help-43

[covert spear]

$\frac{d}{dx}ln(x) = \frac{1}{x}$

[boreal girder]

Bakoles

[open lotus]

ln(x/4)

OHHH

SO THATS HOW THE ANSWER KEY GOT 1/4

[covert spear]

yes

[verbal rover]

Yep

[covert spear]

simple as that

[open lotus]

ty guys i get it now

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[sinful birch]

Hi 👋 I was trying to figure out the groups of order 8, and I have G={e, a, b, c, ab, bc, ac, abc} with a^2=b^2=c^2=0. Then I have f: G -> Z/2Z x Z/2Z x Z/2Z with f(a) = (1, 0, 0), f(b) = (0, 1, 0), and f(c) = (0, 0, 1). I’m trying to show that this is a group isomorphism. But I’m not sure how to figure out what f(ab) for example would be without explicitly stating it (it’s (1, 1, 0)) but I’m not sure how to prove that without it being a given or using the properties of the group homomorphism I am trying to prove. Thanks

[snow finch]

when you define a group you define a set and an operation

and when you define a group homomorphism you need it to be consistent so f(ab) = f(a) f(b)

you implicitly defined an operation because you're doing the direct product of Z2 three times

and in that group 100 * 010 does equal 110

so as long as your homeomorphism takes ab -> 110 then it's consistent

[sinful birch]

Do I just state where it takes everything as part of the definition of my function or is there some way to work it out from where it takes a and b without using the fact that it is a group homomorphism?

[wind tapir]

if f isn't given to be a group homomorphism, then it's not fully defined as a set function

[sinful birch]

Ohhh ok

Thanks everybody

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[near kestrel]

I solved for x1, x2, and x3. Is A inverse just my x vector? Shouldn't it be a matrix.

[subtle helm]

!pdf

[upper bane]

!filetype

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[twin anchor]

The A^(-1) matrix will be the columns you got from each Cramer’s rule put together

[near kestrel]

so [8/5, 1/5, 1]?

thats my solution vector right

[twin anchor]

That’s for one specific vector right

Wait

I don’t think you finished solving for any of your vectors

[near kestrel]

wait am i supposed to replace each column of a with the givin vector

[twin anchor]

yeah

to get each entry

It’s really annoying

[near kestrel]

sob okay thank you

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[pallid sierra]

Could someone explain this to me? In simple terms please lol

[rough roost]

have you learned the fundamental theorem of calculus?

im happy to give you the tldr

[pallid sierra]

Yeah

[rough roost]

ok and what does that say?

[pallid sierra]

Split it?

[rough roost]

you should do that but i was asking what the ftc says

[pallid sierra]

Maybe I don't know it because I am confsued

[rough roost]

so the fundamental theorem of calculus

says

basically

that if you have F' = f

then that int_a^b f dx= F(b)-F(a)

i.e.

integrals and derivatives are opposites of some sort

[pallid sierra]

What's int?

[rough roost]

integral from a to b

[pallid sierra]

Oh Integeral A^b f

Oh

[rough roost]

yeah

[pallid sierra]

So I plug 4 into F?

[rough roost]

well, wait up

[pallid sierra]

Then it would be 6 - 3 ?

[rough roost]

because we dont have just the integral of f in front of us

right

we have the integral of something related to f

so we should split it up so that we can get just the integral of f

right?

[pallid sierra]

Wait what?

If int_a^b is the same as F(a) - F(b). which we are given why not just plug and run?

F(15) = 6
F(4) = 3

3?

[rough roost]

not yet

because we are trying to get int_{4}^{15}(4f(x)-5)dx

right

and ur right

int_{4}^{15}(f(x))dx is 3

[rough roost]

we just need to manipulate this guy until we can use

[rough roost]

that

that make sense?

[pallid sierra]

Kind of? We need to use 3 in place of f(x)?

[rough roost]

basically

the other thing that should be popping into your head right now

is that

if we have $\int_{a}^{b}(h(x)+g(x))dx$ then that is $\int_{a}^{b}(h(x))dx+\int_{a}^{b}g(x))dx$

that means we can split up 4f from the -5

[pallid sierra]

Then 4 = f(x), 5= g(x)?

[rough roost]

i picked bad names

one sec

4f = h

4f(x) is h(x)

-5 is g(x)

[boreal girder]

niko

[rough roost]

there we go

mb

[pallid sierra]

Then our next step is to deriv no?

[rough roost]

yeah, we then need to pull out the 4 as well

but yeah

[pallid sierra]

I am super confuse because if you deriv -5 then it's nothing

[rough roost]

no

integral is anti derive

[pallid sierra]

If you deriv a constant, it becomes 0 no?

Oh

[rough roost]

yes

[rough roost]

right that what this is saying

so what is the "area under the curve" of -5 from 4 to 15

[pallid sierra]

int_a^b f dx= F(b)-F(a)

If this is the case then

a = 4
b = 15

F(15) = 6
F(4) = 3

3

So we times -5 by 3?

bc

dx = F(b) - F(a)
dx = 6 - 3

[rough roost]

wait holdup

lets be clear

we have

$\int_4^{15} 4f(x)-5 dx$

[boreal girder]

niko

[rough roost]

and we split it up

$\int_4^{15}4f(x)dx - \int_4^{15}5dx$

[boreal girder]

niko

[rough roost]

now all we have to do is calculate the two integrals separately

does that make sense?

[pallid sierra]

Ahhhhhhhhhhhh

What does dx represent? 3?

[rough roost]

you shouldnt treat the dx as anything real for right now

the way you should think about the integral is that its the area under the curve from a to b

[pallid sierra]

If it isn't real, how will we calculate the two?

[rough roost]

i mean you think about it as notation

lets think about the second one

what guy do you take the derivative of to get 5?

[pallid sierra]

5x

[rough roost]

great

so using that handy dandy ftc

int from 4 to 15 of 5x is 5x evaluated at 15 minus 5x evaluated at 4

[pallid sierra]

I kind of see?

If we do int 4-15 of 5x then it is 15 - 5x = 4?

