help-0

[fast mantle]

we learn that stuff in year 4-

[alpine sable]

And they are making me take ap geometry 2

I never took the first one

[fast mantle]

that doesnt look like ap geometry 👀

but okay

[alpine sable]

Idk

My teacher said it’s a review

From ap geometry 1

[fast mantle]

the angles formed by a transversal between two parallel lines are interior angles

[alpine sable]

But I never took the first one

[fast mantle]

and if they're on opposite sides of the transversal, they're alt interior angles

get it?

in the figure, 3 and 8, 4 and 5 are alt interior angles

[alpine sable]

Ooh I see

[fast mantle]

mhm

[alpine sable]

Is it because they are opposed of each other and inside

[fast mantle]

yep

alternate exterior angles are 1 7 and 6 2

[alpine sable]

This right

[fast mantle]

yep

[alpine sable]

This right

[alpine sable]

^^^^^

Please help need to finish this I have 10 min

Left

[fast mantle]

option 2

[alpine sable]

Is it because they are on the same line

[fast mantle]

it's bc they are formed in the matching corners ( corresponding corners ) with the transversal.

[alpine sable]

@fast mantle this right

[fast mantle]

this

[alpine sable]

I have 4 questions in 2 mi

N

[fast mantle]

alt interior angles are equal
alt exterior angles are equal
corresponding angles are equal
consecutive angles form a linear pair

try doing it yourself

[alpine sable]

For some reason the assignment didn’t end

Omg it says 10:40 pm

I’m dumb

[fast mantle]

122 instead of 116

[alpine sable]

Do all of them

[fast mantle]

just change the 116s into 122

rest are correct

[alpine sable]

No

The right one

I mean left

[fast mantle]

..?

[alpine sable]

This right

[fast mantle]

yep

[alpine sable]

This right

[alpine flare]

Ys

[alpine sable]

This right

[fast mantle]

yee

[stable grotto]

how many different 6 digit numbers can you compose using 1 1 1 2 2 3 ?

[alpine sable]

This right

[fast mantle]

first is parallel, second isnt

[alpine sable]

This right

@fast mantle this right ???

[fast mantle]

yep it is

[warped phoenix]

hey all i need to use row ops. to "eliminate z in the second equation" for this matrix

[ocean seal]

TheMane3

[warped phoenix]

do I just need to make the z in the second row into a 0?

[merry isle]

me voy a comer a sus madres uwu

[alpine sable]

make as short as possible

i tried factorizing

but it gets 5a() on the top and i cant factorize the bottom

[gray isle]

can you show what you currently have

[alpine sable]

It's in my notebook can't really screenshot that

[gentle prism]

Need help for 18

[gray isle]

can you type it out

(with appropriate use of parentheses)

[devout sigil]

factorise b in the denominator

[devout sigil]

ur numerator seems correct so far

[alpine sable]

(5a(2a^2-5b))/(5b^2-2a^2-b)

i dont know the command to print

[gray isle]

the denominator can be factorised further

[devout sigil]

denominator factorised is
b(5b-2a²)

[gray isle]

its fine as long as you use appropriate notation

[alpine sable]

hmm i noticed

[gray isle]

actually you inserted an extra - sign in there that you shouldn't have

[devout sigil]

and notice 5b-2a²=-(2a²-5b)

[alpine sable]

no

thats a

[devout sigil]

wdym

[alpine sable]

-(2a²-5b) = -2a²+5b

[devout sigil]

-(2a²-5b)=-2a²+5b=5b-2a²

[alpine sable]

yeah

¨tjat was what i meant

but hmm

im not sure how to solve

[gray isle]

after doing some factorisation you should've at least reached something like
$$\frac{5a(2a^2-5b)}{b(5b- 2a^2)}$$

[ocean seal]

ℝamonov

[alpine sable]

i got there

all by myself

i swear

[gray isle]

and then apply -(2a²-5b) = -2a²+5b

[alpine sable]

wait what

[devout sigil]

just factorise -1

[gray isle]

$=\frac{2a}{b} \cdot \underbrace{\frac{(2a^2-5b)}{(5b- 2a^2)}}_{?}$

[alpine sable]

but why would i do that

[ocean seal]

ℝamonov

[devout sigil]

u need a common factor in both the numerator and denominator to simplify the fraction

or else u cannot further simplify the expression

[gray isle]

your goal is to simplify and applying the above property allows you to do so

[alpine sable]

i am so confused

[gray isle]

do you agree that
5b - 2a^2 = -2a²+5b = -(2a^2 - 5b)

[alpine sable]

i am here with
$$\frac{5a(2a^2-5b)}{b(5b- 2a^2)}$$

[ocean seal]

Netocian

[devout sigil]

do u know that -(x-y)=y-x

[alpine sable]

=-x+y=y-x

[devout sigil]

yes

[alpine sable]

indeed

[gray isle]

do you agree that
5b - 2a^2 = -2a^2+5b = -(2a^2 - 5b)

[devout sigil]

now we replace x and y with 2a² and 5b respectively

essentially we r still applying the same concept

[alpine sable]

yes but why would you do that

[gray isle]

your goal is to simplify and applying the above property allows you to do so

[devout sigil]

because we would have a common factor in the numerator and denominator so we can simplify them by “cancelling”them out

[alpine sable]

yes but the both sides will not have the same at the front of ()

[gray isle]

taking a trivial case like
$$\frac{5}{-5}$$
that would simplify to just -1 right?

