n! would have your product going down all the way to 1
#help-10
royal basin
idk what "actual meaning" you're looking for.
dense stirrup
I mean how is nP5 represented
royal basin
do you know the general formula/definition ${}^n P_k = n(n-1)\dots (n-k+1)$?
warm shaleBOT
Ann
dense stirrup
yes
royal basin
yeah so
dense stirrup
got it
royal basin
take k=5 here
severe reef
This is the same thing
You can observe this, or ykw just ut sub in some values and compare yourself
The denominator multiplies all numbers from 1 to n-r
And the numerator is product of all numbers from 1 to n
So all numbers upto n-r are cancelled
dense stirrup
When to use which formula?
severe reef
When to use which formula?
They're both the same
But I like n!/(n-r)!
Because its compact
royal basin
When to use which formula?
whichever is more convenient
but forbid yourself from stressing out over it
royal basin
how would you solve that question
same way as you showed in your image
dense stirrup
same way as you showed in your image
I want to see how the solution would go if I use this formula
dense stirrup
this one
severe reef
I want to see how the solution would go if I use this formula
n(n-1)(n-2)(n-3)(n-4)(n-5)!/(n-5)! = 42*nP3
severe reef
All numbers from n-5 are common in numerator and denominator
turbid magnet
The answer is supposed to be B = 2A - a. I dont even know how to get started math without numbers confuses me😅
obtuse pebbleBOT
The answer is supposed to be B = 2A - a. I dont even know how to get started mat...
turbid magnet
,rotate
warm shaleBOT
proven zephyr
what have you tried?
high lily
I dont even know how to get started math without numbers confuses me
the same way you'd do math with numbers
instead just leave stuff you can't combine / simplify as they are
pretty mucht he same rules apply
turbid magnet
what have you tried?
I dont know how to isolate a, i watched a video but it was a different equation structure
high lily
if the question was
$$7 = \frac{3 + b}{11}$$
how would you solve that?
warm shaleBOT
ℝamonov
high lily
(for b)
turbid magnet
im not sure
high lily
have you solved any equations before
turbid magnet
Like this? Maybe, i dont remember
high lily
,tex .algebra lesson
warm shaleBOT
ℝamonov
high lily
start with that for the basics
turbid magnet
I think i know these ones but this one is a different form
turbid magnet
**ℝamonov**
.
high lily
the same ideas applies
clear condor
what did you do for the other ones then
high lily
applying the same operations to both sides to simplify the equation and work towards isolating your desired variable
though a fraction being initially present isn't in the above examples,
you could employ the idea of multiplying both sides by something to rid yourself of fractions
warm shaleBOT
turbid magnet
Wrong image
turbid magnet
Like this?
,rotate
warm shaleBOT
high lily
same image
high lily
,rotate
warm shaleBOT
turbid magnet
Then what do i do to get -a?
Do i do the same again?
high lily
well the question wants you to get b
Ex1 is applicable here
what could you do to
a + b
that'll give you just b
then do that to both sides of the equation
turbid magnet
what could you do to
`a + b`
that'll give you just b
then do that to both sides ...
Why isnt it ab?
high lily
because you don't have ab
turbid magnet
I meant like, why isnt a+b = ab?
turbid magnet
How do i get the -a so that its b=2A-a?
clear condor
2a=a+b
high lily
what could you do to
a + b
that'll give you just b
if there were numbers involved
what could you do to
x + 7
tha'll give you just x
turbid magnet
But the answer is b = 2A-a
How do i get there?
high lily
answer what i'm asking
don't overthink what i'm asking
what could you do to
x + 7
that'll give you just x
(anything you want, preferable simple)
turbid magnet
> what could you do to
> x + 7
> that'll give you just x
Subtract on both sides?
high lily
that's what you'd do if you had an equation
but for what I'm asking you, just focus on that expression for now
what would you subtract from
x + 7
to get just x?
turbid magnet
7
high lily
yes
and if you had the equation
x + 7 = 123
subtracting 7 from both sides isolates/solves that equation for x
similarly
what could you do to
a + b
that'll give you just b
what could you subtract froma+bthat'll give justb
turbid magnet
Subtracr a
turbid magnet
2A
-a
Wouldnt this just remove the a?
high lily
math is case sensitive
A is not the same thing as a
A and a aren't like terms
can't combine them
turbid magnet
So i do it with b?
