Light bulb switching brain teaser

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.400
Let's say we have
100 light bulbs.

00:00:03.400 --> 00:00:04.980
Well, let me actually
just draw them.

00:00:04.980 --> 00:00:08.860
So I have one light
bulb there.

00:00:08.860 --> 00:00:12.100
I have another light bulb.

00:00:12.100 --> 00:00:14.760
And I have 100 of them.

00:00:14.760 --> 00:00:16.490
100 light bulbs.

00:00:16.490 --> 00:00:20.380
And what I'm going to do is--
Well actually, before I even

00:00:20.380 --> 00:00:23.290
start turning these light bulbs
on and off, let me let

00:00:23.290 --> 00:00:25.260
you know that they
are all off.

00:00:25.260 --> 00:00:26.520
So they start off.

00:00:30.440 --> 00:00:33.110
Now, the next thing I'm going
to do is I'm going to number

00:00:33.110 --> 00:00:34.110
the hundred light bulbs.

00:00:34.110 --> 00:00:35.590
I'm going to number them
one through 100.

00:00:35.590 --> 00:00:37.750
So the first light bulb
is light bulb one.

00:00:37.750 --> 00:00:39.780
The second light bulb
is light bulb two.

00:00:39.780 --> 00:00:42.610
All the way to light bulb 100.

00:00:42.610 --> 00:00:46.250
And what I'm going to do is,
first I'm going to go and I'm

00:00:46.250 --> 00:00:53.810
going to switch essentially
every light bulb.

00:00:53.810 --> 00:00:58.210
So if they all start off, I'm
going to turn them all on.

00:00:58.210 --> 00:01:00.690
So let me do it.

00:01:00.690 --> 00:01:07.110
So on my first pass-- let's
call this pass one-- so in

00:01:07.110 --> 00:01:08.810
pass one, I'm going to
turn all of these on.

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On, on, on.

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They're all going
to be turned on.

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On.

00:01:15.130 --> 00:01:18.430
And then in pass two, what I'm
going to do is I'm going to

00:01:18.430 --> 00:01:22.820
switch only every other
light bulb.

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So, for example I'll say, OK, I
won't switch light bulb one.

00:01:27.430 --> 00:01:29.470
I'll only switch
light bulb two.

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So light bulb one
will stay on.

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Light bulb two will be off.

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Light bulb three will be on.

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Light bulb four will be off.

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And so essentially, every light
bulb-- if you look at

00:01:40.770 --> 00:01:43.500
their numbers-- that
is a multiple of

00:01:43.500 --> 00:01:45.010
two, will be switched.

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So 100 will be switched,
so that'll also be off.

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Then I'm going to come-- and
ignore this right here-- and

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I'm going to switch every
third light bulb.

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So what's going to happen?

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This one's going to-- let me
switch colors arbitrarily--

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this one's going to stay on.

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This one's going to stay off.

00:02:03.330 --> 00:02:05.320
And I'm going to switch
the third light bulb.

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So this one was on.

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Now this one will be off.

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The fourth light bulb will
stay off, because I'm not

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touching it.

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The fifth light bulb
would have been on

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and it'll stay on.

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Now the sixth light bulb, in
this case we switched it off,

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and now it'll be on again.

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But I think you get the point.

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Every third light bulb, or if we
look at the numbers of the

00:02:28.330 --> 00:02:31.930
light bulb, every numbered light
bulb would that is a

00:02:31.930 --> 00:02:34.700
multiple of three is going
to be switched.

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And if it was a multiple of
three and two, it would have

00:02:37.320 --> 00:02:40.020
been switched on the
first time and then

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off the second time.

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But I think you're getting
the point.

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But what I'm going to do is,
I'm going to do 100 passes.

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So the first pass, I switch
every light bulb.

00:02:47.830 --> 00:02:49.360
They all started off,
so they're all going

00:02:49.360 --> 00:02:50.700
to be turned on.

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The second pass, I switch every
other light bulb or

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every second light bulb.

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The third pass, I do every
third one, or that's a

00:02:58.960 --> 00:03:00.630
multiple of three.

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And my question to you is, after
100 passes, how many

00:03:16.130 --> 00:03:19.780
light bulbs are still on?

00:03:19.780 --> 00:03:21.165
Or how many are on, period?

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And that is the brain teaser?

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How do you figure out, of
the hundred, which ones

00:03:32.220 --> 00:03:33.020
are going to be on?

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You should be able to do
this in your head.

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You don't have to make an Excel
spreadsheet and actually

00:03:36.890 --> 00:03:39.200
do all the on and
off switches.

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So the first question is, how
many of these are going to be

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on after I do 100 passes?

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And just to make it clear,
what's the 100th

00:03:44.110 --> 00:03:45.050
path going to be?

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Well, I'm only going to switch
every 100th light bulb.

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So whatever this light bulb was
already doing, I'm just

00:03:50.630 --> 00:03:51.360
going to switch it.

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If it was off, it'll come on.

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If it was on, it'll
become off.

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So the first question is, how
many of these are going to be

00:03:57.200 --> 00:03:58.450
on after 100 passes?

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And then the bonus question
is, which of these

00:04:01.080 --> 00:04:02.710
are going to be on?

00:04:02.710 --> 00:04:04.100
And so that's the question.

00:04:04.100 --> 00:04:06.980
Pause it if you don't
want the answer.

00:04:06.980 --> 00:04:07.960
And then try to solve it.

00:04:07.960 --> 00:04:09.850
I think it shouldn't take
you too much time.

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But now I'm going to give
you the answer.

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Or maybe I'll start with
a couple of hints.

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So, when do we know that a light
bulb is being switched?

00:04:21.050 --> 00:04:25.380
So if I'm on the second
pass, I will-- I don't

00:04:25.380 --> 00:04:26.030
want to say turn on.

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It might already be on.

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I will switch every light bulb
that's divisible by two.

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And then if I'm on the third
pass, I'll switch every light

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bulb that's divisible
by three.

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So on every pass,
what am I doing?

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If I'm on pass n-- this
is a hint if you want

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it-- what do I know?

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I know all the light bulbs that
are numbered where n is a

00:04:56.880 --> 00:05:00.480
factor of that light bulb,
that will get switched.

00:05:00.480 --> 00:05:11.380
So we know that switched
if n is a factor of

00:05:11.380 --> 00:05:13.620
the light bulb number.

00:05:17.020 --> 00:05:20.240
And that's just a fancy way of
saying, look, if I'm on pass

00:05:20.240 --> 00:05:24.320
17, I'm going to switch all
the multiples of 17.

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Or I could say, I know that I'm
going to switch light bulb

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51, because 17 is
a factor of 51.

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So that tells you that we're
always going to be switching

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one of these light bulbs on
or off when one of its

00:05:36.960 --> 00:05:39.930
factors is our pass.

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So for example, if we're looking
at light bulb eight.

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This is light bulb eight.

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So when will it be switched?

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So on pass one, we're
definitely going

00:05:56.153 --> 00:05:56.660
to switch it on.

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So pass one, it's going
to be switched on.

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Pass two, it'll be
switched off.

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I know that because two is
divisible into eight.

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Pass three, nothing's
going to happen.

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On pass three, nothing's going
to happen because this isn't a

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multiple of three.

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Pass four, what'll happen?

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It will be switched.

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It'll be switched back on.

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And then pass eight is the next
time we'll touch this

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light bulb, and it'll
be switched back.

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So every time one of its factors
go by, we're going to

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switch this thing.

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And as you can see, in order
for it to be on at the end,

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you have to have an odd
number of factors.

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So that's an interesting
thing.

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So in order for a light bulb to
be on, it has to have odd

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number of factors.

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Now that's an interesting
question.

00:06:53.850 --> 00:06:56.740
What numbers have an odd
number of factors.

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And this is something that I
think they should teach you in

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grade school, and
they never do.

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But it's a really interesting
kind of number theory.

00:07:03.230 --> 00:07:05.540
It's a simple one, but it's
interesting to think about.

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So what numbers are true?

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Let's do all the factors for
some of the starting numbers.

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So all the factors of one.

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Well one, the only factor
is just one.

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So one works?

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One has an odd number
of factors.

00:07:18.190 --> 00:07:20.190
So that means that one
will remain on.

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Because you're only going to
turn it on in the first pass.

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Makes sense.

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Two.

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What are all the
factors of two.

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Well you have one and two.

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So two has an even number.

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You're going to switch it on the
first time, then off the

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second time.

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Then you're never going
to touch it again.

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So this is going to stay off.

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Three.

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Your factors are
one and three.

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Four.

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Your factors are one,
two and four.

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Interesting.

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Here we have three factors.

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We have an odd number
of factors.

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So four is going to stay on.

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We're going to turn it on in our
first pass, we're going to

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turn it off on our
second pass.

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And we're going to turn it on
again in our fourth pass.

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Let's keep going.

00:08:00.660 --> 00:08:01.622
So five.

00:08:01.622 --> 00:08:05.290
The factors of five
are one and five.

00:08:05.290 --> 00:08:05.770
Six.

00:08:05.770 --> 00:08:08.210
The factors are one,
two, three and six.

00:08:08.210 --> 00:08:10.540
It's an even number, so they're
going to be off when

00:08:10.540 --> 00:08:11.790
we're done with it.

00:08:11.790 --> 00:08:12.360
Seven.

00:08:12.360 --> 00:08:14.340
it's one and seven.

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Eight.

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We just did that.

00:08:15.620 --> 00:08:17.790
It's one, two, four and eight.

00:08:17.790 --> 00:08:19.310
Still going to be off.

00:08:19.310 --> 00:08:20.860
Nine.

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Let's see.

00:08:21.190 --> 00:08:21.800
Nine.

00:08:21.800 --> 00:08:25.450
The factors are one,
three and nine.

00:08:25.450 --> 00:08:26.090
Interesting.

00:08:26.090 --> 00:08:28.700
Once again we have an odd
number of factors.

00:08:28.700 --> 00:08:34.880
So the light bulb number nine
is also going to be on when

00:08:34.880 --> 00:08:36.230
everything's done.

00:08:36.230 --> 00:08:37.400
Let's keep going.

00:08:37.400 --> 00:08:40.830
I don't know, I actually did
this at our mental boot camp

00:08:40.830 --> 00:08:41.419
with some of the kids.