[rough roost]

wait its

int 4 to 15 of 5

which is 5(15)-5(4)

[pallid sierra]

This is the 5x right?

turned 5(15) and 5(4)

[rough roost]

yeah

exactly

[pallid sierra]

So why did we get the 3?

Couldn't we just have split, anti-deriv one of them (5)

then put the 4 and 15 in the place of x?

[rough roost]

wait the 3 comes into place in the first integral

we just calculated the second one

so we know that if we integrate f from 4 to 15

we get 3

so

how do we get an integral of f from $\int_4^{15}4f(x)dx$

[boreal girder]

niko

[pallid sierra]

We do the antideriv?

or do we plug 3?

[rough roost]

first we have to pull out the 4

so we have

$\int_4^{15}4f(x)dx=4\int_4^{15}f(x)dx$

[boreal girder]

niko

[pallid sierra]

then we just need to find f(x) and anti deriv it?

[rough roost]

now were set

[pallid sierra]

and what of the 4? does it come to play after?

we just multiply it

[rough roost]

bc you calculated the integral of f

you got it was 3

so this integral is 4*3

[pallid sierra]

12

but couldn't we just find 3 then multiply it by $\int_4^{15}, the 4?

$\int_4^{15}

and save us the trouble or is that just over-generalizing

[rough roost]

i wouldnt call it multiplying here

you have to be a bit careful with integrals

they are really multiplying per se

like

$\int_4^{15}f(x)dx$ is not

[boreal girder]

niko

[rough roost]

f(x) times dx time the integral sign thing

its something else

[pallid sierra]

Ah

Is 12 the final answer?

[rough roost]

no so we have

$\int_4^{15} 4f(x)-5 dx= \int_4^{15} 4f(x)dx - \int_4^{15}5dx= 4\int_4^{15} f(x)dx - \int_4^{15}5dx=4*3-(5(15)-5(4))$

[boreal girder]

niko

[rough roost]

does that make sense

[pallid sierra]

No

I thought we were finished with 4int4-15fxdx

It became 12

and 12 isn't used there

[rough roost]

thats the 4 * 3

that you see there

[pallid sierra]

Ah

and the 5(15) - 5(4) is from the second part of 5x?

then we get 55? then 43

[rough roost]

i think its -43

[pallid sierra]

Let em rrite down the steps to you

So

First part
int_4^15
F(4) - F(15) = 6 - 3 = 3
4 * 3 = 12

Second Part
5 -> 5x
-5(15-11) = -55

-55 - 12 = 43

Yes?

I also write it as formula but I doubt it makes much sense

[rough roost]

second part is a little off

[pallid sierra]

How so?

[rough roost]

so

the second part was

$-\int_4^{15}5dx$

[boreal girder]

niko

[pallid sierra]

It's negative?

[rough roost]

u see the neg 5

here

i pulled out the minus sign

[pallid sierra]

Oh

[rough roost]

mb

[pallid sierra]

All g, I get it

that's why it's -43 no?

[rough roost]

yeah

[pallid sierra]

Ok cool

[rough roost]

so we get 12 - (55)

=-43

[pallid sierra]

For the original question, is 14b the same way?

[rough roost]

you mean the second part

the int from 12 to 12 thing?

[pallid sierra]

Yeah

[rough roost]

well

do you know what the integral means?

[pallid sierra]

Oh

It's already at 12?

[rough roost]

well, the integral of from a to be is what

the "area under the curve" right

[pallid sierra]

So it would be the point under the curve no? It's the same

[rough roost]

yeah but theres no area if its just a single point

it has no width !

another way this ties in

is

its $\int_{12}^{12}f(x)dx=F(12)-F(12)=0$

[boreal girder]

niko

[pallid sierra]

I thought so

and there's nothign ti split

so it's just 0?

[rough roost]

yeah

[pallid sierra]

Nice

I hope they repeat that question on the exam lol

[rough roost]

maybe for free points

[pallid sierra]

I hope so

but if there aws something to split, would be not be 0?

[rough roost]

i mean

[pallid sierra]

We would have to go through the entire process with it just being 0?

[rough roost]

the integral from a to a of anything* is 0

[pallid sierra]

Nice

Ok thanks for your help

[rough roost]

ofc

[pallid sierra]

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[molten badger]

Given a pyramid S.ABCD with the base ABCD is a trapezoid $AB || CD$, $AB=2CD$. Let $(P)$ be a plane doesn't go through S and intersect lateral side $SA,SB,SC,SD$ at $M,N,P,Q$ respectively.If $\overrightarrow{SA}=2\overrightarrow{SM}$ and $\overrightarrow{SC}=5\overrightarrow{SP}$.Find out the minimum value of $T=2(\frac{SB}{SN})^2+(\frac{SD}{SQ})^2$

[boreal girder]

Alexis_Fx

[molten badger]

I solved this problem but it's quite long, this's my work and I don't even show everything like how I determine N and Q, or how I get the ratio. I wonder if there's a faster way to do it

[compact pewter]

@molten badger Has your question been resolved?

[molten badger]

on second thought maybe it's not that long, I just hate writing solution

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[remote yoke]

kind of confused with how these questions are worded

[subtle helm]

What does “opposite slope” mean

Opposite as in opposite sign?

[remote yoke]

idk

💔

because if it is opposite sign

then it won’t be parallel

[subtle helm]

Nvm

[kind viper]

i do think that the slopes are meant to be m and -m

which given the lines are parallel can only mean both slopes are 0

[remote yoke]

Ohh

[kind viper]

confusing as hell if you ask me

[remote yoke]

Yeah 😖

[kind viper]

#21 is also weird cause i don't really see the 2nd error unless ||they want alan to include the edge-case of vertical vs horizontal||

[remote yoke]

i was also confused about that

because the only error i saw was 1/m

which is supposed to be -1/m

idk where the other one is and i don’t think we learnt about the edge-case of vertical vs horizontal

[kind viper]

0 slope vs. undefined slope

a horizontal line and a vertical line are still perpendicular

[remote yoke]

Ohh

so do i answer something like “it isn’t the only case”???

like it isn’t if and if only

[kind viper]

("if and only if" is a bit of a fixed phrase in math jsyk)

i think it is best to be explicit in naming what exactly the guy missed.