high lily
wdym
turbid magnet
I subtract both sides by a and to b
high lily
a + b - a = b
is the reasoning for subtracting a from both sides
wdym by and to b
$$2A = a + b$$
subtracting $a$ from both sides
$$2A \red{-a} = a+ b \red{-a}$$
$$2A - a = b$$
warm shaleBOT
ℝamonov
warm shaleBOT
high lily
no
chuck -a on each side of the equation (once) as demonstrated above in red
here you're subtracting a from the right side of the equation twice
turbid magnet
**ℝamonov**
How is a in the middle gone in the answer?
high lily
a - a = ?
turbid magnet
**ℝamonov**
But there isnt a-a its just a*+b-a
high lily
what's with that * on the a
turbid magnet
To point which a im talking about😅
turbid magnet
a + b - a = a - a + b
Im confused?
high lily
commutative property of addition
are you implying that to evaluate
123456789 + 98765432123456789 - 123456789
you would first need to add
123456789 and 98765432123456789
together to get
,w 12345656789 + 98765432123456789
warm shaleBOT
high lily
before subtracting 123456789
turbid magnet
you would first need to add
123456789 and 98765432123456789
together to get
No?
high lily
so how would you simplify that numerical example
i mean you've essentially already answered this earlier
subtracting 7 from x+7 gives x
and you also answered that subtracting a from a+b gives b
which is the reason why you'd subtract a from both sides of the equation in the first place
turbid magnet
Yes, i know that but the part where im confused is where do i do that to get 2a-a
high lily
MaTh Is CAse SEnsiTivE
subtracting $a$ from both sides
$$2A \red{-a} = a+ b \red{-a}$$
$$2A - a = b$$
warm shaleBOT
ℝamonov
turbid magnet
`2A - a`
can't be combined and stays as is
Oh ok. Makes sense, also how come does it simplify to b?
high lily
a + b - a = b
Yes, i know that
turbid magnet
I meant as in, whats the logic behind it that it simplies just to b?
high lily
commutative property of addition
a + b - a = a + b + (-a) = a + (-a) + b
the bolded part is 0
which is what allows you to simplify
123456 + 5467468435798 - 123456
to 5467468435798
without tedious addition/subtraction
turbid magnet
I dont understand sorry😅
high lily
lets go back to a numerical example
x + 7 = 123
when i asked earlier
you knew that subtracting 7 from x+7 gives just x
thus to solve for x, you can subtract 7 from both sides
that action is represented as
$$x + 7 \ \red{-7} = 123 \ \red{-7}$$
warm shaleBOT
ℝamonov
high lily
do you have any issues with that
turbid magnet
No?
turbid magnet
Yea
high lily
because
you knew that subtracting 7 from x+7 gives just x
and what does the right side simplify to
turbid magnet
Oh and then x-7=x?
turbid magnet
How come?
high lily
where's x - 7 = x coming from
turbid magnet
what does that left side of that equation, the `x + 7 - 7`
simplify to?
X+7-7
X+7 = x-7
turbid magnet
Oh
high lily
x + 7 - 7 = x
THAT IS ALL
DO NOT ADD STUFF
DO NOT REMOVER STUFF
DO NOT MAKE ANY CHANGES TO THAT WHATSOVER
turbid magnet
DO NOT ADD STUFF
DO NOT REMOVER STUFF
DO NOT MAKE ANY CHANGES TO THAT WHATSOVER
But it doesnt make sense that it just x? Does this apply to any number and how much there are of?
high lily
how doesn't it make sense
you've said yourself that subtracting 7 from x+7 gives x
x + 7 - 7 = x
is the equation that reprents that
the left side indicates subtraction for 7 from x+ 7
overall the equation is saying that is equal to x
i don't understand how you have an issue with that
echo lava
Yo
high lily
are you implying that
1234 + 42 - 42
gives you something other than 1234?
turbid magnet
i don't understand how you have an issue with that
I dont, its just. Its just that it doesnt make any sense to me😅, i dont understand the logic behind it, i just know as it is
turbid magnet
are you implying that
1234 + 42 - 42
gives you something other than 1234?