00:08:41.419 --> 00:08:43.200
And they immediately said,
the distance between

00:08:43.200 --> 00:08:45.380
one and four is three.

00:08:45.380 --> 00:08:50.870
The distance between four
and nine is five.

00:08:50.870 --> 00:08:52.810
And maybe the distance between
nine and the next number is

00:08:52.810 --> 00:08:54.500
going to be seven.

00:08:54.500 --> 00:08:56.260
It increases by odd numbers.

00:08:56.260 --> 00:08:57.110
What's nine plus seven?

00:08:57.110 --> 00:08:57.890
Let's just try that out.

00:08:57.890 --> 00:08:59.060
16.

00:08:59.060 --> 00:09:00.940
What are the factors of 16?

00:09:00.940 --> 00:09:07.640
They're one, two, four,
eight and 16.

00:09:07.640 --> 00:09:08.100
Interesting.

00:09:08.100 --> 00:09:08.870
That worked, right?

00:09:08.870 --> 00:09:12.430
From nine to 16, you incremented
by seven.

00:09:12.430 --> 00:09:15.220
From four to nine, you
incremented by five.

00:09:15.220 --> 00:09:16.530
So it seems like we
have a pattern.

00:09:16.530 --> 00:09:19.080
But can you see something even
more interesting about the

00:09:19.080 --> 00:09:23.960
numbers one, four,
nine and 16.

00:09:23.960 --> 00:09:25.770
And you could try all the
numbers between nine and 16,

00:09:25.770 --> 00:09:28.380
and you'll see that they have
an even number of factors.

00:09:28.380 --> 00:09:31.030
But what's interesting about
all of these numbers?

00:09:31.030 --> 00:09:33.830
Why do they have an odd
number of factors?

00:09:33.830 --> 00:09:36.720
In all of these other cases,
every factor is paired with

00:09:36.720 --> 00:09:37.450
another number.

00:09:37.450 --> 00:09:38.920
One times two is two.

00:09:38.920 --> 00:09:40.770
One time six is six.

00:09:40.770 --> 00:09:41.850
Two times three is six.

00:09:41.850 --> 00:09:42.990
There's always a pair.

00:09:42.990 --> 00:09:45.360
Except for these numbers.

00:09:45.360 --> 00:09:46.670
There's no pair.

00:09:46.670 --> 00:09:47.530
Why isn't there?

00:09:47.530 --> 00:09:49.740
One times four is four.

00:09:49.740 --> 00:09:53.650
But two times two
is also four.

00:09:53.650 --> 00:09:55.860
So we only write the two once.

00:09:55.860 --> 00:09:57.560
Three times three is nine.

00:09:57.560 --> 00:09:59.480
Four times four is 16.

00:09:59.480 --> 00:10:01.730
So all the lights that are going
to be on are actually

00:10:01.730 --> 00:10:02.890
perfect squares.

00:10:02.890 --> 00:10:06.040
That's why they have an
odd number of factors.

00:10:06.040 --> 00:10:08.810
So what's our question?

00:10:08.810 --> 00:10:12.300
So this problem of what light
bulbs are going to be on,

00:10:12.300 --> 00:10:15.170
boils down to how many perfect
squares are there

00:10:15.170 --> 00:10:16.760
between one and 100?

00:10:16.760 --> 00:10:18.580
And you could just
list them out.

00:10:18.580 --> 00:10:20.950
And you could say, oh well
the perfect squares are

00:10:20.950 --> 00:10:24.070
one, four, nine, 16.

00:10:24.070 --> 00:10:25.430
And you could try to
think of them all.

00:10:25.430 --> 00:10:27.680
Or you could say, well how many
numbers can I square and

00:10:27.680 --> 00:10:30.890
get a number less than
or equal to 100?

00:10:30.890 --> 00:10:35.270
Well 100 is equal
to 10 squared.

00:10:35.270 --> 00:10:38.120
So you can only square the
numbers between one and 10 to

00:10:38.120 --> 00:10:39.220
get a perfect square.

00:10:39.220 --> 00:10:43.130
So there's only 10 perfect
squares between one and 100.

00:10:43.130 --> 00:10:45.010
Hopefully you didn't lose that,
but if that confuses

00:10:45.010 --> 00:10:47.180
you, just list them all out.

00:10:47.180 --> 00:10:50.780
Given that 100 is the largest
number, it's the whole largest

00:10:50.780 --> 00:10:51.470
perfect square there.

00:10:51.470 --> 00:10:53.330
And that's 10 squared.

00:10:53.330 --> 00:10:56.370
The only other perfect squares
in our range we're talking

00:10:56.370 --> 00:10:58.160
about are one squared, two
squared, three squared, all

00:10:58.160 --> 00:10:58.950
the way up to 10 squared.

00:10:58.950 --> 00:10:59.600
And we could do that.

00:10:59.600 --> 00:11:01.290
Four squared is 16.

00:11:01.290 --> 00:11:03.270
Five squared is 25.

00:11:03.270 --> 00:11:08.250
36, 49, 64, 81, and 100.

00:11:08.250 --> 00:11:13.290
So the light bulbs with these
numbers on it are the ones

00:11:13.290 --> 00:11:15.670
that will stay lit when
they're all done.

00:11:15.670 --> 00:11:18.150
Anyway, hope you enjoyed that.


Parametric equations 4

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.170
In the last video, we started
with these parametric

00:00:03.170 --> 00:00:06.860
equations: x is equal to
3 cosine of t and y is

00:00:06.860 --> 00:00:08.440
equal to 2 sine of t.

00:00:08.440 --> 00:00:11.630
And doing a little bit of
algebra, we were able to remove

00:00:11.630 --> 00:00:15.410
the parameter and turn it into
an equation that we normally

00:00:15.410 --> 00:00:16.590
associate with an ellipse.

00:00:16.590 --> 00:00:25.340
We got x squared over 9 plus y
squared over 4 is equal to 1.

00:00:25.340 --> 00:00:26.370
And we graphed it.

00:00:26.370 --> 00:00:27.320
And I'll do that.

00:00:27.320 --> 00:00:29.860
Actually I have it from the
previous-- We graphed it and

00:00:29.860 --> 00:00:33.270
we got something that looked
like that right there.

00:00:33.270 --> 00:00:35.460
And we said well why
don't we always just

00:00:35.460 --> 00:00:37.300
represent it like this?

00:00:37.300 --> 00:00:42.030
Well this right here-- this
equation just in terms

00:00:42.030 --> 00:00:44.740
of x and y --it just
represents the shape.

00:00:44.740 --> 00:00:48.510
It doesn't represent the actual
path that-- Let's say that t

00:00:48.510 --> 00:00:51.480
represents an object's point--
an object's position at

00:00:51.480 --> 00:00:52.370
some place and time.

00:00:52.370 --> 00:00:53.300
And we went over that.

00:00:53.300 --> 00:00:56.450
And as time progressed,
we went around the

00:00:56.450 --> 00:00:57.380
ellipse, in that case.

00:00:57.380 --> 00:01:00.400
And as time goes forward, we
kept going around this ellipse.

00:01:00.400 --> 00:01:04.540
But this begs the question: is
this the only set of parametric

00:01:04.540 --> 00:01:07.870
equations that, when we do a
little bit of algebra,

00:01:07.870 --> 00:01:09.890
gives this path?

00:01:09.890 --> 00:01:12.660
Can we think of another set
that gives-- that would

00:01:12.660 --> 00:01:15.350
also, when you do the
algebra, get here?

00:01:15.350 --> 00:01:17.250
And I think we can.

00:01:17.250 --> 00:01:18.820
So let's try a couple.

00:01:18.820 --> 00:01:21.050
I mean, we could do a
whole bunch of them.

00:01:21.050 --> 00:01:22.500
Actually, maybe we'll do a
whole bunch of them, just to

00:01:22.500 --> 00:01:24.380
see that there-- And what
you'll see is that there's

00:01:24.380 --> 00:01:26.550
actually an infinite number
of parametric equations

00:01:26.550 --> 00:01:27.690
that have this path.

00:01:27.690 --> 00:01:38.180
So one could be x is
equal to 3 cosine of 2t.

00:01:38.180 --> 00:01:44.830
And the other one is y
is equal 2 sine of 2t.

00:01:44.830 --> 00:01:48.370
Like we did in the last video,
let's solve for cosine of 2t

00:01:48.370 --> 00:01:50.320
and solve for sine of 2t.

00:01:50.320 --> 00:01:52.660
So we just have to divide both
sides of-- Well, both sides

00:01:52.660 --> 00:01:54.960
of this top equation by 3.

00:01:54.960 --> 00:02:00.700
You get x over 3 is
equal to cosine of 2t.

00:02:00.700 --> 00:02:03.770
Divide both sides of
this equation by 2.

00:02:03.770 --> 00:02:09.680
You get y over 2 is
equal to sine of 2t.

00:02:09.680 --> 00:02:12.100
And we'll use the same trig
identities we did before.

00:02:12.100 --> 00:02:17.910
When we did this problem, we
said that sine squared of t

00:02:17.910 --> 00:02:22.290
plus cosine squared
of t is equal to 1.

00:02:22.290 --> 00:02:24.020
Well if that's true,
so is this true.

00:02:24.020 --> 00:02:25.440
Let me see.

00:02:25.440 --> 00:02:29.820
If that's true, then we can
also say that sine squared

00:02:29.820 --> 00:02:36.140
of 2t plus cosine squared
of 2t is equal to 1.

00:02:36.140 --> 00:02:40.100
In fact, sine squared of
anything here plus cosine

00:02:40.100 --> 00:02:43.210
squared of anything, as long as
it's the same anything as we

00:02:43.210 --> 00:02:46.300
put here-- I can't have a 2t
here and a 3t here, but if this

00:02:46.300 --> 00:02:48.805
is a 2 and this is a 2 --then
that's always going

00:02:48.805 --> 00:02:49.640
to be equal to 1.

00:02:49.640 --> 00:02:50.620
You can just fill in the black.

00:02:50.620 --> 00:02:53.170
Because this is always going
to be the same angle.

00:02:53.170 --> 00:02:58.815
So if this is true, we can then
just take this and substitute

00:02:58.815 --> 00:03:03.745
it in for cosine of 2t and take
the y over 2 and substitute

00:03:03.745 --> 00:03:05.740
it in for sine of 2t.