[devout sigil]

one of them would become negative

[alpine sable]

0

[ocean seal]

ℝamonov

[alpine sable]

yes

[alpine sable]

hmm

[gray isle]

the same idea applies here

[alpine sable]

wait

so

give me a second

[devout sigil]

to simplify, we rather get a negative expression but is simplified in the terms that we have totally reduced the original fractions

[alpine sable]

you want
$$\frac{5a*-(2a^2-5b)}{b(5b-2a)}$$

[ocean seal]

Netocian

[alpine sable]

?

[gray isle]

no

[alpine sable]

you want
$$\frac{-5a(2a^2-5b)}{b(5b-2a)}$$

[ocean seal]

Netocian

[gray isle]

you just inserted a minus sign in the numerator and also dropped the square on the a in the denominator

[alpine sable]

im so dead

i will not make

this

[gray isle]

5b - 2a^2 = -2a^2+5b = -(2a^2 - 5b)
simply replace
5b-2a^2 in the denominator with (-(2a^2 - 5b))

[alpine sable]

im not native

[gray isle]

i mean this is basic substitution, try not to overthink it that much

[alpine sable]

$$\frac{5b - 2a^2 = -2a^2+5b = -(2a^2 - 5b)}$$

[ocean seal]

Netocian
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[gray isle]

replace expressions with expressions of equivalent value, (inserting parentheses if needed)
don't add or remove things for no reason

[alpine sable]

ohh wait

you want
$$\frac{5a(-(2a^2-5b))}{b(5b-2a)}$$

[ocean seal]

Netocian

[gray isle]

no

[alpine sable]

oh

[gray isle]

you're making the same mistake

[alpine sable]

im so

dumb

[gray isle]

look carefully at what you have

[alpine sable]

why would you have 5b - 2a^2
it's 2a^2-5b
you should do
-5b+2a^2

right?

[gray isle]

you're starting with:
$$\frac{5a(2a^2-5b)}{b(5b- 2a^2)}$$
and considering stuff like
$$5b - 2a^2 = -(2a^2-5b)$$
$$2a^2-5b = -(5b - 2a^2) $$
you can choose to do stuff like replace $5b - 2a^2$ with $(-(2a^2-5b))$ \
or replace $2a^2-5b$ with $(-(5b - 2a^2))$

[ocean seal]

ℝamonov

[gray isle]

it doesn't matter which expression you're replacing/substituting as long as what you're replacing it with has the same value

alternatively if you know
$$2a^2-5b = -(5b - 2a^2)$$
dividing both sides by $(5b-2a^2)$ would also lead you to
$$\frac{2a^2-5b}{5b-2a^2} = - 1$$

[ocean seal]

ℝamonov

[alpine sable]

$$\frac{5a(-(2a^2-5b))}{b(-(5b-2a^2))}$$

[ocean seal]

Netocian

[gray isle]

only do one substitution.

although mathematically valid, substituting both is counterproductive here

[alpine sable]

alright you want me to break it up or what?

[gray isle]

break what up

[alpine sable]

hmm

fine

ill do one substituin at the at the time

[gray isle]

you can choose to do stuff like replace $5b - 2a^2$ with $(-(2a^2-5b))$
or replace $2a^2-5b$ with $(-(5b - 2a^2))$
\
do only do one of these, don't do both

[alpine sable]

but

[ocean seal]

ℝamonov

[alpine sable]

so i choose one side to do that to?

$$\frac{5a(-(2a^2-5b))}{b(5b-2a^2)}$$

[ocean seal]

Netocian

[gray isle]

no

[alpine sable]

im so done in math right?

[gray isle]

you're overthinking this

[alpine sable]

i shouldnt consider starting cs

in future

[gray isle]

or you're just not typing, your math properly

because it really looks like you're just inserting a minus sign into the numerator

[alpine sable]

im not native in english

that's hard words

for me

i'm from sweden btw

[gray isle]

$k \neq -k$

[ocean seal]

ℝamonov

[alpine sable]

yeah

-x != x

indeed

[gray isle]

yeh but you're breaking that rule with what you're doing

$2a^2-5b \neq -(2a^2-5b)$

[ocean seal]

ℝamonov

[gray isle]

$2a^2-5b = (-(5b - 2a^2))$

[ocean seal]

ℝamonov

[alpine sable]

i realized that while writing that but i thought you gave me that one anyways

[gray isle]

wdym

[alpine sable]

$$\frac{5a(-(5b - 2a^2))}{b(5b-2a^2)}$$

[ocean seal]

Netocian

[gray isle]

that's better

[alpine sable]

ooo

i seee something

5b-2a^2 5b - 2a^2
we can do some magic with those

right

[gray isle]

yeh

[alpine sable]

@gray isle whats next

[gray isle]

$\frac{5}{5} =, ?$

[ocean seal]

ℝamonov

[alpine sable]

1

but

[gray isle]

$\frac{\smiley}{\smiley} =, ?$

[ocean seal]

ℝamonov

[alpine sable]

alrigth

alright

but

how

would i implement

in this

[gray isle]

commutative/associative properties of multiplication

[alpine sable]

should i do it

uhm

separetly?