Yea
high lily
some value - that same value = 0
high lily
1234 + 42 - 42 = 1234
because 42 - 42 = 0
obtuse pebbleBOT
@turbid magnet Has your question been resolved?
timid silo
\textbf{Question:} Let $f_0 = 0, f_1 = 1$ and $f_n = f_{n-1}+f_{n-2}$ for $n\ge 2$. Use mathematical induction to prove that if $x^2 = x +1$ then $x^n =f_n x +f_{n-1}$ for $n \ge 2$
\vs{3 mm}
Let's take $\map P n$ to be the statement that $x^ = f_n x+ f_{n-1}$. We can start by induction on $n$
\vs{3 mm}
\textbf{Basis step:} For $n =2$ we have $f_2 = f_1 + f_0 = 1$ and we have [
x^2 = x + 1 = f_2 x + f_1 ]
\textbf{Inductive step:} Suppose that $\map P k$ is true for $k \ge 2$, then we have [
x^k = f_k x + f_{k-1}
]
We want to show that $\map P{k+1}$ is also true:
\begin{align*}
x^{k+1} &= x^k \cdot x = (f_k x + f_{k-1})x\
&=f_k x^2 +f_{k-1}x \
&=f_k(x+1) + f_{k-1}x \
&=f_k x + f_k + f_{k-1}x \
&=(f_k + f_{k-1})x + f_k
\end{align*}
obtuse pebbleBOT
\textbf{Question:} Let $f_0 = 0, f_1 = 1$ and $f_n = f_{n-1}+f_{n-2}$ for $n\ge ...
warm shaleBOT
timid silo
I must be very close to getting to the conclusion, but im not seeing how just yet?
flat gate
maybe you got something wrong here
warm shaleBOT
redstoneplayz09
flat gate
·
timid silo
i guess just
brisk grove
look at how the recurrence with f is defined
brisk grove
yes
flat gate
you've already done it
brisk grove
ur latex is nice tho
flat gate
you can do it, trust yourself
timid silo
but anyways ty red, been such a massive help recently!
obtuse pebbleBOT
flat gate
Sometimes you can get confused accidentally
timid silo
wild swallow
no
wild swallow
you can
brisk grove
$\textbf{nice}$
warm shaleBOT
redstoneplayz09
timid silo
is my preamble the same as lex's? <@118663865264242688>
disappear for 2 weeks and delete your discord account and then come back and get invited to a 
server and you might get it
obtuse pebbleBOT
Available help channel!
brisk grove
obtuse pebbleBOT
<:sully:651816820122189834>
brisk grove
.close
obtuse pebbleBOT
obtuse pebbleBOT
<:sully:651816820122189834>
Show your work, and if possible, explain where you are stuck.
obtuse pebbleBOT
Available help channel!
timid silo
toby is that u
yeah
pulsar wing
What this question is asking in b. part? For a. It was -4 right?
obtuse pebbleBOT
What this question is asking in b. part? For a. It was -4 right?
edgy gorge
What this question is asking in b. part? For a. It was -4 right?
Yea it was -4, for part b, they're asking rate of change of y w.r.t x, which is again dy/dx i.e. the slope again
Since they're asking for derivative in part a, I assume you're familiar with derivatives and that you should know that rate of change of y w.r.t x is the derivative of y w.r.t x
pulsar wing
Yes I know but that sound weird to me. Why they asking the slope again?
When derivate can be used to find slope as well of parabolas
edgy gorge
Yes I know but that sound weird to me. Why they asking the slope again?
idk their wish, maybe to confuse you
high lily
to reinforce the relation between derivative and slope
edgy gorge
makes sense
pulsar wing
This chapter I was reading was intro to derivates maybe thats why.
fallow rampart
Solve for x
obtuse pebbleBOT
Solve for x
fallow rampart
Not sure where I go from here / if I got here correctly
gilded needle
looks fine so far (you're missing a 5 at the end of the last line)
royal basin
what's what 71n?
fallow rampart
Yeah I just noticed that
royal basin
where did seventy-one, or the letter n, come from
fallow rampart
Sorry 7ln5
royal basin
7 ln(5).
fallow rampart
These are natural logs yeah
royal basin
you should make your 1's and l's more distinguishable.
unless you enjoy people being unable to read your work.
fallow rampart
Thanks
gilded needle
to continue, just recognize that the left hand side is x times some number
fallow rampart
I was wondering if dividing the right side by 3ln5 would be the correct next step
gilded needle
well, dividing both sides by that
fallow rampart
Yes
gilded needle
would indeed be a good plan
fallow rampart
royal basin
should work
gilded needle
the expression looks ok, but i get a different numerical result:
>> (log(311)-7*log(5))/(3*log(5))
ans =
-1.14455538217513
(above is matlab, where log means natural log)
fallow rampart
I got 0.047 when I checked in the original equation
But I think I put it in the calculator wrong
This looks better
gilded needle
maybe an operator precedence issue
fallow rampart
gilded needle
looks good
fallow rampart
What is with the .00000003 ?
gilded needle
just a floating point precision issue
gilded needle
sure, yw
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
hollow schooner
in the answer it shows sin^2x/Root2 how can i transform my answer into that
obtuse pebbleBOT
in the answer it shows sin^2x/Root2 how can i transform my answer into that
@hollow schooner Has your question been resolved?