00:03:05.740 --> 00:03:10.770
And we get the sine of 2t
squared becomes y over 2

00:03:10.770 --> 00:03:17.010
squared plus cosine of
2t is x over 3 squared.

00:03:17.010 --> 00:03:17.780
It equals 1.

00:03:17.780 --> 00:03:20.750
And that is exactly what
we had to begin with.

00:03:20.750 --> 00:03:24.170
That is the same thing as x
squared over 9-- I'm just

00:03:24.170 --> 00:03:29.680
switching the order here --plus
y squared over 4 is equal to 1.

00:03:29.680 --> 00:03:34.600
So both of these parametric
equations, both sets of

00:03:34.600 --> 00:03:38.830
parametric equations have
the exact same path.

00:03:38.830 --> 00:03:40.920
So how are they different?

00:03:40.920 --> 00:03:42.010
Let's see.

00:03:42.010 --> 00:03:45.650
If we were to take-- So in
the previous video, we

00:03:45.650 --> 00:03:46.550
took this yellow one.

00:03:46.550 --> 00:03:47.880
Actually I rewrote it here.

00:03:47.880 --> 00:03:50.370
And we did a little table
and we plotted the points

00:03:50.370 --> 00:03:52.600
that it hits as you
go around the circle.

00:03:52.600 --> 00:03:56.470
So let's do the same
thing for our new set.

00:03:56.470 --> 00:04:00.000
So just since I lost it, let me
write the two equations here.

00:04:00.000 --> 00:04:06.870
It was x is equal to 3
cosine of 2t and y is

00:04:06.870 --> 00:04:10.520
equal to 2 sine of 2t.

00:04:10.520 --> 00:04:11.495
Let me make a little table.

00:04:18.570 --> 00:04:20.576
There we go.

00:04:20.576 --> 00:04:23.334
And we're going to have a t.

00:04:23.334 --> 00:04:26.430
And we're going to
have an x and a y.

00:04:26.430 --> 00:04:29.460
And let's pick the
same values of t.

00:04:29.460 --> 00:04:34.780
0 pi over 2 and pi.

00:04:34.780 --> 00:04:37.860
When t is 0, cosine
of 2 times 0.

00:04:37.860 --> 00:04:42.450
That's cosine of 0, which
is one, times 3, it's 3.

00:04:42.450 --> 00:04:46.340
When t is equal to pi over
2, 2 times pi over 2 is pi.

00:04:46.340 --> 00:04:48.400
Cosine of pi over 2 is minus 1.

00:04:48.400 --> 00:04:52.180
Minus 1 times 3 is minus 3.

00:04:52.180 --> 00:04:56.600
When t is equal to pi, cosine
of 2 times pi is 2 pi.

00:04:56.600 --> 00:04:59.850
Cosine of 2 pi is the same
thing as cosine of 0.

00:04:59.850 --> 00:05:01.120
So that's 1.

00:05:01.120 --> 00:05:03.330
1 times 3 is 3.

00:05:03.330 --> 00:05:05.430
And then on the y side.

00:05:05.430 --> 00:05:07.245
When t is equal to 0,
that's the same thing

00:05:07.245 --> 00:05:08.740
as sine of 2 times 0.

00:05:08.740 --> 00:05:13.480
Sine of 0 times 2 is 0.

00:05:13.480 --> 00:05:14.550
Pi over 2.

00:05:14.550 --> 00:05:16.330
Sine of 2 times pi over 2.

00:05:16.330 --> 00:05:18.070
That's the same
thing as sine of 4.

00:05:18.070 --> 00:05:19.140
That's also 0.

00:05:19.140 --> 00:05:21.720
0 times 2 is 0.

00:05:21.720 --> 00:05:22.820
And then pi.

00:05:22.820 --> 00:05:24.070
Sine of 2 pi.

00:05:24.070 --> 00:05:25.200
Well, that's also 0.

00:05:25.200 --> 00:05:27.430
So here we get a bunch of 0's.

00:05:27.430 --> 00:05:28.980
So what happened here?

00:05:28.980 --> 00:05:30.110
Let me do it in a
different color.

00:05:30.110 --> 00:05:35.260
So in our first situation-- So
they both-- We established that

00:05:35.260 --> 00:05:38.325
both this set of parametric
equations and this set of

00:05:38.325 --> 00:05:42.160
parametric equations, they both
have the same overall, I guess

00:05:42.160 --> 00:05:43.670
we could say, shape
of their path.

00:05:43.670 --> 00:05:45.340
They're both this ellipse.

00:05:45.340 --> 00:05:49.410
But what we saw in the previous
videos, when you plot these

00:05:49.410 --> 00:05:55.780
points at time t equals
0, we were right.

00:05:55.780 --> 00:06:00.550
This point right here
is that right there.

00:06:00.550 --> 00:06:07.490
And then at t is equal to pi
over 2, we got right there.

00:06:07.490 --> 00:06:13.270
And then at t is equal to
pi, we got over here.

00:06:13.270 --> 00:06:16.240
And we did that to establish
that the direction of motion.

00:06:16.240 --> 00:06:18.840
If these parametric equations
really are describing some

00:06:18.840 --> 00:06:21.960
type of motion, it would
be counterclockwise.

00:06:21.960 --> 00:06:25.860
And what happens in this
scenario-- So when t is

00:06:25.860 --> 00:06:30.410
equal to 0, we're still
at the point 3, 0.

00:06:30.410 --> 00:06:33.900
So they at least-- At t equals
0, you can kind of view them

00:06:33.900 --> 00:06:35.890
as starting at the same
point, if we assume we're

00:06:35.890 --> 00:06:37.390
starting a t equals 0.

00:06:37.390 --> 00:06:39.230
You don't have to
start at t equals 0.

00:06:39.230 --> 00:06:41.740
I did that a little bit in the
last video, but you can start

00:06:41.740 --> 00:06:45.180
at t equals minus a google
or t equals minus infinity.

00:06:45.180 --> 00:06:47.130
So you don't necessarily
have a start point.

00:06:47.130 --> 00:06:49.970
But if we assume t equals 0 as
a start point, you could say

00:06:49.970 --> 00:06:52.810
that they both started there.

00:06:52.810 --> 00:06:56.720
Then at-- let me pick a new
color that I haven't used

00:06:56.720 --> 00:07:01.650
yet --then at t is equal to
pi over 2-- Where are we?

00:07:01.650 --> 00:07:03.910
We're at minus 3, 0.

00:07:03.910 --> 00:07:04.890
We get all the way-- Oh.

00:07:04.890 --> 00:07:07.330
That's the same color
I used before.

00:07:07.330 --> 00:07:08.170
Let me see this.

00:07:08.170 --> 00:07:10.490
Let me do this color.

00:07:10.490 --> 00:07:12.080
We're here.

00:07:12.080 --> 00:07:15.730
Now notice: in the first one,
when we went from t equals 0

00:07:15.730 --> 00:07:20.340
to t equals pi over 2, we
went from here to there.

00:07:20.340 --> 00:07:23.700
We went kind of a quarter of
the way around the ellipse.

00:07:23.700 --> 00:07:26.720
But now when we went from
t equals zero to pi over

00:07:26.720 --> 00:07:28.280
2, where did we go?

00:07:28.280 --> 00:07:28.430
we?

00:07:28.430 --> 00:07:30.530
Went halfway around
the ellipse.

00:07:30.530 --> 00:07:36.240
We went all the way from there,
all the way over there.

00:07:39.190 --> 00:07:42.550
And likewise, when we went from
t equals pi over 2 to t equals

00:07:42.550 --> 00:07:46.320
pi with this set of parametric
equations, we went another

00:07:46.320 --> 00:07:47.510
quarter of the ellipse.

00:07:47.510 --> 00:07:50.100
We went from there to there.

00:07:50.100 --> 00:07:52.790
But here, when we go from t
equals pi ever 2 to t equals

00:07:52.790 --> 00:07:56.910
pi, we go all of this way.

00:07:56.910 --> 00:08:01.100
We go back to the beginning
part of our ellipse.

00:08:01.100 --> 00:08:05.450
So what you see is that this
set of parametric equations

00:08:05.450 --> 00:08:08.565
has the exact same shape of
its path as this set of

00:08:08.565 --> 00:08:09.660
parametric equations.

00:08:09.660 --> 00:08:13.640
Except it's going around it
at twice as fast of a rate.

00:08:13.640 --> 00:08:19.270
For every time when t increases
by pi over 2 here, we go by--

00:08:19.270 --> 00:08:22.210
we kind of go a quarter
way around the ellipse.

00:08:22.210 --> 00:08:25.830
But when t increases by pi
over 2 here, we go halfway

00:08:25.830 --> 00:08:27.290
around the ellipse.

00:08:27.290 --> 00:08:29.870
So the thing to realize-- and I
know I've touched on this

00:08:29.870 --> 00:08:33.000
before --is that even though
both of these sets of

00:08:33.000 --> 00:08:35.780
parametric equations, when you
do the algebra, they can

00:08:35.780 --> 00:08:39.040
kind of be converted
into this shape.

00:08:39.040 --> 00:08:44.470
You lose the information about
where our particle is as it's

00:08:44.470 --> 00:08:47.050
rotating around the ellipse or
how fast it's rotating

00:08:47.050 --> 00:08:47.670
around the ellipse.

00:08:47.670 --> 00:08:50.580
And that's why you need
these parametric equations.

00:08:50.580 --> 00:08:52.590
We can even set up a parametric
equation that goes in

00:08:52.590 --> 00:08:54.120
the other direction.

00:08:54.120 --> 00:08:57.730
Instead of having these-- and I
encourage you to play with that

00:08:57.730 --> 00:08:59.740
--but if you instead of this,
if you just put a minus

00:08:59.740 --> 00:09:01.480
sign right here.

00:09:01.480 --> 00:09:04.500
Cosine of minus t and
2 sine of minus t.

00:09:04.500 --> 00:09:07.610
Instead of going in that
direction, it would

00:09:07.610 --> 00:09:08.860
go in this direction.

00:09:08.860 --> 00:09:11.430
It would go in a
clockwise direction.