[gray isle]

you can pretty much multiply in any order you want and you can choose the one most convenient to you

[alpine sable]

$$\frac{-(5b - 2a^2))}{-(-5b+2a^2)}$$

[ocean seal]

Netocian

[gray isle]

no

where did the rest of your stuff go

and it feels like you just went backwards

[alpine sable]

this?
$$\frac{5a(-(5b - 2a^2))}{b(5b-2a^2)}$$

[ocean seal]

Netocian

[gray isle]

in this case you can do some manipulation to get:
$$=-\frac{5a}{b} \cdot \frac{5b-2a^2}{5b-2a^2}$$

[alpine sable]

i dont know how to proceed from there that's it

[ocean seal]

ℝamonov

[gray isle]

and/or identify that 5b-2a^2 is a common factor in both the numerator and denominator and you can cancel them, assuming that the expression isn't 0

[alpine sable]

I SEE WHAT YOU DID THERE

awesome

that was what i needed

let me

continure from

there

a second

[alpine sable]

where did the minus come from btw

in the front

[gray isle]

$k \times -1 = -k$

[ocean seal]

ℝamonov

[gray isle]

$-\frac mn = \frac{-m}{n} = \frac{m}{-n}$

[ocean seal]

ℝamonov

[alpine sable]

that was correct

but

but i have a question

or ill wait with it

and try it first

before asking

[glass elm]

Hi, really sorry to disturb! (please answer later also no problem, i don't mean to interrupt)
x^2 = y and y^2 = x. These two curves, At which point will the curves orthogonally cut each other?

[alpine sable]

$$\frac{5a}{b} /cdot \frac{-(5b - 2a^2)}{-(-5b+2a^2)}$$
$$\frac{5a}{b} /cdot \frac{5b - 2a^2}{-5b+2a^2}$$
$$\frac{5a}{b} /cdot \frac{5 - 2}{-5+2}$$
$$\frac{5a}{b} /cdot \frac{3}{-3}$$
$$\frac{5a}{b} /cdot -1$$
$$\frac{-5a}{b}$$

[ocean seal]

Netocian

[alpine sable]

@gray isle

[gray isle]

no

\cdot not /cdot

also 5b - 2a^2 isn't the same as 5-2

nor is -5b + 2a^2 the same as -5 + 2

and you overcomplicated it way too much

and also worked backwards for some reason

[alpine sable]

no i removed the b and a^2

from the both

substitues

and it did work

@gray isle

[gray isle]

wdym by removed

[alpine sable]

you can remove b and a^2 from the both sides if they appear there and have * infront of them

basic laws

[gray isle]

you can't just remove those b and a^2 like that

not like that

[alpine sable]

yes

like that

in sweden at least

[gray isle]

not anywhere

[alpine sable]

in sweden

we're built different

lol

[gray isle]

its generally something that you shouldn't be doing

[alpine sable]

dear ramonov

if a code works dont change it

i apply it here aswell

[gray isle]

really?

[alpine sable]

yeas

[gray isle]

if i'm telling you that its unreliable...

[alpine sable]

it works

[gray isle]

then you probably shouldn't do it

[alpine sable]

my teacher taught me it

[gray isle]

ok how would you simplify
$$\frac{a+b}{a+c}$$ then

[ocean seal]

ℝamonov

[alpine sable]

you cant do it here

only if

$$\frac{a+b}{2a+c}$$ then

[ocean seal]

Netocian

[gray isle]

how would you simplify that then...

[alpine sable]

oh wait bad example

$$\frac{2a+b}{2a+c}$$ then

[ocean seal]

Netocian

[alpine sable]

1*b/c

[gray isle]

no

[alpine sable]

yes

[gray isle]

no

pls

[alpine sable]

yes

[devout sigil]

no

[gray isle]

are you serious about getting help or not

[alpine sable]

it works right?

[gray isle]

no

in general it doesn't

that's the point i'm making

[alpine sable]

give me a second

[silver viper]

then let a = 1, b = 2, c = 3
then you are saying that
$$\frac{2\cdot1 + 2}{2\cdot1 + 3} = \frac{2}{3}$$

[ocean seal]

kiid

[alpine sable]

Lmao

[gray isle]

simplification of fractions is all about common factors

not terms that just happen to appear in both the denominator and numerator

like what do you think happens? that they erase each other from existence?

$\frac{1+1+1}{1+1} \wthonk \frac{1}{}$

[ocean seal]

ℝamonov

[silver viper]

wait lol whats \wthonk

[glass lichen]

Yeah whats wrong with that? \s

[gray isle]

$\frac{ab}{ac} = \underbrace{\blue{\frac aa}}_{1} \cdot \frac bc = \frac bc$

[ocean seal]

ℝamonov

[crisp grove]

that's actually dope

[gray isle]

created something worse than divisibility by 0

[crisp grove]

[blazing rose]

how do i do this

this is the answer i got but its wrong somehow

[silver viper]

integration by parts?

[blazing rose]

yeah

i did that

had to do it twice

heres my work

[ancient saddle]

Hello, just leave du in terms of dx, it's incorrect to isolate the dx. You should end up with du = 2 dx

[crisp grove]

u know the cross method of doing by parts multiple times?

@blazing rose

[blazing rose]

no whats that

[blazing rose]

but isnt it mathematically correct if i do it?

[crisp grove]

basically u make 2 column, u in one v in one

[silver viper]

3rd step why did u multiply the integral by 1/2

[ancient saddle]

Yes, but you have to replace du, not dx

[crisp grove]

and keep taking derivative of u column and integration of v column

[blazing rose]

because 1/2du = dx

[ancient saddle]

Just like you did before that

[blazing rose]

oh i understand

[crisp grove]

also I get the answer $(x^2+7)\sin{x}+(2x+1)\cos{x}-2\sin{x}$

[blazing rose]

oh thats a neat trick

[ocean seal]

Ryuzaki

[glass lichen]

Tabular is just a way to organize ibp nothing new

[gray isle]

derivative of 2x+7 isn't 1/2

[blazing rose]

its 2, but i divided both sides by 2

[crisp grove]

@glass lichen yeah ik just makes it easier

[tidal lion]

hi

[blazing rose]

show work?