Available help channel!
late stump
obtuse pebbleBOT
late stump
part b
i have 2 parametric equations (tho i dont think theyre right?)
and i would assume that the line from P1 to P2 at t is CD(t) - AB(t)
but after that i have no clue where to go
the line segment being perpendicular to AB and CD makes me think we would use the cross product (i think?)
plain owl
and i would assume that the line from P1 to P2 at t is CD(t) - AB(t)
It would be good to take two different values as in CD(t1) and AB(t2)
late stump
oh maybe thats why i was so stuck
plain owl
Because then it would represent two independent points on two lines
plain owl
Yes
late stump
okay great
how would i go about finding the ts where the p1p2 is perpendicular to vec ab & cd
plain owl
You know that whatever that perpendicular vector is, it's dot product with direction vectors of each of the lines is 0
This gives you two equations in two variables
obtuse pebbleBOT
@late stump Has your question been resolved?
late stump
.close
thorny geode
What's the best way to find the interval of convergence?
obtuse pebbleBOT
What's the best way to find the interval of convergence?
raw hinge
?
obtuse pebbleBOT
@thorny geode Has your question been resolved?
violet sentinel
What's the best way to find the interval of convergence?
fringe rivet
$c) sec^{2}x-1=\frac{sin^{2}x}{1-sin^{2}x}$
obtuse pebbleBOT
$c) sec^{2}x-1=\frac{sin^{2}x}{1-sin^{2}x}$
warm shaleBOT
deviousglxy
fringe rivet
I attempted this
it didnt verify that it was equivalent
prove the trigonometric identity.
that the question
violet sentinel
ok so yeah
sec^2(x) - 1 is a pythagorean identity
hint Use sin^2(x) + cos^2(x) = 1
fringe rivet
What step did I do wrong
in the picture
or is it all right i just need to add more
violet sentinel
noting wrong per se but you overcomplicated it
oh wait
I don't see how you get form this line to this line
fringe rivet
so cos moves over and 1-1 is 0
Also move the six x and the other to the other side
I forgot the -1
violet sentinel
so you multiplied the right side denominator on both sides?
fringe rivet
ya
violet sentinel
ok so yeah
what you actually have at that point is
$\left(\left(\frac{1}{cos(x)}\right)\left(\frac{1}{cos(x)}\right) - 1\right)\left(\left(1 - sin(x)(sin(x)\right)\right) = sin(x)sin(x)$
warm shaleBOT
mellowdramallama
fringe rivet
Whats LHS
fringe rivet
MM so the work I solved for it wrong
or can I use it
for it to be easier
since there are less terms
violet sentinel
if you use pythagorean identity this a 3 step process
violet sentinel
MM so the work I solved for it wrong
yes from that point it was incorrect
fringe rivet
if you use pythagorean identity this a 3 step process
3 steps
intresting
tell
tell
tell
violet sentinel
sec^2(x) - 1 is a pythagorean identity
use this
violet sentinel
*hint* Use sin^2(x) + cos^2(x) = 1
you can derive it using this
(try dividing both sides by cos^2(x))
fringe rivet
k
violet sentinel
how about this
fringe rivet
$\frac{1}{cosx}(sinx)(sinx)-1=(sinx)(sinx)(cosx)$
warm shaleBOT
deviousglxy
violet sentinel
$sin^2(x) + cos^2(x) = 1$
$\\\frac{sin^2(x)}{cos^2(x)} + 1 = \frac{1}{cos^2(x)}$
$\\tan^2(x) + 1 = sec^2(x)$
$\\tan^2(x) = sec^2(x) - 1$
warm shaleBOT
mellowdramallama
fringe rivet
how did you get sin^2x+cos^2x=1
violet sentinel
that's a known identity
something that you're told and memorize (although it's not hard to derive on your own)
violet sentinel
so you can directly substitute sec^2(x) - 1 for tan^2(x)
then tan^2(x) = sin^2(x)/cos^2(x)
then since sin^2(x) + cos^2(x) = 1, then cos^2(x) = 1 - sin^2(x)
so then we get sin^2(x)/(1 - sin^2(x))
see? 3 steps 🙂
fringe rivet
thats neat
violet sentinel
yep
knowing sin^2(x) + cos^2(x) = 1 will help you go a long way
here's a video on the proof if you need it
fringe rivet