[remote yoke]

Ahh okay

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[winged dock]

A father had three sons namely, Kailash, Harish and Prakash. All are above 65 years in age. Prakash happens to be the eldest while Kailash as youngest. As per the health history, it is estimated that the probability that Kailash survives another 5 years is 0.8 Harish survives another 5 years is 0.6 and Prakash
survives another 5 years is 0.5 The probabilities that Kailash and Harish survive another 5 years is 0.46, Harish and Prakash survive another 5 years is 0.32 and Kailash and Prakash survive another 5 years is 0.48. The probability that all three sons survive another 5 years is 0.26. What shall be the probability that at least one of them survives another 5 years?

[winged dock]

This question was given to me by a friend of mine

I am not sure how to begin since the probabilities don't seem to matcy

[kind viper]

don't seem to match in what sense?

did you try to start filling out the venn diagram and get a negative number somewhere?

call the events of each son surviving the next 5 years K, H and P by their initials

we are given the following:

• Pr(K) = 0.8
• Pr(H) = 0.6
• Pr(P) = 0.5
• Pr(K&H) = 0.46
• Pr(H&P) = 0.32
• Pr(K&P) = 0.48
• Pr(K&H&P) = 0.26
• goal: Pr(K or H or P)

@winged dock can you tell us where the mismatch is

[winged dock]

Yes they got a negative number

[kind viper]

where

[mild sky]

Tbh from the story told in the question we would expect the probablities of each of them dying are independent.

[kind viper]

nah who knows

[kind viper]

maybe they're close friends and would go through a lot of grief at each others deaths

[kind viper]

yup that matches me

the question is wrong

[winged dock]

The answer is supposed to be 0.9

[kind viper]

garbage in, garbage out.

the setup is self-contradictory, end of story really

i mean you can put bullshit numbers into it and add together all the numbers in this picture and get 0.9 but that's not math anymore

[winged dock]

Thanks for the help

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[upper bane]

<@&268886789983436800>

[bleak dock]

<@&268886789983436800>

[compact pewter]

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[twin idol]

ok what's with these botted accounts?

[bleak dock]

no eye deer

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[real schooner]

According to the picture, if the distance FD is 2x - 3, and the distance BF is 3x + 2, what is the value of x? I’m stuck on how to figure out how to do the value of x.

[ebon peak]

are we told anything else?

[real schooner]

No, its just that from the problem

[ebon peak]

is it a scale diagram?

[real schooner]

No

[ebon peak]

are we told anything at all about any of the other points

[real schooner]

the distance of BZ is equal to 16 units

thats about it

[young raft]

still insufficient info

[ebon peak]

what do we know about BF if BZ is equal to 16

[compact pewter]

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.

[real schooner]

Nothing, I asked the creator and he stated to review bisectors and the midpoint theorem

[ebon peak]

you haven't told us anything about midpoints in the context of the question

are you sure this is the entire original context

[real schooner]

yeah

wait

this is the entire problem

the 1C is 16

[young raft]

okay thank you

next time send that right away

right okay

since they say BC is bisected by AE at point F

what does that tell you about the length of FC and length of BF

[real schooner]

that they are the same length?

[young raft]

yup

[young raft]

you are also given that D is the midpoint of FC. so, since length of FD is 2x-3, what is the length of FC?

[real schooner]

I’m throwing a guess here but is it 2x-3 two times?

[young raft]

yup

which is?

expand 2*(2x-3)

[real schooner]

4x-6

[young raft]

yes

[young raft]

so you have that FC is 4x-6 and BF is 3x+2 and you know they are the same length

thus,
4x-6 = 3x+2

from this you can solve for x

[real schooner]

x = 8?

[young raft]

yes!

[real schooner]

thank you so much ❤️😭

[young raft]

do you need help with the other part where BZ is 16?

or you think you can finish it off on your own

[real schooner]

I think I can finish it on my own but I’m not too sure

[young raft]

try it and come back if you are stuck

[real schooner]

okay

Is 3x+2 going to be used while solving the problem since BF is 3x+2?

[young raft]

indeed

[real schooner]

sorry if I sound dumb but I’m genuinely stuck on it 😭

[young raft]

that's okay

[young raft]

since you've found that x = 8, what is 3x+2?

[real schooner]

OH

26

[young raft]

yes

soo

BZ is 16, BF is 26, so ZF is ...

[real schooner]

10?

[young raft]

yes!

that's it

[real schooner]

thank you! i got confused with the pic because zf was longer than bz

but then i remembered that the problem is unique to anyone who took it

so thank you so much!

[compact pewter]

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[chilly monolith]

I was doing this math problem but i got stuck at number c

[topaz silo]

This A level or gcse?

[winged lion]

number c

[chilly monolith]

just gcse

[topaz silo]

I recognise those edexcel papers

[chilly monolith]

it is 💔 i have solution but i do not understand why they add all yellow probability while it ask for red only

[topaz silo]

Is there a specific part u need help on or just the whole thing

[chilly monolith]

just c tho i understand other ones

i still can make sense til 2/7 but i do not know why it added all yellow probability

[topaz silo]

Alr bear with me I'm cracking out the ipaf

ipad

[chilly monolith]

take your time

[topaz silo]

can u send the completed diagram of parts a and b

I mean I can work it out but this will speed it along

@chilly monolith

[chilly monolith]

got it

[topaz silo]

tyty

[winged lion]

As I read it, its the solution for the color being the same

Although I am not sure, is left side supposed to be the final solution?