[ancient saddle]

I think the method Ryuzaki mentioned can prevent you from making these little mistakes

[blazing rose]

where did the 1 come from in (2x+1)

alright ill redo it

[crisp grove]

should be 7

typo

[blazing rose]

oh

[gray isle]

its 2, but i divided both sides by 2
wdym

[alpine sable]

@gray isle sorry for dissapering

im back

and yes

that example was bad

aswell

[blazing rose]

u = 2x+7, du = 2dx, du/2 = dx

[gray isle]

in the ibp form you're using that right part would be
$$-\int v \dd{u}$$
right?

[silver viper]

its -int v du not -int v dx

[ocean seal]

ℝamonov

[blazing rose]

whats ibp

oh

LMAO

yeah the right is that

which is why i put the 2nd ibp in the bracket

so i dont get confused with the signs

[gray isle]

and note that du = 2dx

its -int v du not -int v dx

[blazing rose]

ohhhhh

alright thanks hopefully i get the right answer rn

[hearty orbit]

I tried writing the binary representation here, but im not sure how to prove by MI, do we have to use strong MI here?

[tidal lion]

i need help whats 2 + 5 x 4 - 2 + 4 x 25

[twin pine]

prioritize multiplication

[tidal lion]

what

[golden nymph]

How old are you

You can’t use discord if less than 13 y’know

[tidal lion]

nah i was just kidding

im 16

[golden nymph]

,calc 2 + 5 * 4 - 2 + 4 * 25

[ocean seal]

Result:

120

[hearty orbit]

uh i actually asked a qn but u interrupted me

[golden nymph]

Kid in a non-occupied channel smh

[hearty orbit]

but i will post it in another channel

[golden nymph]

No

Hey

[tidal lion]

[golden nymph]

Illegal

[hearty orbit]

for me to post in another channel?

[golden nymph]

The channel is free now

For you

I mean

So no need

[hearty orbit]

ok ill ask it again

cuz ppl may not see it

I tried writing the binary representation here, but im not sure how to prove by MI, do we have to use strong MI here?
#help-0 message

[golden nymph]

I think I heard “strong” induction before but don’t remember what it meant

[hearty orbit]

it's like instead of just assuming P(k), then showing P(k+1) is true, we assume P(0) and P(1) and ... P(k+1), then we show P(k+2) is true

[golden nymph]

Oh

All my guess is that you can try adding 1 to a general number and show it results in another number (in binary) or sth

[red kestrel]

Can anyone explain this please I don’t understand

[golden nymph]

And that’s why I don’t like #help-0

Those who don’t read the rules seem to like it

Try a non-occupied channel

[alpine sable]

@red kestrel what steps do you not understand?

[red kestrel]

The last 2 steps

[alpine sable]

so from $$\sum^{100}{m=1}m²-\sum^{99}{m=0}m²$$ to $$\sum^{99}{m=1}m²+100²-0²-\sum^{99}{m=1}m²$$ i'm guessing

[ocean seal]

Al3dium

[red kestrel]

Yes

[alpine sable]

since it seems you got helped in another channel with the previous

okay so

[red kestrel]

I’m an idiot

[alpine sable]

?

did you get it now somehow?

[red kestrel]

Still not

[alpine sable]

oh okay

so

so just to be clear on which parts goes with which: $${\color{green}{\sum^{100}{m=1}m²}}-{\color{blue}{\sum^{99}{m=0}m²}}$$ blue corresponds to the blue and green to green. $${\color{green}{\sum^{99}{m=1}m²+100²}}{\color{blue}{-0²-\sum^{99}{m=1}m²}}$$

[ocean seal]

Al3dium

[alpine sable]

so let's start with the green.

[red kestrel]

Oh basically it splits

[alpine sable]

would you be able to expand out the terms each one of $\sum_{k=1}^4 k²$?

[alpine sable]

yeah, the green part above is equal to the green below and so on with the blue.

[ocean seal]

Al3dium

[red kestrel]

And why m=n to make it identical ?

[alpine sable]

that's the previous step? thought we were clear on that one

but yeah to have identical index

[alpine sable]

yes? to have identical indexes, but this is the previous step

thought we said you only needed to know about the last one

[red kestrel]

m=n+1 makes sense

[alpine sable]

i have to be afk for a few minutes, meanwhile think about this

[red kestrel]

But you can’t just say m=n in the second bit

[alpine sable]

yeah

at the end of the day, it's just a variable

m doesn't work differently from n

[red kestrel]

m^2 is not equal to n^2

[alpine sable]

okay i'm back

how is it not? would you say the solutions of the eqn $x²-1=0$ are different from those of $y²-1=0$?

[ocean seal]

Al3dium

[alpine sable]

@red kestrel

[red kestrel]

Oh yeah I understand now

[alpine sable]

you understand how m=n is a completely valid step?

[red kestrel]

It is just different symbol

[alpine sable]

yeah.

okay good

now back to the very last step

[alpine sable]

think about this ^, what do you think?

[blazing rose]

how do i split the integral into 2?

[harsh girder]

∫(f+g) = ∫f+∫g

[blazing rose]

so i need to multiply out everything in the integral and then split it up?

[harsh girder]

yes

[blazing rose]

alright

and do i keep the 14

on both integrals

yep i do

thanks

[vapid herald]

could I get some help

[alpine sable]

okay

what have you tried so far?

[vapid herald]

im gonna be honest im not really sure where to start I know the dashed line on the other side of the rectangle is 2 and I know the bottom dash line is also 2

then 1 on the opposing side of course

[alpine sable]

have you then tried to express $\vec{v}$ coordinates?

[ocean seal]

Al3dium

[vapid herald]

it would be (2,1,2)

[alpine sable]

usually it depends on how you express the axis, i'm guessing we'll go with the x-axis being at the right part and the left part being y-axis

but yeah that's correct

and now any ideas on how we can try to do $\vec{w}$ coordinates? considering we know $w=4$ and the $45^\circ$ located between the y axis and the vector?