00:09:11.430 --> 00:09:14.750
So one thing that you've
probably been thinking from the

00:09:14.750 --> 00:09:18.620
beginning is OK, I was able to
go from my parametric equations

00:09:18.620 --> 00:09:22.210
to this equation of ellipse
in terms of just x and y.

00:09:22.210 --> 00:09:24.020
Can you go back the other way?

00:09:24.020 --> 00:09:26.330
Could you go from this to this?

00:09:26.330 --> 00:09:30.300
And, I think you might realize
now, that the answer is no.

00:09:30.300 --> 00:09:33.400
Because there's no way, just
with the information that

00:09:33.400 --> 00:09:36.560
you're given here, to know
that you should go to this

00:09:36.560 --> 00:09:40.230
parametric equation or this
parametric equation or any of

00:09:40.230 --> 00:09:42.100
an infinite number of
parametric equations.

00:09:42.100 --> 00:09:50.540
I mean anything of the form x
is equal to 3 cosine of really

00:09:50.540 --> 00:09:56.450
anything times t and y is equal
to 3 times cosine of-- As long

00:09:56.450 --> 00:09:58.590
as it's the same anything-- I
drew the two squiggly

00:09:58.590 --> 00:10:00.240
marks the same.

00:10:00.240 --> 00:10:03.250
--as long as these two things
are the same, then you

00:10:03.250 --> 00:10:07.620
will have-- they'll both
converge to this shape.

00:10:07.620 --> 00:10:09.760
So if you have the shape, you
don't know what parametric

00:10:09.760 --> 00:10:10.930
equation you can go back to.

00:10:10.930 --> 00:10:13.380
You could make up one,
but you don't know which

00:10:13.380 --> 00:10:14.970
one it'll go back to.

00:10:14.970 --> 00:10:17.560
And just to kind of hit the
point home of kind of making

00:10:17.560 --> 00:10:18.800
up a parametric equation.

00:10:18.800 --> 00:10:21.460
Sometimes that's asked of you.

00:10:21.460 --> 00:10:23.780
So let's just do a
very simple example.

00:10:23.780 --> 00:10:27.570
Let's start with a
normal function of x.

00:10:27.570 --> 00:10:31.440
So let's say that we have
the equation y is equal

00:10:31.440 --> 00:10:33.990
to x squared plus x.

00:10:33.990 --> 00:10:35.690
I just made it up.

00:10:35.690 --> 00:10:38.340
And let's say that you were
asked to turn this into

00:10:38.340 --> 00:10:40.220
a parametric equation.

00:10:40.220 --> 00:10:43.500
And this is often a very hard
thing for people because there

00:10:43.500 --> 00:10:44.650
is not one right answer.

00:10:44.650 --> 00:10:47.260
You can turn into an arbitrary
number of parametric equations.

00:10:47.260 --> 00:10:49.540
I just may do something
really crazy and arbitrary.

00:10:49.540 --> 00:10:53.550
I could say x is equal-- When I
have y defined explicitly in

00:10:53.550 --> 00:10:56.170
terms of x like this, I can
make x into anything

00:10:56.170 --> 00:10:56.660
in terms of t.

00:10:56.660 --> 00:11:01.620
I could say it's cosine of t
minus the natural log of t.

00:11:01.620 --> 00:11:02.620
I just made that up.

00:11:02.620 --> 00:11:04.380
That's just a random thing.

00:11:04.380 --> 00:11:08.550
But then if x is this, then y
is going to be equal to-- I

00:11:08.550 --> 00:11:14.300
just substitute this back in
--cosine of t minus ln of t

00:11:14.300 --> 00:11:21.240
squared plus cosine
of t minus ln of t.

00:11:21.240 --> 00:11:21.800
And we're done.

00:11:21.800 --> 00:11:24.550
We just converted this into a
parametric equation or a set

00:11:24.550 --> 00:11:25.730
of parametric equations.

00:11:25.730 --> 00:11:32.660
I could've also just
written x is equal to t.

00:11:32.660 --> 00:11:34.010
And then what would y equal?

00:11:34.010 --> 00:11:40.310
y would be equal to
t squared plus t.

00:11:40.310 --> 00:11:42.040
You might say well what's
the difference between this

00:11:42.040 --> 00:11:44.240
parametric equation and
this parametric equation?

00:11:44.240 --> 00:11:47.250
Well, they're both
going to have the same

00:11:47.250 --> 00:11:48.870
shape of their paths.

00:11:48.870 --> 00:11:50.790
It's going to be something
like a parabola.

00:11:50.790 --> 00:11:54.520
But the rate and the direction
with which they progress

00:11:54.520 --> 00:11:57.290
on those paths will be
very very different.

00:11:57.290 --> 00:11:58.500
It's actually a very
interesting thing

00:11:58.500 --> 00:11:58.920
to think about.

00:11:58.920 --> 00:12:01.860
There are some paths that you
could take where-- Let's say

00:12:01.860 --> 00:12:04.610
that the-- You could set
up a parametric equation.

00:12:04.610 --> 00:12:07.080
Let's say that the shape is--
You know everything we've done

00:12:07.080 --> 00:12:09.540
so far, we've always been going
in one direction, but you could

00:12:09.540 --> 00:12:12.350
have a-- There are scenarios--
And maybe if I have time,

00:12:12.350 --> 00:12:13.610
I'll make a video.

00:12:13.610 --> 00:12:17.410
And this isn't what-- Let's say
the shape of the path is some

00:12:17.410 --> 00:12:21.060
type of, I dunno, let's say
it's a circle of some kind.

00:12:21.060 --> 00:12:23.240
I'm just going to--
That's not this.

00:12:23.240 --> 00:12:25.130
I'm doing a completely
different example right now.

00:12:25.130 --> 00:12:28.750
I'm just kind of-- Sometimes,
with the ellipses we had paths

00:12:28.750 --> 00:12:32.400
that went counterclockwise and
then paths that went clockwise.

00:12:32.400 --> 00:12:35.480
You can also have paths that
kind of isolate in between,

00:12:35.480 --> 00:12:37.690
move around, move back and
forth along the thing.

00:12:37.690 --> 00:12:39.470
So there's all sorts of
parametric equations

00:12:39.470 --> 00:12:40.580
that you can define.

00:12:40.580 --> 00:12:43.490
And you can say if t
equals from this to this,

00:12:43.490 --> 00:12:45.330
you'll use this set of
parametric equations.

00:12:45.330 --> 00:12:46.980
If it's another set of
t's, use another one.

00:12:46.980 --> 00:12:49.450
So there's all sorts of crazy
things you can do to say what

00:12:49.450 --> 00:12:51.510
happens as you move
along the path.

00:12:51.510 --> 00:12:54.190
So the difference-- Not that
this is the path of that.

00:12:54.190 --> 00:12:55.820
This is actually
more of a parabola.

00:12:55.820 --> 00:12:58.130
But the difference between
this and this is how you

00:12:58.130 --> 00:13:01.000
move along the shape.

00:13:01.000 --> 00:13:03.820
Anyway, I hope you found
that vaguely useful.


Parametric equations 3

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:04.220
Let's see if we can remove the
parameter t from a slightly

00:00:04.220 --> 00:00:06.180
more interesting example.

00:00:06.180 --> 00:00:14.130
So let's say that x is equal
to 3 times the cosine of t.

00:00:14.130 --> 00:00:19.570
And y is equal to 2
times the sine of t.

00:00:19.570 --> 00:00:22.470
We can try to remove the
parameter the same way we did

00:00:22.470 --> 00:00:25.970
in the previous video, where we
can solve for t in terms of

00:00:25.970 --> 00:00:28.120
either x or y and then
substitute back in.

00:00:28.120 --> 00:00:29.300
And I'll do that.

00:00:29.300 --> 00:00:31.575
But I want to do that first,
just to show you that it kind

00:00:31.575 --> 00:00:34.740
of leads to a hairy or
an unintuitive answer.

00:00:34.740 --> 00:00:37.220
So if we solve for--
let's solve for t here.

00:00:37.220 --> 00:00:39.940
We could do it either one,
they're equally complex.

00:00:39.940 --> 00:00:42.940
So if we solve for t here,
we would say divide

00:00:42.940 --> 00:00:44.060
both sides by 2.

00:00:44.060 --> 00:00:48.970
You'd get y over 2 is
equal to sine of t.

00:00:48.970 --> 00:00:52.150
And then you would take the
arcsine of both sides, or the

00:00:52.150 --> 00:00:56.730
inverse sine of both sides, and
you would get-- I like writing

00:00:56.730 --> 00:01:01.710
arcsine, because inverse sine,
people often confuse it with an

00:01:01.710 --> 00:01:04.600
exponent, taking it to
the negative 1 power.

00:01:04.600 --> 00:01:08.510
Arcsine of y over
2 is equal to t.

00:01:08.510 --> 00:01:10.050
Actually, let me do that
little aside there.

00:01:10.050 --> 00:01:12.630
I should probably do it at the
trigonometry playlist, but it's

00:01:12.630 --> 00:01:13.710
a good thing to hit home.

00:01:13.710 --> 00:01:16.570
Because I think
people get confused.

00:01:16.570 --> 00:01:24.840
So arcsine of anything,
let's say, y.

00:01:24.840 --> 00:01:29.910
The other way of writing
that is sine minus 1 of y.

00:01:29.910 --> 00:01:31.320
These two things are
equivalent, when

00:01:31.320 --> 00:01:32.420
they're normally used.

00:01:32.420 --> 00:01:36.200
But I don't like using this
notation most of the time,

00:01:36.200 --> 00:01:37.890
because it can be ambiguous.

00:01:37.890 --> 00:01:47.150
This could mean sine of y to
the negative 1 power, which

00:01:47.150 --> 00:01:50.990
equals 1 over sine of y.

00:01:50.990 --> 00:01:55.540
And arcsine and this are
definitely not the same thing.

00:01:55.540 --> 00:01:57.750
So you want to be very careful
there to make sure that you

00:01:57.750 --> 00:01:59.750
don't get confused when someone
writes an inverse

00:01:59.750 --> 00:02:00.560
sine like this.

00:02:00.560 --> 00:02:02.365
But they're not actually
taking sine of y to

00:02:02.365 --> 00:02:03.700
the negative 1 power.