[topaz silo]

it says the final solution is 27/32

I'm a bit lost where 5/9 has come from

why is it not 3/9

Ohhhh

[chilly monolith]

5/9 is where you get 2 yellow from bag a, add it to bag b, adding to original 3 yellow ones

[winged lion]

They added two counters

[topaz silo]

nvm I am bugging

[topaz silo]

Yeah I realised now 😭

[winged lion]

Although they didnt specify the color, weird

Could be also mixed

[topaz silo]

Embarassing from me lets ignore that

[chilly monolith]

but the thing is i dont get why would they add all yellow probability when they ask probability for all red

[topaz silo]

Because right

Ur finding the probability they are all red GIVEN they are all the same colour

Which means

,, \frac{P(red)}{P(red)+P(yellow)}

[boreal girder]

AnitaG

[winged lion]

Oh

[chilly monolith]

[topaz silo]

As the denominator encompasses all probabilities in which they are the same colour

[winged lion]

Now we are even

I cant read either

It was a conditional probability

[topaz silo]

Tbh I'm a bit surprised conditional probability is even in gcse

[chilly monolith]

heng

[winged lion]

What is gcse?

[topaz silo]

heng?

[chilly monolith]

its included in almost every probability question 💔

[topaz silo]

The exams 16 y/o sit in the UK

[winged lion]

Oh

[chilly monolith]

oh im on igcse

[topaz silo]

Is it not the same thing

Just a dif exam board?

[covert crater]

igcse is for international students iirc

[chilly monolith]

igcse is for the international ones? idk man

[chilly monolith]

yeh umjm like i still dont get it help adlkajfdsjlsdf

[topaz silo]

That's odd bc where I went sixth form all the internal students did igcse despite it being an English school

[topaz silo]

Okay so say we have 2 events A and B

[chilly monolith]

mhmm

[topaz silo]

Guys how do I do union and overlap signs in latex

brb google search

[winged lion]

\cap
\cup

[topaz silo]

which is which

[winged lion]

\cup is union

\cap is intersection

[topaz silo]

,, P(A|B) = \frac{P(A\capB)}{P(B)}

ermmm

[winged lion]

Dont glue the commands

[topaz silo]

,, P(A|B) = \frac{P(A \cap B)}{P(B)}

[boreal girder]

AnitaG

[topaz silo]

Okay calmm

[winged lion]

You can also use \mid for |

[topaz silo]

Oh I didn't know that

@chilly monolith so P(A|B) is just short hand for the probability that A has happened given that B has happened

And above is the formula

[chilly monolith]

catno

[topaz silo]

I like to think of it as a ratio of both specific things happening to all the possible outcomes in B

If that makes sense

In this case our 'B' is all 3 coins being the same colour

and our 'A' is all 3 coins being red

[chilly monolith]

mhmm

[topaz silo]

'A' is encompassed in 'B'

[chilly monolith]

like

a depends on b?

[topaz silo]

Sort of yeah

[chilly monolith]

cry~5

[topaz silo]

We can't just find P(A) bc that will be lower than what we want

We KNOW that they are all the same colour

So we're no longer interested in all outcomes

Just the ones where it's all the same colour

[chilly monolith]

then isnt it just we finding event b

[topaz silo]

Then from that set we want to know what proportion of probability is of them being just red

[topaz silo]

We need to find both

Bc they r both in the formula

Idk does this make sense

[chilly monolith]

i think

i still cant use formulas

its not allowed for O lev

[topaz silo]

Just think that u need to ignore all the outcomes where they are not all the same colour

Think of it like this:

The total probability of all outcomes is 1

[chilly monolith]

like right? what im thinking is if you find All red + All yellow, we are just finding B which is probability that is all 3 of them are the same color

[topaz silo]

So in a way every probability P(X) can be written as P(X)/1

But if we are looking at a smaller subset of all outcomes

It should no longer be 1

We're looking at a different proportion of outcomes

[topaz silo]

Yess

[chilly monolith]

ohh mhmm i think i kinda gets it

[topaz silo]

It's tricky to explain 😭

[chilly monolith]

so whole properbibity is all red+all yellow instead of 1

[topaz silo]

Yesss

Bc its a GIVEN probability

[chilly monolith]

BUT THE THING IS RIGHT

[topaz silo]

What thing

[chilly monolith]

IF I WANT ALL RED, I NEED TO - THE ALL YELLOW FROM ALL RED+ALL YELLOW WHICH LEAD ME TO 2/7 AGAIN

cry~5

[topaz silo]

THIS is what ur looking to find

Bc the top is the outcomes where they are ALL RED

And the bottom is the outcomes where they are ALL THE SAME

[chilly monolith]

OHHHHH

SO

(2/7)/(2/7+ all yellow) ?

[topaz silo]

Now I'm lost

[chilly monolith]

2/7 is all red probability

[topaz silo]

On the left hand side they have found P(red) + P(yellow)

[topaz silo]

Yes

So u do 2/7

divided by

the thing in the MS

(the sum of P(red) & P(yellow)

[topaz silo]

Okay wait I get it now what u were trynna say

Yes thats correct

[chilly monolith]

OHHHHHH

GOT IT

THANK YOU SO MUCH

I LOVE YOU

[topaz silo]

No problem

😭

That was intense

[chilly monolith]

cry~5

i know right i missed the concept of the whole probability being all red + all yellow instead of 1

thats why i couldnt understand it

[topaz silo]

Tbh I think it's gotta be one of the hardest things in gcse

Some ppl still struggle w it at A level

[chilly monolith]

it was 2025 june paper

[topaz silo]

Well thank god u didn't have to sit that paper then

[chilly monolith]

im sitting this year oct/nov and MY MATH B EXAM IS TMR SO IM COOKED LMAO

THANK YOU SOOOOOOO MUCH

now i get it

[topaz silo]

Your welcome

Btw its '.close' to close this

[chilly monolith]

.close

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[sullen jewel]

Write down the total matrix for the following system of equations.

I'm not quite sure what to do here? I assume that I have to find x_1 for the last row?

[twin idol]

No x1 is multiplied by 0 if it's not multiplied by a constant

You find x1 first, then use it to find x2 or 3, then you'll find x3 from the last equation

[sullen jewel]

Ah alright, I'm not very experienced with matrixes. Would you recommend this is a good video to understand the question more? https://www.youtube.com/watch?v=yRwQ7A6jVLk&ab_channel=TheOrganicChemistryTutor

[twin idol]

Hmm

Sure

[sullen jewel]

I'll take a look and come back if I still don't get it, thanks ^^

.close

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[sullen jewel]

So this was the right answer. Does meaning writing down the matrix mean to just make a matrix with the coefficients?