[ocean seal]

Al3dium

[vapid herald]

well would we have to find (4)sin(45)

or just 1^2 + b^2 = 4^2

[alpine sable]

that's for the x-component, yes

and what about the y-component?

[vapid herald]

so just sqrt(15) for the x

y would be just 1

[alpine sable]

no

be careful, the 1 there is showing the y component of the v vector

not the w vector

[vapid herald]

so cos(45) * 4

4

[alpine sable]

and how did you get sqrt(15) for the x-component? you said correctly it was sin(45)*4 which is 2sqrt(2), which can be also written as sqrt(8) if you insist

[vapid herald]

couldnt you use Pythagorean therom?

because its a right angle

[alpine sable]

great job

[alpine sable]

i mean sure but remember that you assumed wrongly that the y-component was 1

[vapid herald]

oh your right

[alpine sable]

so as a recap

x-component: 4sin(45)=2sqrt(2)
y-component: 4cos(45)=2sqrt(2)

and now, don't overthink this, what about the z component?

can you notice the z component from the drawing?

[vapid herald]

0 right?

[alpine sable]

yes.

so $\vec{v}=(2,1,2)$ and $\vec{w}=(2\sqrt{2},2\sqrt{2},0)$

[ocean seal]

Al3dium

[alpine sable]

okay now that we have our vectors

[vapid herald]

theres a formula for angles between 2 vectors

[alpine sable]

,tex \anglevectors

yes

[ocean seal]

Al3dium

[alpine sable]

okay i have this written for 2 dimensions

but it can be applicable to 3 dimensions

including the z component

you can give this a read if you want.

[vapid herald]

yeah I know how to find magnitude

[alpine sable]

great

and do you know how to find the dot product between v and w?

[vapid herald]

I think so let me try it and see what I get

[alpine sable]

sure

[vapid herald]

I got arccos( (4sqrt(2) + 2sqrt(2)) / 12)

[alpine sable]

correct.

[vapid herald]

how would I find the degree value of that by hand?

[alpine sable]

you can find it by hand, yes

no need for calculator on this one

[worldly plover]

I have a question

[vapid herald]

yeah I am unable to use a calculator

[worldly plover]

Can you solve please

[vapid herald]

on the real test

[alpine sable]

@worldly plover this channel is occupied, please read #❓how-to-get-help

[alpine sable]

okay

then try simplifying $\arccos(\frac{4\sqrt{2}+2\sqrt{2}}{12})$

[ocean seal]

Al3dium

[alpine sable]

and see where you are able to get up to

[vapid herald]

just sqrt(2) / 6 right?

i mean

2

[alpine sable]

you mean the inside?

if so, then yes

[vapid herald]

yes

[alpine sable]

$\arccos(\frac{\sqrt{2}}{2})$

[ocean seal]

Al3dium

[alpine sable]

do you know how to evaluate this considering the unit circle?

i.e known values and angles

[vapid herald]

that should be 45 degrees right?

[alpine sable]

correct.

good job

[vapid herald]

thank you so much for the help, im pretty terrible at visualizing the vectors in 3 dimensions

[alpine sable]

with practice and a good drawing it usually helps

you'll get the hand of it with time

[vapid herald]

thank you again

[dusky notch]

Hello all! Im working on directional derivatives and gradients and I’m making a mistake in part b and c but I’m not sure where or why. Any help is appreciated, thanks!

[vapid herald]

sorry to barge in again I just have 1 more quick question, for @alpine sable

for this one would finding the dashed line that connects the z axis to the v vector would using Pythagorean theorem be wrong here then?

[alpine sable]

no, it would be totally fine

[vapid herald]

could you explain why in this case its fine?

[alpine sable]

i didn't say anything that pythag was not allowed in the other case

[alpine sable]

i was saying you wrongfully assumed that in this drawing, the y component of w was 1

while it wasn't

1 was the y component of v, not w

[vapid herald]

OH

[alpine sable]

the 1 in that drawing represented from the origin to the end of the vector v and not w

[vapid herald]

I think I understand now the dashed lines connecting the vector correlate with that vector

[alpine sable]

if the 1 was for the w vector, then surely you could've used pythag

[alpine sable]

yup

[vapid herald]

okay wow that cleared up so much, thank you so much again sorry for barging in!

[alpine sable]

no worries!

[quartz osprey]

hello

i require assistance

so im pretty much at the point where i find concavity and im stuck

i dont know if i set the bottom to 0 or top to 0 and even if i set the bottom to -0 it cannot be 0 without imaginary numbers

hmm os i tried 0 and sqrt 6 and it was wrong

[small trellis]

Why not set the entire thing to 0?

And I mean the second derivative

[quartz osprey]

well to = 0 the top has to be 0

so i set the top part to 0 right

and i ended up with like sqrt 6

and 0

well sqrt 6 mirhgt be wrong

nvm its not

[small trellis]

You need two values

[quartz osprey]

yea 0 and sqrt 6

[small trellis]

👍

[quartz osprey]

i kinda dont know where to go now

or how to approach this problem

[small trellis]

So, if it concaves upwards, then the left-most value will be smaller than the turning point, and the right-most value will be greater than the turning point

right?