00:02:03.700 --> 00:02:07.970
On the other hand, if someone
were to write sine squared of

00:02:07.970 --> 00:02:13.270
y, this is unambiguously the
same thing as sine

00:02:13.270 --> 00:02:15.230
of y squared.

00:02:15.230 --> 00:02:18.050
In fact, I wish this was the
more conventional notation

00:02:18.050 --> 00:02:20.770
because it wouldn't make people
think, oh, 2 and minus 1

00:02:20.770 --> 00:02:23.860
there, and of course, that's
just sine of y squared.

00:02:23.860 --> 00:02:25.380
So it can be very ambiguous.

00:02:25.380 --> 00:02:27.950
And of course, if this was a
negative, this would be a minus

00:02:27.950 --> 00:02:34.480
2, and then this really would
be 1 over sine of y squared.

00:02:34.480 --> 00:02:37.930
That's why, just a long-winded
way of explaining why I

00:02:37.930 --> 00:02:41.120
wrote arcsine, instead of
inverse sine right there.

00:02:41.120 --> 00:02:42.830
Needless to say, let's
get back to the problem.

00:02:42.830 --> 00:02:45.090
So we've solved for
t in terms of y.

00:02:45.090 --> 00:02:47.410
Now we can substitute
back here.

00:02:47.410 --> 00:02:49.780
And we've got an expression
for x in terms of y.

00:02:49.780 --> 00:02:56.135
So we get x is equal to 3
times the cosine of t.

00:02:56.135 --> 00:02:59.510
But we just solved for t. t
is this thing right here.

00:02:59.510 --> 00:03:07.030
So it's the cosine of
arcsine of y over 2.

00:03:07.030 --> 00:03:10.380
And we have eliminated the
parameter, but this is a very

00:03:10.380 --> 00:03:12.680
non-intuitive equation.

00:03:12.680 --> 00:03:13.710
We could have done
the other way.

00:03:13.710 --> 00:03:16.190
We could have solved for y in
terms of x and we would have

00:03:16.190 --> 00:03:18.200
gotten the sine of
the arccosine.

00:03:18.200 --> 00:03:21.840
It would have been equally
hairy or non-intuitive.

00:03:21.840 --> 00:03:24.400
But either way, we did remove
the parameters so I guess

00:03:24.400 --> 00:03:26.720
we could mildly pat
ourselves on the back.

00:03:26.720 --> 00:03:28.110
But that's not the
purpose of this video.

00:03:28.110 --> 00:03:30.280
The purpose of this video is to
see if there's any way we can

00:03:30.280 --> 00:03:35.150
remove the parameter that leads
to a more intuitive equation

00:03:35.150 --> 00:03:36.790
involving x and y.

00:03:36.790 --> 00:03:39.250
And what we're going to do is,
I guess you can call it a bit

00:03:39.250 --> 00:03:41.140
of a trick, but it's something
that shows up a lot.

00:03:41.140 --> 00:03:42.850
Especially when you deal
with polar coordinates.

00:03:42.850 --> 00:03:45.250
And you might want to watch
my polar coordinate videos,

00:03:45.250 --> 00:03:47.545
because this essentially
touches on that.

00:03:47.545 --> 00:03:52.800
But if I said-- let me rewrite
them. x is equal to 3 cosine of

00:03:52.800 --> 00:03:58.940
t and y is equal
to 2 sine of t.

00:03:58.940 --> 00:04:01.780
So what we can do is
just think, well, how

00:04:01.780 --> 00:04:02.620
can we write this?

00:04:02.620 --> 00:04:04.110
And you know, cosine
of t and [? kind ?]

00:04:04.110 --> 00:04:05.270
of t, how can we relate them?

00:04:05.270 --> 00:04:07.460
And the first thing that comes
to my mind is just the unit

00:04:07.460 --> 00:04:10.300
circle, or to some degree, the
most basic of all of the

00:04:10.300 --> 00:04:11.890
trigonometric identities.

00:04:11.890 --> 00:04:19.320
And that is that the cosine
squared of t plus the sine

00:04:19.320 --> 00:04:21.980
squared of t is equal to 1.

00:04:21.980 --> 00:04:24.210
This comes from
the unit circle.

00:04:24.210 --> 00:04:26.000
I explained it in the unit
circle video, and that's

00:04:26.000 --> 00:04:28.840
because the equation for the
unit circle is x squared plus

00:04:28.840 --> 00:04:30.330
y squared is equal to 1.

00:04:30.330 --> 00:04:33.300
The cosine of the angle is the
x coordinate, the sine of the

00:04:33.300 --> 00:04:35.180
angle is the y coordinate,
and so on and so forth.

00:04:35.180 --> 00:04:36.100
But this is our trig identity.

00:04:36.100 --> 00:04:37.690
You don't have to think about
it too much right now.

00:04:37.690 --> 00:04:40.550
Just, I guess, know that it's
true and watch some of the

00:04:40.550 --> 00:04:43.330
other videos if you want
it proven that it's true.

00:04:43.330 --> 00:04:46.600
But if we can somehow replace
this cosine squared with some

00:04:46.600 --> 00:04:50.880
expression in x, and replace
the sine or the sine squared

00:04:50.880 --> 00:04:53.400
with some expression of
y, we'd be done, right?

00:04:53.400 --> 00:04:55.490
And then we would
have it equaling 1.

00:04:55.490 --> 00:04:56.680
And that shouldn't be too hard.

00:04:56.680 --> 00:04:58.160
We can rewrite this.

00:04:58.160 --> 00:05:02.040
We can set cosine of t equal to
something in x, and we can set

00:05:02.040 --> 00:05:03.780
sine of t equal in
something in y.

00:05:03.780 --> 00:05:05.400
So let's do that.

00:05:05.400 --> 00:05:08.100
We divide both sides
of the equation by 3.

00:05:08.100 --> 00:05:12.165
You get x over 3 is
equal to cosine of t.

00:05:12.165 --> 00:05:16.400
And if you divide both sides of
this equation by 2, you get y

00:05:16.400 --> 00:05:20.590
over 2 is equal to sine of t.

00:05:20.590 --> 00:05:25.120
And then we can use this
trigonometric identity.

00:05:25.120 --> 00:05:28.190
Instead of the cosine of t,
we can substitute x over 3.

00:05:28.190 --> 00:05:31.780
Instead of the sine of t, we
can substitute y over 2.

00:05:31.780 --> 00:05:39.810
And you get x over 3 squared--
that's that, right there,

00:05:39.810 --> 00:05:45.490
that's just cosine of t
squared-- plus y over 2

00:05:45.490 --> 00:05:51.730
squared-- that's just sine of t
squared-- is equal to 1.

00:05:51.730 --> 00:05:57.190
And now this is starting to
look a lot better than this.

00:05:57.190 --> 00:05:59.330
This, I have no
idea what this is.

00:05:59.330 --> 00:06:01.850
But hopefully if you've watched
the conic section videos, you

00:06:01.850 --> 00:06:04.110
can already recognize that this
is starting to look

00:06:04.110 --> 00:06:05.660
like an ellipse.

00:06:05.660 --> 00:06:06.870
We can simplify
it a little bit.

00:06:06.870 --> 00:06:11.010
We could say this is equal to x
squared over 9 plus y squared

00:06:11.010 --> 00:06:13.070
over 4 is equal to 1.

00:06:13.070 --> 00:06:16.380
And if we were to graph this
ellipse-- we will actually

00:06:16.380 --> 00:06:19.220
graph it-- we get--
let me draw my axis.

00:06:19.220 --> 00:06:21.910
I'm using this blue color
a little bit too much,

00:06:21.910 --> 00:06:23.920
it's getting monotonous.

00:06:23.920 --> 00:06:26.530
OK, let me use the purple.

00:06:26.530 --> 00:06:29.210
So that's our x-axis.

00:06:29.210 --> 00:06:29.860
That's our y-axis.

00:06:32.820 --> 00:06:35.610
The major axis is in the
x direction because the

00:06:35.610 --> 00:06:38.010
denominator here is
larger than that one.

00:06:38.010 --> 00:06:42.150
And it's the semi-major
radius-- this is going

00:06:42.150 --> 00:06:44.240
to be the square root
of this, it's 3.

00:06:44.240 --> 00:06:47.360
1, 2, 3 in that direction.

00:06:47.360 --> 00:06:49.720
1, 2, 3.

00:06:49.720 --> 00:06:51.470
I know I'm centered in
0, because neither of

00:06:51.470 --> 00:06:52.140
these are shifted.

00:06:52.140 --> 00:06:54.540
You should watch the conic
section videos if this

00:06:54.540 --> 00:06:56.160
sounds unfamiliar to you.

00:06:56.160 --> 00:06:58.540
And the semi-minor radius
is the square root

00:06:58.540 --> 00:07:00.210
of 4, so that's 2.

00:07:00.210 --> 00:07:02.665
So they get 1, 2.

00:07:02.665 --> 00:07:05.410
And 1, 2.

00:07:05.410 --> 00:07:09.570
Let me see if I can
draw this ellipse.

00:07:09.570 --> 00:07:17.020
So it looks something
like that.

00:07:17.020 --> 00:07:19.870
There you go.

00:07:19.870 --> 00:07:24.860
So just like that, by
eliminating the parameter t, we

00:07:24.860 --> 00:07:28.170
got this equation in a form
that we immediately were able

00:07:28.170 --> 00:07:29.360
to recognize as ellipse.

00:07:29.360 --> 00:07:31.970
When I just look at that,
unless you deal with parametric

00:07:31.970 --> 00:07:34.680
equations, or maybe polar
coordinates a lot, it's not

00:07:34.680 --> 00:07:37.170
obvious that this is the
parametric equation

00:07:37.170 --> 00:07:38.110
for an ellipse.

00:07:38.110 --> 00:07:40.050
But this, once you learn
about conic sections,

00:07:40.050 --> 00:07:41.100
is pretty clear.

00:07:41.100 --> 00:07:41.410
It's an ellipse.

00:07:41.410 --> 00:07:43.490
And it's easy to
draw that ellipse.

00:07:43.490 --> 00:07:48.400
But in removing the t and from
going from these equations up

00:07:48.400 --> 00:07:53.090
here, and from going from that
to that, like in the last

00:07:53.090 --> 00:07:55.050
video, we lost information.