[compact pewter]

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[sullen jewel]

So this was the right answer. Does meaning writing down the matrix mean to just make a matrix with the coefficients?

[kind viper]

maybe show the full question just to be sure?

[upper bane]

is total matrix = augmented matrix?

[twin idol]

Yeah. The matrix of coefficients augmented with the vector of coefficients of RHS of each equation.

https://cdn.discordapp.com/attachments/844681108473118750/1407349006291566633/1000057838007226480.png?ex=68a5c766&is=68a475e6&hm=f55ce6772387adcbe8d3c276debc5561ccaa3c5f3c7af996fc0b46371311b584&

In this format

The left side is the coefficients of the variables

The right side is the RHS of the equations

[boreal girder]

VulcanOne

[lavish gull]

i have to say, this is the first time i have heard of an augmented matrix being referred to as a total matrix

[sullen jewel]

I do have to say that this is translated with Google Translate, so very possible that is the reason it's not using proper language

So if I understand correctly, we can use these matrices to solve a set of equations?

[uneven bluff]

yep we can 👍🏻

[lavish gull]

yes

[uneven bluff]

just like you can use the gauss-jordan method (it’s basically the same method)

[lavish gull]

if you've heard of Gaussian elimination, this is the time to use it

[twin idol]

Gauss-Jordan for Reduce-Row-Echelon-Form

[upper bane]

just a quick check: gauss-jordan is gaussian elim, right?

[uneven bluff]

yes

[upper bane]

aye thanks

[uneven bluff]

it depends on the country you’re from how you call it

[upper bane]

just wanted to make sure

[sullen jewel]

Give the matrix in reduced echelon form and give the solution to the system of equations.

I'm doing something wrong with the first row at column 2 and 3, however I'm not sure what it is I did wrong

[lavish gull]

this part is incorrect

[upper bane]

i also have to make note that if you are asked to put the matrix in reduced echelon form, the final matrix is incorrect

(if by reduced echelon form you mean reduced row echelon form

[sullen jewel]

Yeah the final matrix is also incorrect, it says that number 3 and 6 is wrong

I'm just not too sure on what I did wrong

[upper bane]

ok then in that case

the current form of the matrix you have is called row echelon form

this form is mostly sufficient to solve for the system of equations

but you are asked to put it in reduced row echelon form

that means, above and below every 1 on the left side of the matrix, all numbers must be 0

now, the number below the 1 in row 1 is 0, correct

but the number above the 1 in row 2 is not 0

use row 2 to clear out the number above the second 1

[sullen jewel]

Meaning I have to find a way to make the -3 into a 0?

[upper bane]

yes. do so with row 2

[sullen jewel]

So just wondering, if I were to add row 2 to row 1, I would get [1 -2 | 5] in the first row

Do I just add it 3 times?

[upper bane]

yes!

well spotted

[sullen jewel]

Feels a bit illegal lol

Alright so now I got the right matrix

[upper bane]

nah it's not

yup

from here you can just quote the solution

[upper bane]

from your original solution, however, this part is wrong

the equation you should have formed from the row 1 in your original picture up top is x - 3y = 6

remember there's a -3 in row 1 col 2

that would give you x - 3(-1) = 6 => x + 3 = 6
x = 3

[sullen jewel]

I can't say that x+y = 3, y = -1?

[upper bane]

no. you can only say that if your top row is [1 1 3]

but your second column (which represents the coefficient of y) is -3

not 1

[sullen jewel]

Awesome I got it right

I think I understand now

[upper bane]

nice

[sullen jewel]

If I were to go back these numbers show the coefficient as you said of each coordinate in the function

Meaning that it actually just says x + 0 = 3 and 0 + y = -1, correct?

[upper bane]

to be more explicit

you can say x + 0y = 3 and 0x + y = -1

oops, my kb has a tendency to bounce the 1 key

sorry

[sullen jewel]

Ah alright

Do you by chance know if mathisfun has an article with all these kinds of methods? I tried looking but couldn't find much more than the basic introduction

[upper bane]

i'm sorry but i'm not familiar with that site

i use math libretexts

(and my textbooks)

[sullen jewel]

Ah I apologize, I thought it was well known

[upper bane]

i know that site

but i don't use it beyond basic stuff nowadays

[sullen jewel]

So with matrices, could we also find out how many solutions one has?

[upper bane]

of course

in particular, when completely row reduced (RREF), the last row tells you the number of solutions

[sullen jewel]

So in my last question, would it just be 1?

[upper bane]

yes, there is only one solution for this matrix

there are two common situations to look out for

both of them are of the form (last row)
[0 0 0 .... b]
where b is the rightmost column, and everything else is 0

if b is also 0, then there is at least one free variable (meaning a variable that can take on any value). such a system has infinitely many solutions

on the other hand, if b is not 0, the system is inconsistent (has no solutions)

[sullen jewel]

Yeah so b in my question would be -1?

[upper bane]

yes

but in your case b doesn't really matter because you have a 1 to the left

[sullen jewel]

I see I see, thanks a lot for the help, makes stuff a lot more clear now 😄

[upper bane]

i'm glad it's clearer for you now

any other questions?

[sullen jewel]

not for now, but who knows later again, thanks!

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[upper bane]

wot

[cerulean bramble]

OP deleted their own message

The bot needs to pin something as reference so that the channel be open; so when it gets deleted, it automatically closes the channel

This is the biggest

[upper bane]

i'm aware. i'm more "wot"ing the OP

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[little isle]

A lighthouse, facing north, sends out a fan-shaped beam of light extending from north-east to north-west. An observer on a steamer, sailing due west, first sees the light when he is 5km away from the lighthouse and continues to see it for 30√2 minutes. what is the speed of the streamer

[little isle]

is 5km the horizontal distance between the lighthouse and the streamer or the actual slanted dist

[kind viper]

it is the actual slanted distance for sure.