Ill add that to the arsenal of trig identity
is there like a
chart of all of them
I can just save
fringe rivet
yep exactly 🙂
I saved it
violet sentinel
awesome
fringe rivet
thats gonna save so much time
I got a question tho
how did sin^2x/cos^2x turn to cos^2x + sin^2x
violet sentinel
it didn't
fringe rivet
then tan^2(x) = sin^2(x)/cos^2(x)
then since sin^2(x) + cos^2(x) = 1
violet sentinel
two separate ideas
- tan^2(x) = sin^2(x)/cos^2(x) turns tan^2(x) into a fraction
- sin^2(x) + cos^2(x) = 1 ==> cos^2(x) = 1 - sin^2(x) just transforms the denominator
violet sentinel
turn cos^2(x) into 1 - sin^2(x)
$sin^2(x) + cos^2(x) = 1$
$\\ cos^2(x) = 1 - sin^2(x)$
warm shaleBOT
mellowdramallama
fringe rivet
Ok thanks a lot
violet sentinel
yep no problem 🙂
fringe rivet
.close
oblique pebble
Could require some help with 18b.
Translation:
Assume w = (1+ti)(1-ti), where t is real.
a) Show that when t varies, then w is on a circle S in the complex plane.
b) Show that the argument-angle theta for w is determined by tan(theta/2) = t.
obtuse pebbleBOT
Could require some help with 18b.
Translation:
Assume w = (1+ti)(1-ti), where t...
oblique pebble
Just clueless, honestly. Dunno if there’s any geometrically way to show it, but I’ve tried for example setting tan(theta) = Im(z)/Re(z), but it doesn’t really help me, eh?
obtuse pebbleBOT
@oblique pebble Has your question been resolved?
oblique pebble
<@&286206848099549185>
obtuse pebbleBOT
@oblique pebble Has your question been resolved?
hybrid gull
Maybe try expanding out w?
obtuse pebbleBOT
@oblique pebble Has your question been resolved?
flat gate
Could require some help with 18b.
Translation:
Assume w = (1+ti)(1-ti), where t...
Oh, you're still learning complex. I'm here again(
firstly, you seemed to write sth wrong in your translation(
The title is division of two, but the translation is multiplication of two.
so let's return to your problem
let's write the expansion expression
at this step, you can find something interesting
It means that the modulus of the complex is 1
so S is the unit circle which its center is the coordinate origin
OK, let's do part b
arg(w)=theta
so we can get
the question asks us to solve t out
let's compare the forms of the tan theta
you can get that t equals tan theta/2
flat gate
<@&286206848099549185>
Are you still there?/Er du fortsatt der?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
fringe rivet
Two roads intersect at 90^\circ. At 9:00, two cars leave the intersection on different roads at speeds of 80 km/h and 100 km/h. At 9:15, a traffic helicopter is right above the slower car, at a height of 1500 m. Determine the angle of depression and the distance from the helicopter to the faster car.
obtuse pebbleBOT
Two roads intersect at 90^\circ. At 9:00, two cars leave the intersection on d...
fringe rivet
The angle is 89 degrees
fringe rivet
drawing a diagram here will do wonders here
violet sentinel
yeah you're on the right track. Basically you'll need to find the hypotenuse distance between the two cars. That'll be one of the sides. Another side is going to be the height from the slow car
then you can use tan
violet sentinel
ah yeah so the distance will just be your hypotenuse
fringe rivet
between the car and the plane
violet sentinel
correct. One moment
violet sentinel
so one thing to notice is that it's 1500m above
but everything else is in km
so you need to convert it to 1.5km
here's what I drew up.
Blue lines are roads that meet at a 90degree angle
Green line represents the distance between the cars
Red is the height of the helicopter
Purple is the distance from the helicoptor to the faster car
obtuse pebbleBOT
@fringe rivet Has your question been resolved?
pearl charm
I have a few matrix questions
obtuse pebbleBOT
I have a few matrix questions
tough bolt
Hi need help?