00:07:55.050 --> 00:07:58.330
We lost, one, what is the
direction that we move

00:07:58.330 --> 00:08:00.850
in as t increases?

00:08:00.850 --> 00:08:04.180
And we also don't know what
point on this ellipse we

00:08:04.180 --> 00:08:07.230
are at any given time, t.

00:08:07.230 --> 00:08:10.750
So to do that, let's
make our little table.

00:08:15.710 --> 00:08:17.525
So let's take some values of t.

00:08:20.370 --> 00:08:26.510
So we'll make a little
table. t, x, and y.

00:08:26.510 --> 00:08:27.960
It's good to pick values of t.

00:08:27.960 --> 00:08:30.620
Remember-- let me rewrite the
equations again, so we didn't

00:08:30.620 --> 00:08:35.250
lose it-- x was equal to 3
cosine of t, and y is

00:08:35.250 --> 00:08:38.010
equal to 2 sine of t.

00:08:38.010 --> 00:08:41.250
It's good to take values of t
where it's easy to figure out

00:08:41.250 --> 00:08:43.900
what the cosine and sine are,
and without using a calculator.

00:08:43.900 --> 00:08:47.130
We're assuming the t is in
radiance, just for simplicity.

00:08:47.130 --> 00:08:49.560
So let's pick t is equal to 0.

00:08:49.560 --> 00:08:51.550
t is equal to pi over 2.

00:08:51.550 --> 00:08:53.240
That's 90 degrees in degrees.

00:08:53.240 --> 00:08:55.650
And t is equal to pi.

00:08:55.650 --> 00:08:57.750
And so what is x when
t is equal to 0?

00:08:57.750 --> 00:09:03.980
Well, cosine of 0 is
1 times 3, that's 3.

00:09:03.980 --> 00:09:06.810
What's x, when t is
just pi over 2?

00:09:06.810 --> 00:09:10.490
Cosine of pi over 2 is 0.

00:09:10.490 --> 00:09:13.780
0 times 3 is 0.

00:09:13.780 --> 00:09:17.100
And what's x equal when
t is equal to pi?

00:09:17.100 --> 00:09:19.300
Cosine of pi is minus 1.

00:09:19.300 --> 00:09:22.380
Minus 1 times 3 is minus 3.

00:09:22.380 --> 00:09:23.750
Fair enough.

00:09:23.750 --> 00:09:25.770
Now let's do the y's.

00:09:25.770 --> 00:09:27.740
When t is 0 what is y?

00:09:27.740 --> 00:09:29.060
Sine is 0, 0.

00:09:29.060 --> 00:09:31.480
So 2 times 0 is 0.

00:09:31.480 --> 00:09:34.890
When t is pi over 2,
sine of pi over 2 is 1.

00:09:34.890 --> 00:09:37.380
1 times 2 is 2.

00:09:37.380 --> 00:09:41.570
And when t is pi, sine of
pi-- that's sine of 180

00:09:41.570 --> 00:09:42.770
degrees-- that's 0.

00:09:42.770 --> 00:09:45.420
2 times 0 is 0.

00:09:45.420 --> 00:09:46.720
So let's plot these points.

00:09:46.720 --> 00:09:49.410
When time is 0, we're
at the point 3, 0.

00:09:49.410 --> 00:09:53.040
So 3, 0-- 3, 0 is right there.

00:09:53.040 --> 00:09:57.090
This is t equals 0.

00:09:57.090 --> 00:10:01.230
When t increases by pi over 2,
or if this was seconds, pi over

00:10:01.230 --> 00:10:04.380
2 seconds is like 1.7
something seconds.

00:10:04.380 --> 00:10:10.030
So at t equals pi over 2,
we're at the point 0, 2.

00:10:10.030 --> 00:10:12.700
We're right over here.

00:10:12.700 --> 00:10:16.380
So this is at t is
equal to pi over 2.

00:10:16.380 --> 00:10:20.820
And then when t increases a
little bit more-- when we're

00:10:20.820 --> 00:10:24.820
at t is equal to pi-- we're
at the point minus 3, 0.

00:10:24.820 --> 00:10:26.760
We're here.

00:10:26.760 --> 00:10:31.160
So this is t is equal to
pi or, you know, we could

00:10:31.160 --> 00:10:35.090
write 3.14159 seconds.

00:10:35.090 --> 00:10:37.080
3.14 seconds.

00:10:37.080 --> 00:10:39.830
And actually, you know, I want
to make the point, t does not

00:10:39.830 --> 00:10:41.620
have to be time, and we don't
have to be dealing

00:10:41.620 --> 00:10:42.090
with seconds.

00:10:42.090 --> 00:10:43.360
But I like to think
about it that way.

00:10:43.360 --> 00:10:46.700
I like to think about, maybe
this is describing some object

00:10:46.700 --> 00:10:50.230
in orbit around, I don't
know, something else.

00:10:50.230 --> 00:10:51.380
So now we know the direction.

00:10:51.380 --> 00:10:56.650
As t increased from 0 to pi
over 2 to pi, we went this way.

00:10:56.650 --> 00:10:58.000
We went counterclockwise.

00:10:58.000 --> 00:11:02.280
So the direction of t's
parametric equations

00:11:02.280 --> 00:11:04.090
is in that direction.

00:11:04.090 --> 00:11:07.150
And you might be saying,
Sal, you know, why'd we

00:11:07.150 --> 00:11:08.450
have to do 3 points?

00:11:08.450 --> 00:11:10.110
We could have just done
2, and made a line.

00:11:10.110 --> 00:11:11.850
If we just had that point and
that point, you might have

00:11:11.850 --> 00:11:14.680
immediately said, oh, we
went from there to there.

00:11:14.680 --> 00:11:16.500
But that really wouldn't
have been enough.

00:11:16.500 --> 00:11:19.150
Because maybe we got from
here to there by going

00:11:19.150 --> 00:11:20.470
the other way around.

00:11:20.470 --> 00:11:23.220
So giving that third point lets
us know that the direction is

00:11:23.220 --> 00:11:25.610
definitely counterclockwise.

00:11:25.610 --> 00:11:31.200
And so what happens if we just
take t from 0 to infinity?

00:11:31.200 --> 00:11:35.470
What happens if we bound t?
t is greater than 0 and

00:11:35.470 --> 00:11:37.200
less than infinity.

00:11:37.200 --> 00:11:39.130
Well, we're just going
to keep going around

00:11:39.130 --> 00:11:40.820
this ellipse forever.

00:11:40.820 --> 00:11:41.760
Multiple times.

00:11:41.760 --> 00:11:44.600
Keep writing over and
over, infinite times.

00:11:44.600 --> 00:11:47.480
If we went from minus infinity
to infinity, then we would

00:11:47.480 --> 00:11:50.480
have always been doing it, I
guess is the way to put it.

00:11:50.480 --> 00:11:54.590
Or if we just wanted to trace
this out once, we could go from

00:11:54.590 --> 00:11:58.440
t is less than or equal to-- or
t is greater than

00:11:58.440 --> 00:11:59.370
or equal to 0.

00:11:59.370 --> 00:12:03.900
All the way to t is less
than or equal to 2 pi.

00:12:03.900 --> 00:12:07.090
And in this situation,
t really is the angle

00:12:07.090 --> 00:12:08.010
that we're tracing out.

00:12:08.010 --> 00:12:11.220
If we were to think of this
in polar coordinates, this

00:12:11.220 --> 00:12:13.010
is t at any given time.

00:12:13.010 --> 00:12:14.570
And I just thought I would
throw that out there.

00:12:14.570 --> 00:12:17.700
It isn't always, but in
this case it really is.

00:12:17.700 --> 00:12:21.510
When you go from 0 to 2 pi
radius, you've made 1 circle.

00:12:21.510 --> 00:12:24.280
But this is about parametric
equations and not trigonometry.

00:12:24.280 --> 00:12:26.090
So I don't want to focus
too much on that.

00:12:26.090 --> 00:12:27.850
But anyway, that was neat.

00:12:27.850 --> 00:12:30.340
When we started with this,
if I just showed you those

00:12:30.340 --> 00:12:32.330
parametric equations, you'd
have no idea what

00:12:32.330 --> 00:12:33.040
that looks like.

00:12:33.040 --> 00:12:36.470
But by recognizing the trig
identity, we were able to

00:12:36.470 --> 00:12:38.930
simplify it to an ellipse,
draw the ellipse.

00:12:38.930 --> 00:12:40.880
And then by plotting a couple
of points, we were able to

00:12:40.880 --> 00:12:45.000
figure out the direction at
which, if this was describing

00:12:45.000 --> 00:12:47.380
a particle in motion, the
direction in which that

00:12:47.380 --> 00:12:49.140
particle was actually moving.

00:12:49.140 --> 00:12:51.530
Anyway, hope you enjoyed that.


Basic multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.970 --> 00:00:02.890
Let's learn to multiply.

00:00:02.890 --> 00:00:07.990
M U L T I P L Y.

00:00:07.990 --> 00:00:10.960
And the best way I think to do
anything is just to actually do

00:00:10.960 --> 00:00:14.040
some examples, and then talk
through the examples, and try

00:00:14.040 --> 00:00:15.690
to figure out what they mean.

00:00:15.690 --> 00:00:21.330
In my first example
I have 2 times 3.

00:00:21.330 --> 00:00:24.020
By now you probably
know what 2 plus 3 is.

00:00:27.300 --> 00:00:28.470
That's equal to 5.

00:00:28.470 --> 00:00:30.860
And if you need a bit of a
review you could think of

00:00:30.860 --> 00:00:34.840
if I had 2-- I don't
know-- 2 magenta--

00:00:34.840 --> 00:00:36.600
this color-- cherries.

00:00:36.600 --> 00:00:41.880
And I wanted to add
to it 3 blueberries.

00:00:41.880 --> 00:00:44.950
How many total pieces of
fruit do I now have?

00:00:44.950 --> 00:00:47.920
And you'd say, oh,
1, 2, 3, 4, 5.

00:00:47.920 --> 00:00:55.460
Or likewise, if I had our
number line, and you probably

00:00:55.460 --> 00:00:57.710
don't need this review,
but it never hurts.