[little isle]

alright

lemme check once tho by solvin

ill let ya kno

im gettin it wrong

i need help

[half pond]

hint: what shape is the light region

[little isle]

its a right triangle

[upper bane]

might wanna sketch it out

as always

[half pond]

nice. now consider where and how the steamer crosses the light area

[half pond]

assuming that speed is constant, how do you compute it

[little isle]

using dist = speed*time and then pythagorean

but why did we assume that the figure is like this

[half pond]

question stated that the light region is fan shaped. ofc, the angle of the fan is assumed to be 90 deg bc it extends northeast and northwest

[little isle]

noo

like this

why can it be liike ts

this is also n-e to n-w

[half pond]

i agree

if the question doesnt require a specific angle or orientation, you can solve it leaving two variables

however

in navigation, northeast and northwest are unambiguous

they refer to the "diagonals"

[little isle]

so we will assume that it means this ..

[half pond]

assuming that the question follows navigation conventions and wants a specific number, then yes

[little isle]

alright

thanx a ton

.close

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[dusky quartz]

Why does $\sum_{k=1}^{n} \frac {k}{(k+1)(k+2)(k+3)} = \frac {n(n+1)}{4(n+2)(n+3)}$

[boreal girder]

laestia

[winged lion]

What answer do you expect?

[molten badger]

telescoping

[dusky quartz]

is there a way to reliably transform sums into resultant form or something

[winged lion]

Yeah, you can apply pfd and hope telescoping bring that result

[dusky quartz]

like above question

pfd as in partial differentiation?

[winged lion]

partial fraction decomposition

[dusky quartz]

oh

1/(k+1)-1/(k+2)=1/(k+1)(k+2)
1/(k+1)(k+2)-1/(k+3)
(k+3-(k+1)(k+2))/(......)
k+3-k^2-3k-2/...
-(k^2+2k-1)/...

something like this?

[winged lion]

$\f{k}{(k+1)(k+2)(k+3)} = \f{A}{k+1}+\f{B}{k+2}+\f{C}{k+3}$ where $A,B$ and $C$ are real constants to be found.

[dusky quartz]

oh

damn

ok thanks

.close

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[kind hollow]

hello! Could someone try to explain to me what the hell happened after dv/dt? Like, what? Also yeah I'm too lazy to wait 2 years to understand this

[kind hollow]

actually, I would like to also know what that squiggly thing is above a,v, and e

[rigid perch]

the tilde appears to be their marker for vectors

so they expand the velocity into its expression in spherical coordinates

[kind hollow]

so e is a vector right? in a 3D space?

[rigid perch]

which includes both the coordinates (R, theta, phi) and the unit basis vectors which point in the increasing direction of those coordinates and orthogonal to each other. unlike cartesian basis vectors these vectors point in a different direction at each point. so as the object moves the vectors themselves change

so they have to apply the product rule to each term. the first term is done in the picture. the time derivative of each unit basis vector is also expanded in terms of the other basis vectors (the appropriate expression for that can be proven geometrically)

[kind hollow]

Okay let's pause and have some questions here

first of all, what is theta and phi?

[rigid perch]

the angular coordinates in spherical coordinates

[kind hollow]

so like one angle is based on the x axis and the other angle on the z axis?

(I've never seen spherical coords)

[rigid perch]

yes

R is the distance from the origin (shown as rho in the figure), theta is the polar angle (counterclockwise distance from the x-axis in the xy-plane, same as in plane polar coordinates) and phi is the angle with the positive z-axis

[kind hollow]

that was lucky of me, cuz why isnt there an angle based on the y axis? is it because we dont need more info about a point on the sphere?

I just wanna clear out potential confusions in the future btw, I prolly wouldnt need to know the answer to that but what the heck

[long vigil]

for a single point, no

but for an object, you'd need 3 angles called yaw pitch and roll

iirc

[kind hollow]

hmmmm oke thank you

In our case, R is the length of our vector, right?

[rigid perch]

actually for this setup judging from the math they are using the convention where phi is measured from the xy-plane instead (so it's complementary to the angle in the picture)

[rigid perch]

it's the length of the position vector

[kind hollow]

yeah i can work with that too

[kind hollow]

but we are interested in the velocity vector are we not?

[rigid perch]

slightly complicated because it also involves cartesian and cylindrical

[kind hollow]

ohhh thats why you mentioned product rule

[rigid perch]

so the position vector is given by $R \tilde{e}_R$ and the velocity vector is found by applying product rule to that

[boreal girder]

cloud

[kind hollow]

wait how does e_θ and e_φ work? like how do we determine their directions? ive never seen a vector based on angle

[rigid perch]

they are based on curvilinear coordinates. the idea is that for all 3 basis vectors they point in the direction of increasing that coordinate while keeping the other coordinates constant

which means they point in a different direction at each location

using a bit of trigonometry we can convert between the cartesian basis vectors i,j,k and these ones

[kind hollow]

I'm sorry, but I don't understand the intuition behind it

[kind hollow]

is it easy to show the process?

[compact pewter]

@kind hollow Has your question been resolved?

[rigid perch]

https://www.desmos.com/3d/ssjnddcuah

the three surfaces represent R = constant, theta = constant, phi = constant for the current coordinates (feel free to change the coordinates to see them move around

the three vectors are color coded to correspond to their surface (so red is for both the surface R = const and the vector e_R). these vectors are perpendicular to the surface corresponding to their own coordinate and tangent to the other two surfaces

[kind hollow]

interesting

it is very very trippy

i think my main question is, why do we represent R, theta, and phi as surfaces?

actually i think i get it

its like all the possible places for a coord to be true

and the three surfaces will always intersect at one point

[rigid perch]

yes. for cartesian coordinates these surfaces will be planes parallel to the coordinate plane

[kind hollow]

I think I get the gist of how the vectors work

why is it better to represent a 3D vector like that, instead of using our i,j,k vectors?