[quartz osprey]

uhh

[small trellis]

And vise versa for it concaving downwards

[quartz osprey]

well yea

but like im finding where its concave up and down

so i set the second derivitve = to 0 whch is 0 when the top is 0

[small trellis]

Sorry, the turning point will be greater than the smaller point, but smaller than the greater point

[quartz osprey]

i just dont know what you are talking about

if its + its concave up if its - its concave down

[small trellis]

I think you need to find 3 X points

Since there are two concaves

[quartz osprey]

well yea

i did

[small trellis]

Ok

[quartz osprey]

its right ther

[small trellis]

I think that's what you need to follow

[silver current]

how to go from this

to this

the sqrt will become sqrt(-2ab)

how is that equal to |a-b|

[stable grotto]

how many different 6 digit numbers can we compose using 1 1 1 2 2 3 ?

The answer I guess is $6!/3!*2!$ but I need also exact explanation why

[ocean seal]

Filga

[celest anvil]

wait idk how to answer the question

y has infinitely many solutions

[alpine sable]

have you tried doing what the questions says if you get infinite sols?

@celest anvil

[quartz osprey]

yea im stuck on this

i dont know what to do

found second dervitive set it = to 0

got the inflection points

did the chart to test wether its concave up or down

and still get it wrong

like what do i do

[celest anvil]

i got (x,(1-5y)/5)

[dusky notch]

<@&286206848099549185>

[mossy kelp]

how would i approach this without doing hundreds of multiplications

the rest

[mossy kelp]

<@&286206848099549185>

[elder viper]

hundreds of multiplications?

what are the options

[mossy kelp]

well yeah right?

[elder viper]

A = {2, 4, 8, 10}, B = {5, 10, 15, 20}, C = {3,6,7,15}

BxAxC

12 is clearly not in {5, 10, 15, 20} so I think that would not be good

[mossy kelp]

b should be 5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20

[elder viper]

how

oh

[mossy kelp]

we dont have any evidence that the sequence is an increase of 5 each step

when thats the case we only increase by 1

[elder viper]

then the first one looks good

just do that for all the other ones then??

am not sure to understand what multiplication you're referring to

[elder viper]

what about it

isnt that just a cartesian product

[alpine sable]

that's correct

[mossy kelp]

cartesian product?

[elder viper]

i think ure overthinking this

[mossy kelp]

yeah probably lmao

[elder viper]

so the first element in ( . , . ) is in A

and the second one is in B

now you just do that with your thing and you add an extra spot

[undone agate]

Can someone help? Not sure what to do for #1

[celest anvil]

it said wrong

[alpine sable]

helppp

[alpine sable]

@celest anvil you solved for x as a function of y, it should be otherwise, y=(1-5x)/5.

[fresh quail]

hi

can i ask here

[red ledge]

Guys, There's something I don't understand..

[celest anvil]

thanks

[red ledge]

the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

Then how in this picture, the sum of the lengths of the 2 sides EQUALS the third one?

Shouldn't they be larger than the 3rd one?

[small bear]

Vectors are not lengths

[red ledge]

How's that?

What does that mean, then?

Isn't a vector a length + direction?

[small bear]

Well you wanted to know why the triangle inequality works. The inequality works when you take the length of the vectors, the vectors themself do not contain the length

[red ledge]

What are vectors?

Make me understand bro

[small bear]

Google that, I'd say a set of coordinates

The coordinates can describe both the length and the direction of the vector

[red ledge]

But I still don't understand

How can 2 sides = 1 side

Aren't they larger?

[vague coral]

vector arent length again

[red ledge]

That's my question..

Then what is PQ + QR

What does it mean to add both of them?

[small bear]

To add the respective coordinates

[red ledge]

Um

What does it mean?

Is there an easy problem u are not explaining it to me?

Don't assume I understand vectors, I'm just new..

Is it like a force?

Okay, What is the difference between line and a vector?

Is the difference that the vector has a direction?

[small bear]

A vector can describe a lot of things in physics and math. However think of them as a pair of coordinates, starting from (0, 0) and going to (a, b). This is for 2D of course

[red ledge]

But line has just a length?

[small bear]

When we say the vector (3, 4), we think of the line starting from (0, 0) and pointing towards (3, 4)

[red ledge]

But don't we need like a graph in order to use (0, 0) such things?

[small bear]

Algebra is fine

[red ledge]

Like I can't comprehend it when u say, use (0,0

Okay Okay

[small bear]

What is difficult to comprehend about (0, 0)?

[red ledge]

It's magnitude

Why didn't u tell me magnitude instead of length

When I kept saying length, U could've just told me that it's magnitude and not length at all

Right?

[small bear]

I am not clairvoyant, and both of those can be used interchangeably

[red ledge]

U just said to me they're not lengths, and u are telling me that those can be used interchangeable?

Does that mean that vectors can contain lengths?

It's magnitude and direction

That's it... But I don't really understand what is a magnitude anyway.

[small bear]

Vectors are a set of coordinates.
We can find their length via their coordinates, we do that by taking the squares of all coordinates, summing them up, and then taking the square root

Now, do vectors and length of vectors seem like the same thing to you?

[red ledge]

I think a vector is a length and a direction according to what you say

Like, A vector has in it a length

It's just a length and direction!

What could it be then?

Maybe I can't imagine what is a set of coordinates!

But I imagine it just a line on a graph!

And that line has a length

Isn't like set of coordinates are like points?

[small bear]

Well that isn't the case, you might want to read up on what vectors really are

[red ledge]

And we connect them to make a line on the graph?

U can't explain visually what a vector is?

Like as a concept?

U just give me a mathemical definition..

Set of coordinates, Set of coordinates

Is there something other than that?!

[gentle ingot]

It can also be described as two different values each pertaining to a different dimension

That's kinda the same thing as what was said

An ordered pair

But maybe think of it as like (h,k) as like hx+ky. h is the x value and k is the y value

[red ledge]

What is coordinates in real life?

Is it not in real life? Are they only on a graph? Can they exist only on a graph?

So we have to make them on a graph, Are they like points?

