that

I like to assume that is a surface integral instead

[stoic imp]

dawg is a line integral i believe

[viscid shale]

🥀

If it was a line integral, the "orientation given by a normal pointing at z > 0" wouldnt have any sense

[stoic imp]

say wallahi bro say wallahi

[stoic imp]

ok lets assume is a surface integral alright

[stoic imp]

ok

[stoic imp]

why

[viscid shale]

cause all* volumes are

[stoic imp]

i don't understand

@viscid shale

[viscid shale]

also, dont ping people at random

Do you understand what this means?

[modest jackal]

Not to interrupt but is that the integration symbol

😭

[viscid shale]

yea im lazy 🥀

[stoic imp]

no

[viscid shale]

Do you understand what gauss's divergence theorem is?

Or do you know just the formula for it

[stoic imp]

i understand more less yeah

[viscid shale]

try to give a quick explanation of it

[stoic imp]

given a solid in R3, we can write the surface integral of the boundary of the solid as the triple integral over the solid

the surface given by the boundary of the solid, should be separable into different intervals

[viscid shale]

as this is written, youre missing a few important details, but ok

[viscid shale]

By this

it means if you grab a lot of different surfaces, they can act as the boundary to a solid if they all end up closing when joined

[stoic imp]

yeah precisely

[viscid shale]

follow on this line of messages then #help-39 message

[stoic imp]

am I missing something? how would you describe gauss in your own words?

[stoic imp]

why do you say we have an non closed surface

[viscid shale]

this is a non closed (open) surface

people in topo are going mad

[stoic imp]

but S is made out of the intersection of a solid cylinder with a line ?

[viscid shale]

Again, not a line

[stoic imp]

a plane?

[viscid shale]

yes
"plane", a surface

[stoic imp]

but we dont know what f(x,y) is dawg

[viscid shale]

we dont care anyways

[stoic imp]

care to elaborate?

[viscid shale]

Keep reading down on the line of messages i already linked

[stoic imp]

what I dont understand is why you say S is non closed surface

all we know about S is that it is the intersection of a solid cylinder and a surface

[viscid shale]

non closed*

[stoic imp]

why so why you say S is open?

[viscid shale]

topologists be mad
Because its part of a surface generated by an equation* z = f(x,y)

[stoic imp]

how does this give that S is open surface btw

?

[viscid shale]

same idea we had a few weeks ago about curves compared to y = f(x) equations

surfaces that have z = f(x,y) can be written as s(u,v) = (u,v,f(u,v))

which implies they are injective in their first and second component

[stoic imp]

so since they are injective in the first and second component

then the parametrization is interely injective in any closed interval

[viscid shale]

they cannot curve into themselves in any way

[stoic imp]

yeah if they would be closed curve, then the endpoints would coincide

[stoic imp]

ok, got it now, good catch saying that S is open surface

what is this doe?

[stoic imp]

yeah the problem is that idk if S is filled or not (is it a solid or no)

[stoic imp]

also, I dont understand the difference between an open and a closed surface

[stoic imp]

in our case, S can be parametrized with a injective parametrization

but how does imply that S is closed or open surface

[sterile gazelle]

hi renato

ur always here fr

[stoic imp]

hey hyzae

[stoic imp]

jajaja maybe I should rest a bit more, like this topics in vector calculus are kind of a pain ngl

[viscid shale]

ima be heading off to sleep

[stoic imp]

.close

[pearl pond]

Channel closed

Closed by @stoic imp

Use .reopen if this was a mistake.

[pearl pond]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[wooden flare]

need some help for this problem (idk how to approach it // what to do first)

[west sapphire]

do you know about normal vectors?

[wooden flare]

yea

[west sapphire]

try considering those

[wooden flare]

but im not given a normal vector

[west sapphire]

but you can find them

[wooden flare]

uh huh...

[west sapphire]

ax + by + cz + d = 0

what's a normal vector to that plane?

[wooden flare]

im not sure

[west sapphire]

does this look familiar?

[wooden flare]

never seen that b4 😭

[west sapphire]

but you have heard of normal vectors to planes?

[wooden flare]

yea

[west sapphire]

and no one showed you how to find them?

[wooden flare]

-# yea

[west sapphire]

ok well it is that simple

just take the coefficients of x,y,z

[wooden flare]

only time ive seen it is like finding the eq. of a plane

like, $\overrightarrow {r} \cdot (p-a) = 0$

[jolly parrot]

･ﾟ✧ 𝓀ℬ ✧ﾟ･

[west sapphire]

yea that's basically what you have here, in disguise

[wooden flare]

mhm ok

[west sapphire]

suppose r = (a,b,c) and p = (x,y,z)

then your formula becomes:

$$(a,b,c) \cdot (x,y,z) - d = 0$$

[jolly parrot]

Bungo

[west sapphire]

where d is some number

and that is equivalent to ax + by + cz - d = 0

[wooden flare]

mhm

and (a, b, c) and (x, y, z) are the normal vectors yea

sorry, (a,b,c) is the normal vector

[west sapphire]

yes

[wooden flare]

ok

[west sapphire]

or more precisely, a normal vector

any nonzero scalar multiple of (a,b,c) is also a normal vector to the same plane

[wooden flare]

ok, so what do we do with the normals

[west sapphire]

how do you check if two planes are parallel or not?

using their normal vectors

[wooden flare]

parallel is always tripping me up for some reason

like, I can do it one way and it works (non-parallel) but the second its parallel everything breaks idky

[west sapphire]

well just check if the normal vectors are parallel

if they are, then so are the planes

[wooden flare]

two vectors are parallel if V_2 is a scalar multiple of V_1 right?

[west sapphire]

yep