[rigid perch]

it usually isn't

but sometimes you have an object which moves in such a way that the representation of its motion is nicer in spherical coordinates

for example if you had an object moving around on the surface of a sphere, its R coordinate would be constant

[kind hollow]

thats true

is there a clear reason why we do it in this instance?

[rigid perch]

well here they are just deriving the appropriate equations of motion in spherical coordinates

which is useful if you have a problem involving that later

[rigid perch]

another scenario is if you had a satellite orbiting the earth, if you set the origin as the center of the earth then the gravitational force always points in the -e_R direction

which would mean F = ma would be cleaner in spherical coordinates, since two components would be 0

[kind hollow]

hmm I see

so we have our basis vectors after the change of coordinates. These vectors represent position

how do we derive the velocity vector's components?

[rigid perch]

basis vectors are the "building blocks" of vectors, meaning every vector is a linear combination of them

by linear combination, i mean some combination of multiplying them by scalars and adding them up

so in cartesian coordinates you could have some position vector
[ \tilde r = x \tilde e_x + y \tilde e_y + z \tilde e_z ]
and some velocity vector
[ \tilde v = v_x \tilde e_x + v_y \tilde e_y + v_z \tilde e_z ]
and so on

[boreal girder]

cloud

[rigid perch]

and in spherical we have the same thing:
[ \tilde r = r_R \tilde e_R + r_\theta \tilde e_\theta + r_\phi \tilde e_\phi, \qquad \tilde v = v_R \tilde e_R + v_\theta \tilde e_\theta + v_\phi \tilde e_\phi ]

[boreal girder]

cloud

[kind hollow]

the r's and v's are scalars right?

[rigid perch]

yes

[rigid perch]

so to find the velocity vector in cartesian all we have to do is take the derivative wrt time. since all of the components are constant it's relatively easy:
[ \tilde v = \odv{\tilde r}t = \odv{}t \big(x \tilde e_x + y \tilde e_y + z \tilde e_z \big) = \dot x \tilde e_x + \dot y \tilde e_y + \dot z \tilde e_z ]

[boreal girder]

cloud

[rigid perch]

so we discover that v_x = x dot and so on in cartesian, relatively straightforward

in spherical the position vector has magnitude $R$ and points in the $e_R$ direction. so it's components are
[ \tilde r = R \tilde e_R ]
and the other two are zero

[boreal girder]

cloud

[kind hollow]

why are the other two zero?

you prolly explained it earlier but I didnt catch on

[rigid perch]

the position vector points from the origin to the point. e_R points in that same direction (directly away from the origin) so they are parallel

if a vector is parallel to a basis vector that means its components are only in that direction

[kind hollow]

hmm I see

so our position vector is just Re_R

I have understood thus far

how do we even represent the velocity now?

do we use the tangent thing you mentioned?

[rigid perch]

to find what the velocity should be we apply the product rule

if the position varies with time then the basis vectors do too (since they vary with position), so we have to take their derivatives

,, \odv{}t \tilde{e}R = \dot{\tilde e}R = \dot \theta \cos \phi \tilde e\theta + \dot \phi \tilde e\phi

[boreal girder]

cloud

[rigid perch]

which can be found geometrically although it's a bit involved

so then applying the product rule we have
[ \tilde v = \odv{}t R \tilde e_R = \dot R \tilde e_R + R \dot{\tilde e}R = \dot R \tilde e_R + R\dot \theta \cos \phi \tilde e\theta + R \dot \phi \tilde e_\phi ]

[boreal girder]

cloud

[rigid perch]

which is the expression in the image

[kind hollow]

Well

that is cool

what area of math do I need to study to understand what R' and (e_R)' are?

I mean, to be able to produce their values like you did

[rigid perch]

the dot just means time derivative

[kind hollow]

yup, our boy newton used that

[rigid perch]

a bit of vector calculus is all you really need to understand it, although typical vector calc courses tend not to cover the basis vectors

so it tends to be taught in subsequent physics or engineering classes

[kind hollow]

I see, thank you for the help

I didnt believe i would actually be able to comprehend anything

.solved

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[frosty reef]

i think I is wrong

ii is also wrong

i think iii is true? since it cant have any positive real solutions if c <= 1 anyway

wait no

i think its also false

it's the opposite that's true

when c >= 1?

[chilly basalt]

Can you show the reasoning?

[frosty reef]

for what

that iii is true or false

[chilly basalt]

Provide a diagram or something

[chilly basalt]

All three of them, I can’t tell if you’re making guesses or not

[frosty reef]

im on desmos rn

because i cant be asked to do a diagram

but yeah as you can see

c is in fact less than 1

and does not have any real solutions

[chilly basalt]

Sure, can you tell me what does c represent in a line?

[frosty reef]

plus if c was bigger than 1

the y intercept

[chilly basalt]

They never say the slope has to be positive

[frosty reef]

i meant for III

[frosty reef]

sure but either way

if theres solutions for c >= 1

then it's not iff

[chilly basalt]

c >=1 two distinct real solutions

@frosty reef

You mistook the question.

[quiet sentinel]

Pretty sure iii is true because if c is greater than 1 it can only have either 1 solution or a negative solution, if c is less than 1 it can still have a high slope cutting through the curve and have 2 positive solutions

[frosty reef]

oh wait

[frosty reef]

oh

[chilly basalt]

Oh wait, asking for positive solutions

Well, Oakley’s explanation is correct

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@frosty reef Has your question been resolved?

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[kind wind]

guys

[kind wind]

in shm

why is that

acceleration=-k displacement

acceleration is directed towards the mean position and directly prop to displacement

been just reading this chap

can yall explain what shm is actually

[paper anchor]

SHM is IG periodic and oscillatory motion

[kind wind]

yea

[paper anchor]

which due to these properties can be represented through a sine or a cosine wave

[kind wind]

ahmm

im not sure abt that

might be because there are graphs later on

[kind viper]

not a sufficient desc.

[kind viper]

yeah, well imagine if you had $a = +kx$ instead.