[gentle ingot]

If you add (1,2) to (3,4) you get (1+3,2+4) or (4,6)

They can be either depending on how you use them, but they're usually points

[gentle ingot]

This is also like how (1x+2y)+(3x+4y)=4x+6y

It's kinda like how the real numbers could be graphed on a number line, but don't have to be

[red ledge]

Um

Yeah

[gentle ingot]

People don't always think of the real number line when they see the number 3

Yeah

It just depends on how they wanna use them

A set of values

[red ledge]

What do u mean real number?

[gentle ingot]

Do you know what the number line is?

[red ledge]

Why do you keep saying **Real ** number

Why don't u just say numbers?

Is there a real number line and there's another number line?

U mean a different kind of numbers, values?

[gentle ingot]

Real numbers make more sense to this context ig. It is like multiple number lines yes

[red ledge]

So vector could not be length?

[gentle ingot]

It can be, but before we get to that do you fully know what a vector is?

[red ledge]

Vectors are limited to specific like quantities that we deal with in life?

[red ledge]

Uh, Do u have a definition other than it's a set of coordinates?

I guess a vector is just a visual way to present a physical concept in real life?

So there's a direction included

[gentle ingot]

I'm not sure if I can find a good way to put it other than that

[red ledge]

So we combined a magnitude and a direction together

[gentle ingot]

https://youtu.be/fNk_zzaMoSs

[red ledge]

And gave them a name a vector

[gentle ingot]

This is a great video

[red ledge]

No don't give it to me mathematically

[gentle ingot]

If you have the time

[red ledge]

So are u trying to say that it's only mathematically?

[gentle ingot]

Wdym?

[red ledge]

So it doesn't make any sense in real life?

[gentle ingot]

No it can. But mathematics may as well be so imbued in real life examples

[red ledge]

This guys explains it mathematically like he's gonna start to say numbers and put them on a graph

[gentle ingot]

Having 2 apples and 3 bananas can be the vector (2,3)

[red ledge]

I'm not trying to say why they put it in mathematics

I'm asking u for the physical definition or the concept

And not the way it's put or used in Mathematics

Like vectors are just representation of something

On papers or graph or whatever we are drawing on

[icy trail]

if you're running then your velocity can be represented as a vector, with the speed and direction... not sure what you're asking?

vectors are a way to quantify something yeah.. they're like level 2 numbers

lol

[red ledge]

Yeah So, Vectors are magnitude and direction both combined

To express them together!

That's it?

So what's the big deal?

[gentle ingot]

That's another way to represent them

[red ledge]

Why can't u explain it? Why can't u say that?

[gentle ingot]

They're all the same thing

[red ledge]

What do u mean?

[red ledge]

How's that? What does that mean?

[icy trail]

you can do cool stuff with them

[gentle ingot]

An ordered pair can be represented with direction and magnitude

[icy trail]

cool stuff being the later episodes of this series 👀

[red ledge]

Huh?

Can a vector just be a direction only?

Okay, Nvm, I got it

[gentle ingot]

R you sure?

This vector is both the ordered pair (2,2) and a vector with direction 45° and magnitude/length sqrt(8)

If you want to know why it equals sqrt(8) I can show u btw

[red ledge]

So a vector has a length when it's put on a graph only?

[gentle ingot]

Uhm you can find a vectors length with out a graph, if that's what you're asking?

[red ledge]

A vector isn't a length bro

[gentle ingot]

???

[red ledge]

No there can't be a length for a vector conceptually

because a vector is magnitude

[icy trail]

length = magnitude

[gentle ingot]

^

[red ledge]

Is length a magnitude as well?

[icy trail]

they mean the same thing

[red ledge]

I thought magnitudes are just quanities like? Like scalar quantites that are just represented only with numbers

That don't need a direction to be satisified with undertanding them

[icy trail]

a length has all those properties too

magnitude is probably more accurate tho so we can call it that instead if you like :)

[red ledge]

Um

Okay can we like postpone this now? Or like put this problem now

I wanna understand something

[gentle ingot]

Mhm

[red ledge]

How can a direction be an angle?

[icy trail]

what would you prefer it to be?

[red ledge]

Like I don't really understand why people describe the direction as angle

[gentle ingot]

Well how would you describe the direction the point (2,2) is from the origin if you had to?

[red ledge]

Okay bro, We all know that direction is like an arrow

[full rampart]

Hi, my algebra is terrible and I'm working on it. How was this simplified?

[gentle ingot]

So you say in an arrow?

[red ledge]

Okay, can u answer this man's question pls

Because I also don't udnerstand it

And we can come back to it..

[gentle ingot]

Are you sure?

[red ledge]

Yeah

[red ledge]

How is this simplified? @gentle ingot

[shut crest]

what plus what = -4

then divide it by 2

[gentle ingot]

Btw I think it's against the rules to ask questions when there are other questions being spoken abt

[full rampart]

Oh sorry

Thank you

[red ledge]

Yeah

But i allow it bc he needs help.. a quick and easy question yeah

But bro wait

[gentle ingot]

It's fine, just do right in the future in you'll be good

[red ledge]

If u divide this equation by 2

x+y = -2

How did u turn it to the final result, 4x + y = 1

?

That's the question.

[shut crest]

wait i thought those were 2 different problems

and you had to solve for x and y

[full rampart]

No sorry

[red ledge]

How is this even simplification, Like it was 2x then it was multiplied by 2 >>> 4x

[full rampart]

Maybe I misunderstood what the instructor was doing then

[gentle ingot]

Uhm do you know how to get each equation in the terms of y= ?

[red ledge]

Guys Is there any solution to it?