00:00:57.710 --> 00:01:01.040
Never hurts to
reinforce the concept.

00:01:01.040 --> 00:01:09.600
And it this is 0,
1, 2, 3, 4, 5.

00:01:09.600 --> 00:01:15.130
If you're sitting 2 to the
right of 0 and in general,

00:01:15.130 --> 00:01:17.580
when we go positive
we go to the right.

00:01:17.580 --> 00:01:20.870
And if you were to add 3
to it, you would move

00:01:20.870 --> 00:01:22.210
3 spaces to the right.

00:01:22.210 --> 00:01:26.040
So if I said, if I just
moved over 3 to the

00:01:26.040 --> 00:01:27.190
right, where do I end up?

00:01:27.190 --> 00:01:30.120
1, 2, 3.

00:01:30.120 --> 00:01:31.450
I end up at 5.

00:01:31.450 --> 00:01:34.950
So either way, you understand
that 2 plus 3 is equal to 5.

00:01:34.950 --> 00:01:37.790
So what is 2 times 3?

00:01:37.790 --> 00:01:41.550
An easy way to think about
multiplication or timesing

00:01:41.550 --> 00:01:45.560
something is it's just a
simple way of doing addition

00:01:45.560 --> 00:01:47.160
over and over again.

00:01:47.160 --> 00:01:50.280
So that you means is, and
it's a little tricky.

00:01:50.280 --> 00:01:52.230
You're not going to add 2 to 3.

00:01:52.230 --> 00:01:54.220
You're going to add-- and
there's actually two

00:01:54.220 --> 00:01:55.770
ways to think about it.

00:01:55.770 --> 00:01:59.375
You're going to add 2
to itself three times.

00:01:59.375 --> 00:02:00.685
Now what does that mean?

00:02:00.685 --> 00:02:07.730
Well, it means you're going
to say 2 plus 2 plus 2.

00:02:07.730 --> 00:02:09.070
Now where did the 3 go?

00:02:09.070 --> 00:02:12.940
Well, how many 2's
do we have here?

00:02:12.940 --> 00:02:16.770
Let's see, I have-- this
is one 2, I have two

00:02:16.770 --> 00:02:18.930
2's, I have three 2's.

00:02:18.930 --> 00:02:21.550
I'm counting the numbers here
the same way that I counted

00:02:21.550 --> 00:02:22.340
blueberries up here.

00:02:22.340 --> 00:02:23.970
I had 1, 2, 3 blueberries.

00:02:23.970 --> 00:02:26.840
I have one, two, three 2's.

00:02:26.840 --> 00:02:33.658
So this three tells me how
many 2's I'm going to have.

00:02:33.658 --> 00:02:36.230
So what's 2 times 3?

00:02:36.230 --> 00:02:41.060
Well, I took 2 and I added
to itself three times.

00:02:41.060 --> 00:02:43.180
So 2 plus 2 is 4.

00:02:43.180 --> 00:02:46.265
4 plus 2 is equal to 6.

00:02:46.265 --> 00:02:48.340
Now that's only one way
to think about it.

00:02:48.340 --> 00:02:50.900
The other way we could have
thought about this is we

00:02:50.900 --> 00:02:55.120
could've said, instead of
having 2 added to itself three

00:02:55.120 --> 00:02:59.020
times, we could've added
3 to itself two times.

00:02:59.020 --> 00:03:01.530
And I know it's maybe becoming
a little bit confusing, but the

00:03:01.530 --> 00:03:04.050
more practice you do it'll
make a little sense.

00:03:04.050 --> 00:03:06.950
So this statement up
here, let me rewrite it.

00:03:06.950 --> 00:03:09.830
2 times 3.

00:03:09.830 --> 00:03:15.780
It could also be rewritten
as 3 two times.

00:03:15.780 --> 00:03:19.880
So 3 plus 3.

00:03:19.880 --> 00:03:22.320
And once again, you're
like, where did this 2 go?

00:03:22.320 --> 00:03:25.440
You know, I had 2 times 3 and
whenever you do addition you

00:03:25.440 --> 00:03:28.130
see I have 2-- oh, I don't
know these-- well, I said

00:03:28.130 --> 00:03:30.470
cherries, but they could be
raspberries or anything.

00:03:30.470 --> 00:03:33.310
And then I had two things, I
have three things and the 2

00:03:33.310 --> 00:03:34.175
and the 3 never disappear.

00:03:34.175 --> 00:03:36.620
And I add them
together, I get 5.

00:03:36.620 --> 00:03:38.720
But here I'm saying that
2 times 3 is the same

00:03:38.720 --> 00:03:40.080
thing as 3 plus 3.

00:03:40.080 --> 00:03:41.210
Where did the 2 go?

00:03:41.210 --> 00:03:44.620
2 in this case, in this
scenario, is telling me

00:03:44.620 --> 00:03:48.580
how many times I'm going
to add 3 to itself.

00:03:48.580 --> 00:03:52.420
But what's interesting is,
regardless of which way I

00:03:52.420 --> 00:03:56.950
interpret 2 times 3, I can
interpret it as 2 plus 2

00:03:56.950 --> 00:04:00.880
plus 2 or adding 2 to
itself three times.

00:04:00.880 --> 00:04:04.390
I can interpret it that way or
I can interpret it as adding

00:04:04.390 --> 00:04:07.100
3 to itself two times.

00:04:07.100 --> 00:04:08.940
But notice, I get
the same answer.

00:04:08.940 --> 00:04:11.330
What's 3 plus 3?

00:04:11.330 --> 00:04:14.120
That is also equal to 6.

00:04:14.120 --> 00:04:16.860
And this is probably the first
time in mathematics you'll

00:04:16.860 --> 00:04:19.340
encounter something very neat.

00:04:19.340 --> 00:04:22.170
Sometimes, regardless of the
path you take, as long as you

00:04:22.170 --> 00:04:25.350
take a correct path you
get the same answer.

00:04:25.350 --> 00:04:27.450
So two people can kind of
visualize it-- as long as

00:04:27.450 --> 00:04:29.780
they're visualizing it
correctly, two different

00:04:29.780 --> 00:04:33.810
problems, but they come up
with the name solution.

00:04:33.810 --> 00:04:38.330
And so you're probably
saying, Sal, when is

00:04:38.330 --> 00:04:42.690
this multiplication
thing even useful?

00:04:42.690 --> 00:04:43.890
And this is where it's useful.

00:04:43.890 --> 00:04:46.890
Sometimes it
simplifies counting.

00:04:46.890 --> 00:04:52.630
So let's say I have a--
well, let's stick with

00:04:52.630 --> 00:04:56.590
our fruit analogy.

00:04:56.590 --> 00:04:59.250
An analogy is just when
you kind of use something

00:04:59.250 --> 00:05:02.090
as-- well, I won't
go too much into it.

00:05:02.090 --> 00:05:03.710
But our fruit example.

00:05:03.710 --> 00:05:05.130
Let's say I had lemons.

00:05:05.130 --> 00:05:07.040
Let me draw a bunch of lemons.

00:05:07.040 --> 00:05:08.810
I'll draw them in rows of 3.

00:05:08.810 --> 00:05:15.050
So I have 1, 2, 3-- well, I'm
not going to count them because

00:05:15.050 --> 00:05:17.926
that'll give our answer away.

00:05:17.926 --> 00:05:21.450
I'm just drawing a
bunch of lemons.

00:05:21.450 --> 00:05:27.070
Now, if I said, you tell me how
many lemons there are here.

00:05:27.070 --> 00:05:30.260
And if I did that you would
proceed to just count

00:05:30.260 --> 00:05:31.210
all of the lemons.

00:05:31.210 --> 00:05:34.120
And it wouldn't take you too
long to say, that oh, there's

00:05:34.120 --> 00:05:39.510
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12 lemons I actually

00:05:39.510 --> 00:05:40.470
already gave you the answer.

00:05:40.470 --> 00:05:43.180
We know that there
are 12 lemons there.

00:05:43.180 --> 00:05:46.390
But there's an easier way
and a faster way to count

00:05:46.390 --> 00:05:48.020
the number of lemons.

00:05:48.020 --> 00:05:51.960
Notice: how many lemons
are in each row?

00:05:51.960 --> 00:05:56.950
And a row is kind of the
side to side lemons.

00:05:56.950 --> 00:05:59.870
I think you know what a row is.

00:05:59.870 --> 00:06:03.330
I don't want to
talk down to you.

00:06:03.330 --> 00:06:06.100
So how many lemons
are there in a row?

00:06:06.100 --> 00:06:09.290
Well, there are 3
lemons in a row.

00:06:09.290 --> 00:06:12.130
And now let me ask you
another question, how

00:06:12.130 --> 00:06:16.170
many rows are there?

00:06:16.170 --> 00:06:21.030
Well, this was one row, and
this is the second row,

00:06:21.030 --> 00:06:26.680
this is the third row, and
this is the fourth row.

00:06:26.680 --> 00:06:30.510
So an easy way to count it
is say, I have 3 lemons per

00:06:30.510 --> 00:06:32.510
row and I have 4 of them.

00:06:32.510 --> 00:06:35.360
So let's say I have
3 lemons per row.

00:06:35.360 --> 00:06:38.350
I hope I'm not confusing you,
but I think you'll enjoy this.

00:06:38.350 --> 00:06:39.800
And then I have 4 rows.

00:06:39.800 --> 00:06:41.765
So I have 4 times 3 lemons.

00:06:46.210 --> 00:06:50.600
And that should be equal to the
number of lemons I have-- 12.

00:06:50.600 --> 00:06:53.720
And just to make that gel
with what I just did

00:06:53.720 --> 00:06:56.000
with the addition,
let's think about this.

00:06:56.000 --> 00:07:01.090
4 times 3-- literally, when you
actually say out the word 4

00:07:01.090 --> 00:07:04.640
times 3, I visualize this.

00:07:04.640 --> 00:07:06.980
I visualize 4 times 3.

00:07:06.980 --> 00:07:09.120
So 3 four times.

00:07:09.120 --> 00:07:11.910
3 plus 3 plus 3.

00:07:11.910 --> 00:07:15.160
And if we did that we
get 3 plus 3 is 6.

00:07:15.160 --> 00:07:16.820
6 plus 3 is 9.

00:07:16.820 --> 00:07:19.410
9 plus 3 is 12.