[boreal girder]

Ann

[paper anchor]

Just look at a pendulum, wrt time it's just a sine wave

[kind viper]

your object would rapidly fuck off to either +∞ or -∞

rather than staying around the origin and swinging back and forth

[paper anchor]

It is a resistive force after all

[kind viper]

wrt to
with respect to to

[paper anchor]

Argh mb

[kind wind]

ahmm well i just started reading the introduction couldnt get through why the negative sign in the equation and why the acceleration directed towards the mean position

im not sure what are u talking

[kind viper]

ok so lets imagine a spring

like a spring with a weight on it

[kind wind]

k k

[kind viper]

horizontal let's say

[kind wind]

k

[kind viper]

and you pull the weight out from its resting position

[kind wind]

ok

[kind viper]

and let's say that outward displacement is positive i.e. we run our axis that way

now think

will the spring try to push the weight further out (positive)

or will it try to pull the weight back in (negative)

[kind wind]

back cuz u removed the weights

[kind viper]

i didn't remove it.

[kind wind]

u didnt ??"

[kind viper]

i simply pulled it, while keeping it connected to the spring.

[kind wind]

if u didnt

[kind viper]

but i didn't detach it from the spring.

[kind wind]

ohhhh

[kind viper]

it's still attached to the spring; the spring is just stretched out

[kind wind]

yea will still go up

[kind viper]

you think the spring will try to stretch out even more?

[kind wind]

no

[kind viper]

or will it rather resist your effort to stretch it

[kind wind]

yea

[kind viper]

and instead try to contract and thereby pull the weight back in

[kind wind]

yea

both

[kind viper]

yeah so does that explain why the diffeq for SHM has a minus sign

[kind wind]

hmm

so it has to do with restoring force

[kind viper]

the key thing about SHM is that the object is always acted on by a force which tries to bring it back to the origin.

[paper anchor]

Yeah I mean whether it be mg, spring or any such force in SHM, it's always resistive to the body's motion

[kind wind]

oh oh

[paper anchor]

Yeah exactly

[kind wind]

k k ill be here with another confusion if i get

[paper anchor]

Like if I dig a hole through Earth and throw you in it, you'd be pulled to the centre, your momentum would force you to go against it, but mg is always towards the centre, which is against your momentum, which is (+)ve, hence that force towards the centre would be (-)ve

[kind wind]

bro

momentum is

basically

mass in motion

why would it even

oppose things

[lime juniper]

because motion can?

[paper anchor]

you're correct, it's not momentum that's resisting, it is being resisted against because of mg

[lime juniper]

go in the opposite direction i mean

[paper anchor]

Yeah that too, there's retarding motion

[lime juniper]

better answer though

[kind wind]

hey

but why directly state that

a prop to -x

[glass lion]

👁️

[kind wind]

acceleration increases as displacement increases

[paper anchor]

because we're talking about the force working against your motion

[kind wind]

understandable

but -ve ahmmm

[kind viper]

bro
momentum is
basically
mass in motion
why would it even
oppose things

can you
please
not
put
so many
message breaks

[kind wind]

k k

[upper bane]

negative to oppose your current direction of motion

[paper anchor]

on contrary, the further you pull a spring, the slower you'll be able to pull, it's a resistive force that increases with displacement, which works against your motion, which is positive, hence we term it as (-)ve

[kind viper]

a lot of the time when you want to signal some kind of pause a well-placed dot-dot-dot ... makes that look and sound a lot better

[upper bane]

as mentioned, if it was positive, as x grows, you fly faster and faster off

[molten badger]

acceleration is second derivative of displacement, you can prove it yourself using math if you don't understand what they said

-# oh I'm green now lol

[upper bane]

grats

[kind wind]

works against my force no???? but shouldnt our direction of motion or force

be positive

[paper anchor]

If you want you can take your motion as negative and the restoring force as positive, it doesn't really matter (for the sake of the argument), they just have to be opposite

[kind wind]

restoring force should be neg and my motion should be positive say

[paper anchor]

Alright

oh wait it thinks I spoke a slur, I meant works against it if that works

[kind wind]

huh

wdym

[paper anchor]

the restoring force works against your motion, hence it's opposite

[kind wind]

ok can i present my views

[paper anchor]

sure

[kind wind]

a prop to x

as x increases

a increses

[paper anchor]

yes

[kind wind]

say that if the displacement occurs in the direction of me pulling the thing

[paper anchor]

yes

[kind wind]

then acceleration produced is to be towards the mean position of the obj's starting position

which imply -a prop x

this thing is just the same no ?

[upper bane]

what and what is the same

[kind wind]

what

[paper anchor]

Yes technically

[kind wind]

-a prop x and a prop to x

[upper bane]

different directions

[lime juniper]

\propto doesnt imply a sign, denotatively

connotatively sometimes it does

[kind wind]

so there is a flaw i guess but what is

[paper anchor]

because see, if a increases with x, then the object would move faster as it's pulled away from the spring, it would continue to move till infinity then

[paper anchor]

There's not, you're fine

[lime juniper]

the uberprojectile

[paper anchor]

Wait what

[upper bane]

that's what i'm confused about

-a being proportional to x is not the same as a being proportional to x, because they're in different directions

afaik

[paper anchor]

Yeah exactly

[upper bane]

the former means that the acceleration works to keep the object oscillating by always pulling it back towards the center

[kind wind]

ok guys once again elaborate some more

i just couldnt wrap my head in

a prop to -x

[upper bane]

the black bar is the starting/mean position.
the top situation is -a proportional to x.
the bottom situation is a proportional to x

[kind wind]

um

[paper anchor]

Finally a figure

[kind wind]

yea

[paper anchor]

Alright mate see

[upper bane]

if you say a is proportional to x, that means that the more you pull something in one direction, the faster it's gonna fly in that direction

[kind wind]

k k

[kind wind]

fr

[upper bane]

so it's gonna just fuck off out of the universe

[kind wind]

makes no sense

fr

[paper anchor]

If I dig a hole through Earth's centre, and throw you in it, then would you start falling faster once you cross the centre, or do you start slowing down

?

[kind wind]

after i cross the centre