[icy trail]

it's not a simplification

[shut crest]

you need to divide both sides by 2, so it becomes x + y = -2

[red ledge]

Yes that's what I thought. There's no way this is a simplification.

[shut crest]

its not?

[full rampart]

they may be two problems. sorry to confuse everyone. going to rewatch

[red ledge]

You said right.

[gentle ingot]

Is this supposed to be a system of equations?

[shut crest]

oh

[red ledge]

Yeah that's what I thought..

[gentle ingot]

You're fine dw

[red ledge]

Okay how can a direction be describing an angle, Like it's very naive...

[shut crest]

its alr

[red ledge]

😒

[gentle ingot]

I think you should move to another chat since I think we're still going over abstracts question

@full rampart

[tiny crown]

@full rampart don't crosspost

[icy trail]

how would you tell someone what direction to go?

[gentle ingot]

^

[icy trail]

say a ship, in a sea with a lighthouse as reference

[red ledge]

So u want this ship to reach the lighthouse?

I would just say to them to sail north

[icy trail]

ok forget the lighthouse

[red ledge]

Or go forward

That's the direction..

[icy trail]

If you wanted it to turn slightly to the left

[red ledge]

I get it that when u imagine it, Like the arrow, U can see the angle

[icy trail]

you'd say turn 10° to the left right?

[red ledge]

But still

[red ledge]

But this is a turn bro! A turn! Like it's not one dimension as a direction

A turn like is like 2d

How do u say direction

Like there should be a new word for it

[icy trail]

its also called the "argument" lol

[red ledge]

My brain can't just get over it or comphrened it?!

[red ledge]

What do u mean by that?

[icy trail]

how is a turn 2d?

[gentle ingot]

Everyone's been here

Dw

[red ledge]

Like look a turn requires much motions more dimensions

Like it takes a lot of physical things

To comphrened than just to go straight

[gentle ingot]

An angle/turn can be multiple dimensions but I think we should just stick with 2 dimensions rn

[red ledge]

Okay so a direction isn't an angle

[icy trail]

a turn is just an angle, from 0-360° you can think about this 0-360 as a number line, a 1d number line

[red ledge]

An angle is totally different scalar or quantity or whatever

[icy trail]

angle is a component of a vector

its a property

[red ledge]

Yeah because it has a magnitued { a number} and a direction

[icy trail]

{an angle}

[red ledge]

That can be represented graphically?

Okay let's just construct arguments because this is vague and I don't understand

So what do u say now?

[icy trail]

like this

[red ledge]

Since an angle is more than 1 dimension

Do we stay need to call it direction?

Like direction isn't the right word to describe it!

[icy trail]

you prefer angle over direction?

that's fine, they're the same

[red ledge]

No I don't know what to call it actually

But I'm sure direction is wrong

Because direction just describes one simple thing

[icy trail]

can you explain how direction is 2d?

[red ledge]

One dimension, one worded single

Abstract concept

[gentle ingot]

If this ship (that's facing forward, at 0° from the origin) had to get to the point with just one movement it must turn a certain angle to be facing in the direction of the point

[red ledge]

I didn't say direction is 2d

[gentle ingot]

(the ship is supposed to be at the origin btw)

[red ledge]

I keep saying that direction is 1 dimensional

[red ledge]

Okay i'll see it

[gentle ingot]

You see it?

[red ledge]

Then why didn't u put it at the origin?

Did u forget it?

Or to make it shown to us?

[red ledge]

I don't understand how can u say the ship facing forward! Like forward is relative I think

the first time I saw it, I thought I'm dumb

And It's just mind

But now that I think about it

U meant north

[gentle ingot]

Is this better?

[red ledge]

But is it right to say forward?

I mean I imagined forward as it could be to the right?

Down

Right?

It's relative...

So how did u say forward!

[gentle ingot]

When in the x,y plane forward means right. It's relative to the dimensions of the plane

[red ledge]

Isn't it more correct to say Up or North?

[red ledge]

Uh, Wait! what! Forward means right?

[gentle ingot]

I guess I can say east

[red ledge]

Wait wait wait!

[gentle ingot]

?

[red ledge]

How does forward mean right in this graph?

Why? based on what?

What justification that allows it to be this way?

I never known that!

[gentle ingot]

There's a lot of deeper math used to justify this but to explain is to go on a pretty deep tangent

Do you want me to go into depth?

[red ledge]

U know what! I just didn't read the whole sentence

i should've just read it but I stopped immediately when u said forward

But found angles and turning included

Okay so u are saying that ship turns into a circle?

is that it?

[gentle ingot]

That last diagram was pretty bad tbh. Forget about it

[red ledge]

Okay

[gentle ingot]

Well do you want me to explain to you coordinate trigonometry so explain why we mean forward is right in this context?

[red ledge]

Um, Maybe we can late that

Like just put it aside

now

[gentle ingot]

Okay

[red ledge]

So we don't consume time

[gentle ingot]

So back to describing direction as an angle?

[red ledge]

I love using late as a verb because I think there's no any substinatial word that convey a meaning better

[red ledge]

Yeah

What did u want to convey by your graph?

What did u mean by your drawing? What was the point?

Like, What did u wanna explain?

[gentle ingot]

For this one?

Sorry had to use the bathroom

[red ledge]

Sure

[gentle ingot]

I was going to try to ask what angle the ship must move to be facing the point

(what direction it must turn)

[red ledge]

Look angle + direction should equal a word

We need a new word for them

[gentle ingot]

Argument

(argument has a different meaning in math than regular English btw)

[red ledge]

Because when I read ur sentence this, I felt like "what angle..." is absurd!

Like how can u describe an angle with what?

Like an angle needs a direction

Right?

But okay going back to ur point