For this one you can just write A B C and D as vectors and this will give you two equations in terms of t
tough bolt
For A, it's equal to (3,2)
tough bolt
Substituting in the equation gets (3,2)=p+t d
tough bolt
Then you can solve for t
tough bolt
First consider the x coordinates
tough bolt
3=3-2t
tough bolt
wait is t = 0?
Yes
pearl charm
well
how will that get us the answer
do we do that for A, B, C, and D?
do we do the same process for B, C, and D?
tough bolt
For A we are done
You can do the same thing for the other ones
You also have to consider the y-coordinates though
pearl charm
yea
what if we get different values for T
from the x and y coordinates
what would we do
tough bolt
I don't get what the first part means
pearl charm
what do u mean
tough bolt
Like it's already written there
pearl charm
hmm?
tough bolt
(x,y)=p+t d
pearl charm
right
pearl charm
I don't get what the first part means
what do u mean by this toh
tough bolt
For the second part, you know how vector scaling and adding visually works right?
pearl charm
i think so
can you go through how I would solve for A?
and i will try B, C, and D
since they all work the same way rigiht?
tough bolt
Yep
pearl charm
oh so it would be a ?
tough bolt
Yes
tough bolt
You know what the inverses represent right?
Also note that matrix multiplication is associative
pearl charm
yes
tough bolt
I'll prove it quickly
pearl charm
ok
tough bolt
So it will hold only if (AB)(A^(-1)B^(-1))=I
Oh wait I typed it wrong
It's (AB)^(-1)=B^(-1)A^(-1)
tough bolt
So only if (AB)(B^(-1)A^(-1))=A(BB^(-1))A^(-1)=AIA^(-1)=AA^(-1)=I
tough bolt
so how does that relate to the question
This will make it more intuitive
At least that's how I think it should work
tough bolt
So for the first one which vector goes in the vague direction of A^(-1)B^(-1)v?
pearl charm
c?
tough bolt
That's B^(-1)v
Maybe not
tough bolt
It's actually a
Because B^(-1) flips v about (3,1) and A^(-1) rotates it π/4 clockwise
tough bolt
The next one's c
tough bolt
The last one's the same as the first one
tough bolt
I got to go to sleep now
tough bolt
Ok
tough bolt
Hmmmmmmmmm...
pearl charm
i just don't even know how to start this one
royal basin
ok hold on
@pearl charm do you know how matrix multiplication works
like in general
pearl charm
yes
royal basin
ok
so you know that when multiplying matrices together, each entry in the product equals the dot product of a row in the first matrix and a column in the second, yes?
pearl charm
yes
tough bolt
Wait you can just use the dot product method
royal basin
right
tough bolt
so you know that when multiplying matrices together, each entry in the product e...
Oh never mind
I got to go sleep
Bye
pearl charm
noooooooo
ok
thx for everything bye
@royal basin so how would that help us for this question
royal basin
so when we multiply out Au, we get a three-dimensional vector in which:
the first entry is u · u,
the second entry is v · u,
and the third entry is (3u+2v) · u
agree or disagree?
royal basin
ok
now can you tell me what u · u is?
the problem gives you the data you need to figure that out.
otherwise, if you need me to prod your memory further, let me know.
royal basin
no, it is not 2.
royal basin
yes, 4. the dot product of a vector with itself is the square of its length.
ok, now what about v · u?
pearl charm
8*cos(120)?
royal basin
yes, corrrect.
cos(120°) is a value you should know from memory or be able to rederive easily.
royal basin
cos(120°) = -1/2 so u · v = -4 yes
ok
now (3u+2v) · u
are you able to calculate that?
pearl charm
no
royal basin
what properties of the dot product do you know?
tardy epoch
i think i know some
Write down those formulas
royal basin
these are formulas to compute the dot product
but not what i have in mind here
do you know that the dot product obeys the distributive law?
royal basin
mildly bad notation but yes.
are you able to compute (3u+2v) · u now that i have primed your mind in the right direction?
royal basin
yes, keep going.
royal basin
yes
royal basin
when i say "what have we calculated just now" i want you to say what the number you've calculated MEANS, not that it is thirty-seven.
we are not yet at the final answer to the whole problem you posted; we are in fact exactly halfway through.
to recap,
so when we multiply out Au, we get a three-dimensional vector in which:
the first entry is u · u,
the second entry is v · u,
and the third entry is (3u+2v) · u
pearl charm
oh ok
royal basin
you have so far successfully calculated u · u, v · u and (3u+2v) · u
pearl charm
right
pearl charm
yes i do
royal basin
ok
royal basin
(4, -4, 4)
pearl charm
yea (4, -4, 4)
royal basin
you cannot drop the parentheses when specifying a vector by its coordinates.
ok.
are you able to calculate Av in the exact same fashion?
royal basin
how did the second component end up as 10?
royal basin
so is it 10 or is it 16?
pearl charm
16
whole nimbus
f
royal basin
ok can you show your equation
royal basin
and throughout all these manipulations, what does the letter x represent?
whole nimbus
total distance
royal basin
that's right.
and 3x/4 + 18 is also the total distance...
so do you see how to turn the expression 3x/4 + 18 into an equation?
whole nimbus
no its how much they travelled
royal basin
do you claim there is a difference between "how much they travelled" and "total distance travelled"
whole nimbus
so do you see how to turn the **expression** 3x/4 + 18 into an equation?
how to turn it to an equation?
royal basin
well,
do you agree or disagree that 3x/4 + 18 and x both represent the total distance travelled by the two trekkers
astral aurora
saw this question on a youtube channel the girl solved it very long
obtuse pebbleBOT
saw this question on a youtube channel the girl solved it very long
astral aurora
how to solve this one?
royal basin
17 and 276 look like they are relatively prime so the modular inverse of 17 mod 276 exists
astral aurora
yes gcd is one
so here i did this
17x=276k+9
x=16k+ (4k+9)/17
so k =2 statified
what should I do next?
royal basin
,calc 16 + (4*2+9)/17
warm shaleBOT
Result:
17
astral aurora
it should be 33
royal basin
,calc 16 * 2 + (4 * 2 + 9)/17
warm shaleBOT
Result:
33
so it works now
royal basin
,calc (17 * 33) mod 276
warm shaleBOT
Result:
9
thank you ANN
obtuse pebbleBOT
@astral aurora Has your question been resolved?
timid silo
\textbf{Question:} Assume that $f$ has a Fourier sine series [
\map f x = \sum_{n=1}^\infty b_n \map \sin{\f{n\pi x}L}, \quad 0 \le x \le L
]
Show formally that [
\f 2 L \int_0^L (\map f x)^2 \dd x =\sum_{n=1}^\infty b_n^2 ]
\textbf{My work:}
\vs{3 mm}
We can begin by examining $\bracks{\map f x}^2$ which would be: \begin{align*}
\bracks{\map f x}^2 &= \sum_{n=1}^\infty b_n \map \sin{\f{n\pi x}L} \sum_{m=1}^\infty b_m \map \sin{\f{m\pi x}L} \
&= \sum_{n=1}^\infty \sum_{m=1}^\infty b_m b_n \map \sin{\f{n\pi x}L} \map \sin{\f{m\pi x}L}
\end{align*}
Now,
[
\int_0^L \bracks{\map f x}^2 \dd x = \sum_{n=1}^\infty \sum_{m=1}^\infty b_m b_n \int_0^L \map \sin{\f{n\pi x}L} \map \sin{\f{m\pi x}L} \dd x]
obtuse pebbleBOT
\textbf{Question:} Assume that $f$ has a Fourier sine series \[
\map f x = \sum_...
warm shaleBOT
timid silo
Unsure on how to compute that integral. Any ideas?
kind hawk
ibp
kind hawk
that will pop out of the ibp
unless I'm misremembering
call the integral I. ibp twice to get I = (something again in terms of I)
then you can rearrange and solve for I. and during that rearranging you have to divide by something that might be zero
timid silo
Okay lets see
warm shaleBOT
rie.mann
timid silo
Last integral is a Kronecker delta
oh what does that mean
tardy epoch
Last integral is a Kronecker delta
Constant times this
tardy epoch
Of what
timid silo
that it is a Kronecker delta
tardy epoch
Reason backwards from what you're supposed to show
Memory from doing this many years ago and multiple times
timid silo
**rie.mann**
following from this:
Albeit not with IBP (unsure on how to do that just yet) but,
\begin{align*} \int_0^L \map \sin{\f{n\pi x}L} \map \sin{\f{m\pi x}L} \dd x &= \f12 \int_0^L 2\map \sin{\f{n\pi x}L} \map \sin{\f{m\pi x}L} \dd x \
&=\f 12 \int_0^L \map \cos{\f{(m-n)\pi x}{L}} - \map \cos{\f{(m+n)\pi x}L} \dd x\
&= \f12 \eval{\f L{(m-n)\pi}\map \sin{\f{(m-n)\pi x}L} - \f{L}{(m+n)\pi} \map \sin{\f{(m+n)\pi x}L}}_0^L \
&=0
\end{align*}
warm shaleBOT
timid silo
haaaaaa i feel like im hardcore latexing with this much shit
timid silo
anyways the m = n case is a simple reduction integral which evaluates to L/2
okay i guess thats it then
thanks boys
.close
obtuse pebbleBOT
timid silo
.reopen
obtuse pebbleBOT
✅
timid silo
Okay im asked now to apply the result from this to the sawtooth wave function and deduce [
\f{\pi^2}6 = \sum_{n=1}^\infty \f1{n^2}
]
warm shaleBOT
timid silo
Any ideas on how to proceed?
the sawtooth wave function is given by [
\map f x = x, \quad -L < x < L ]
with a Fourier of [
\map f x = \f{2L}\pi \sum_{n=1}^\infty \f{(-1)^{n+1}}n \map \sin{\f{n\pi x}L}
]
warm shaleBOT
kind hawk
plug in both sides?
obtuse pebbleBOT
@timid silo Has your question been resolved?
timid silo
plug in both sides?
yeah ty
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
pearl charm
I have matrix questions
obtuse pebbleBOT
I have matrix questions
pearl charm
cursive pulsar
Ok,
pearl charm
<@&286206848099549185>
obtuse pebbleBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
urban patrol
obtuse pebbleBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
I don't know where to start
prime reef
if you are supposed to look at the grid to get information you can find the u and v in terms of x y
prime reef
looks like it
royal basin
u isn't just (2,-2)
the coordinates of u in the basis {x,y} are (2, -2)
this means specifically that u = 2x - 2y
do you understand this y/n
pearl charm
ig
yea
yea
so v would be
-2x + 0y
so -2x
what do i do from here @royal basin
royal basin
u = 2x - 2y
rewrite this so that it has the form ax + by + cu = 0
clearly state what you get for a, b and c
if you are unsure how to state it clearly i can give you a fill-in-the-blanks. all you need to do is ask.
then don't post in this channel lol
pearl charm
didn't ask
oh he deleted the message
wait
to make it in ax + by + cu = 0
should i do it like
cu = -ax - by
therefore
wait no
that doesn't work
royal basin
kinda overthinking it / going in ever so slightly the wrong direction.
royal basin
well you might want to subtract u from both sides to get 2x - 2y - u = 0...
royal basin
yes it does
do you see how to proceed from here
it is not difficult at all it just requires care
royal basin
well, you said it a little backward but yes, that's correct.
pearl charm
right ok
royal basin
do you see how to get d, e and f in the same fashion
pearl charm
ok
-2x - v = 0
d = -2
e = 0
f = -1
oh that was easy
I have another question
how would I do this
royal basin
draw the vectors emanating from a common start point and mark the angles that you are told they make
pearl charm
how would I do that
royal basin
which part of my instructions is unclear?
royal basin
get a pen and paper
place paper on table
place pen in hand
use pen to mark one point on paper
use pen to draw three vectors with that point as the start point for all three
use eyes & brain & visual cortex to read the problem statement and extract from it the angles between your vectors
use your pen to mark said angles on the diagram with little arcs representing them
idk how much more detail i can go into lmao just do it
honest, i can't do it for you
pearl charm
oh ok
i just tried to do it
and i don't really get how it relates to the problem
@royal basin
or like i don't know how to continue from this
@royal basin
royal basin
no need to ping me twice there.
show what you've got.
maybe you are on the right track but maybe you've screwed up the diagram.
pearl charm
how do i do that
i have no way to take a picture
but I think i did the drawing part right
royal basin
... no phone?
pearl charm
no.... not right now
royal basin
if you can't send a picture then i can't verify it
this is what it should look like. does it?
pearl charm
ummm
wait is
u and w
supposed to be a right angle?
oh wait
Ohhh
I messed up
royal basin
sure is.
pearl charm
But yea I got the same thing as that
royal basin
right
pearl charm
yes
royal basin
ok
pearl charm
vector / magnitude of vector
royal basin
bad!
royal basin
a unit vector is a vector whose length is 1
you are confusing the concept of a unit vector per se with the process of finding a unit vector parallel to a given vector
pearl charm
oh right
royal basin
you can use u and w as an orthonormal basis to express v in.
you're gonna need a little bit of trigonometry here.