00:07:19.410 --> 00:07:24.450
And we learned up here, this
part of the video, we learned

00:07:24.450 --> 00:07:29.670
that this same multiplication
could also be interpreted

00:07:29.670 --> 00:07:33.220
as 3 times 4.

00:07:33.220 --> 00:07:36.950
You can switch the order and
this is one of the useful and

00:07:36.950 --> 00:07:41.940
interesting actually, kind of
properties of multiplication.

00:07:41.940 --> 00:07:46.830
But this could also be
written as 4 three times.

00:07:46.830 --> 00:07:49.930
4 plus 4 plus 4.

00:07:49.930 --> 00:07:52.400
You add 4 to itself
three times.

00:07:52.400 --> 00:07:54.430
4 plus 4 is 8.

00:07:54.430 --> 00:07:58.250
8 plus 4 is 12.

00:07:58.250 --> 00:08:04.680
And in the U.S. we always say 4
times 3, but you know, I've met

00:08:04.680 --> 00:08:06.870
people and a lot of people in
my family they kind of learned

00:08:06.870 --> 00:08:09.900
in the-- I guess, you could
call it the English system.

00:08:09.900 --> 00:08:13.870
And they'll often call this
four 3's or three 4's.

00:08:13.870 --> 00:08:15.640
And that in some ways is
a lot more intuitive.

00:08:15.640 --> 00:08:17.810
It's not intuitive the first
time you hear it, but they'll

00:08:17.810 --> 00:08:20.150
write this multiplication
problem or they'll say this

00:08:20.150 --> 00:08:23.340
multiplication problem and
they'll say, what are four 3's?

00:08:23.340 --> 00:08:25.850
And when they say four 3's,
they're literally saying,

00:08:25.850 --> 00:08:27.570
what are four 3's?

00:08:27.570 --> 00:08:31.810
So this is one 3, two 3's,
three 3's, four 3's.

00:08:31.810 --> 00:08:34.270
So what are four 3's
when you add them up?

00:08:34.270 --> 00:08:35.480
It's 12.

00:08:35.480 --> 00:08:38.665
And you could also say,
what are three 4's?

00:08:38.665 --> 00:08:41.520
So let me write this down.

00:08:41.520 --> 00:08:42.740
Let me do it in a
different color.

00:08:42.740 --> 00:08:46.840
That is four 3's.

00:08:46.840 --> 00:08:49.060
I mean literally,
that's four 3's.

00:08:49.060 --> 00:08:52.350
If I told you to say, write
down for four 3's and add them

00:08:52.350 --> 00:08:53.460
up, that's what that is.

00:08:53.460 --> 00:08:57.830
And that is 4 times
3 or 3 four times.

00:08:57.830 --> 00:09:03.080
And this is-- let me do
that in a different color.

00:09:03.080 --> 00:09:09.870
That is three 4's.

00:09:09.870 --> 00:09:13.090
And it could also be
written as 3 times 4.

00:09:13.090 --> 00:09:15.550
And they all equal 12.

00:09:15.550 --> 00:09:17.490
And now you're probably
saying, OK, this is nice.

00:09:17.490 --> 00:09:19.600
It's a cute little trick,
Sal, that you've taught me.

00:09:19.600 --> 00:09:25.140
But it took you less time
to count these lemons

00:09:25.140 --> 00:09:26.500
than to do this problem.

00:09:26.500 --> 00:09:28.880
And well first of all, that's
only right now because you're

00:09:28.880 --> 00:09:30.110
new to multiplication.

00:09:30.110 --> 00:09:34.010
But what you'll find is there
are times and there are

00:09:34.010 --> 00:09:36.240
actually many times-- and I
don't want to use the word

00:09:36.240 --> 00:09:41.195
times too much in a video on
multiplication-- where each row

00:09:41.195 --> 00:09:45.530
of lemons, instead of having 3
maybe they have 100 lemons.

00:09:45.530 --> 00:09:47.770
Maybe there's 100 rows.

00:09:47.770 --> 00:09:50.160
And then it would take you
forever to count all the lemons

00:09:50.160 --> 00:09:52.210
and that's where multiplication
comes really useful.

00:09:52.210 --> 00:09:54.560
Although, we're not going
to learn right now how to

00:09:54.560 --> 00:09:57.500
multiply 100 times 100.

00:09:57.500 --> 00:09:59.350
Now, the one thing that I
want to get you and this

00:09:59.350 --> 00:10:00.030
is kind of a trick.

00:10:00.030 --> 00:10:03.390
I remember my sister just to
try to show how much smarter

00:10:03.390 --> 00:10:06.420
she was than me when I was in
kindergarten and she was in

00:10:06.420 --> 00:10:12.690
third grade, she would say
Sal, what is 3 times 1?

00:10:12.690 --> 00:10:15.100
And I would say, because
my brain would say, oh,

00:10:15.100 --> 00:10:16.450
that's like 3 plus 1.

00:10:16.450 --> 00:10:19.920
And I would say 3 plus
1 is equal to 4.

00:10:19.920 --> 00:10:22.190
And so I'd say, 3 times 1?

00:10:22.190 --> 00:10:23.780
That must be 4 as well.

00:10:23.780 --> 00:10:25.270
And she would say, no, silly.

00:10:25.270 --> 00:10:26.080
It's 3.

00:10:26.080 --> 00:10:28.130
And I was like,
how can that be?

00:10:28.130 --> 00:10:30.220
How can the 3 times some
other number still

00:10:30.220 --> 00:10:31.270
be the same number?

00:10:31.270 --> 00:10:33.320
And think about
what this means.

00:10:33.320 --> 00:10:39.440
You could view this
as three 1's.

00:10:39.440 --> 00:10:40.570
What are three 1's?

00:10:40.570 --> 00:10:44.550
That's one 1 plus another
one 1 plus another 1.

00:10:44.550 --> 00:10:46.010
And that's equal to 3.

00:10:46.010 --> 00:10:49.410
Or you could view
this as 3 one time.

00:10:49.410 --> 00:10:51.130
So what's 3 one time?

00:10:51.130 --> 00:10:53.370
It's almost silly
how easy it is.

00:10:53.370 --> 00:10:54.620
It's just 3.

00:10:54.620 --> 00:10:55.900
That's one 3.

00:10:55.900 --> 00:10:59.780
You could write this as one 3.

00:10:59.780 --> 00:11:03.090
And that's why anything
times 1 or 1 times

00:11:03.090 --> 00:11:05.580
anything is that anything.

00:11:05.580 --> 00:11:08.960
So then 3 times 1 is 3.

00:11:08.960 --> 00:11:10.110
1 times 3 is 3.

00:11:10.110 --> 00:11:16.710
And you know, I could say 100
times 1 is equal to 100.

00:11:16.710 --> 00:11:23.450
I could say that 1 times
39 is equal to 39.

00:11:23.450 --> 00:11:26.730
And I think you're familiar
with numbers this large by now.

00:11:26.730 --> 00:11:27.930
So that's interesting.

00:11:27.930 --> 00:11:30.550
Now there's one other
really interesting thing

00:11:30.550 --> 00:11:31.950
about multiplication.

00:11:31.950 --> 00:11:34.760
And that's when you
multiply by 0.

00:11:34.760 --> 00:11:38.060
And I'll start with the analogy
or the example when you add.

00:11:38.060 --> 00:11:42.210
3 plus 0 you've
hopefully learned is 3.

00:11:42.210 --> 00:11:43.790
Because I'm adding
nothing to the 3.

00:11:43.790 --> 00:11:46.950
If I have 3 apples and I
give you 0 more apples,

00:11:46.950 --> 00:11:48.940
you still have 3 apples.

00:11:48.940 --> 00:11:52.100
But what is 3 and maybe I'm
just fixated on the number

00:11:52.100 --> 00:11:53.240
3 a little bit too much.

00:11:53.240 --> 00:11:53.950
Let me switch.

00:11:53.950 --> 00:11:59.070
What is 4 times 0?

00:11:59.070 --> 00:12:03.040
Well this is saying,
0 four times.

00:12:03.040 --> 00:12:09.110
So what's 0 plus
0 plus 0 plus 0?

00:12:09.110 --> 00:12:11.890
Well, that's 0.

00:12:11.890 --> 00:12:14.000
I have nothing plus nothing
plus nothing plus nothing,

00:12:14.000 --> 00:12:15.390
so I get nothing.

00:12:15.390 --> 00:12:18.940
Another way to think of it,
I could say 4 zero times.

00:12:18.940 --> 00:12:21.190
So how do I write 4 zero times?

00:12:21.190 --> 00:12:23.090
Well, I just don't
write anything, right?

00:12:23.090 --> 00:12:25.770
Because if I write anything, if
I write 1/4 and I don't have

00:12:25.770 --> 00:12:31.250
no 4's-- let me write this.

00:12:31.250 --> 00:12:40.676
This is four 0's, but I
could also write zero 4's.

00:12:40.676 --> 00:12:41.880
And what is zero 4's?

00:12:41.880 --> 00:12:43.630
I'll just write a
big blank here.

00:12:43.630 --> 00:12:44.310
There, I wrote it.

00:12:44.310 --> 00:12:45.370
There are no 4's here.

00:12:45.370 --> 00:12:47.610
So there's just a big blank.

00:12:47.610 --> 00:12:48.710
And that's another fun thing.

00:12:48.710 --> 00:12:50.800
So anything times 0 is 0.

00:12:50.800 --> 00:13:01.390
I could write a huge number,
you know, 5,493,692 times 0.

00:13:01.390 --> 00:13:02.560
What does that equal?

00:13:02.560 --> 00:13:04.560
That equals 0.

00:13:04.560 --> 00:13:06.180
And by the way, what's
this number times 1?

00:13:06.180 --> 00:13:07.215
Well, it's that number again.

00:13:07.215 --> 00:13:12.410
And what's 0 times 17?

00:13:12.410 --> 00:13:15.230
Once again, that is 0.

00:13:15.230 --> 00:13:17.855
Anyway, I think I've
talked for long enough.

00:13:17.855 --> 00:13:20.350
See you in the next video.


Parametric equations 2

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.880
In the last video, we used a
set of parametric equations to

00:00:03.880 --> 00:00:08.240
describe the position of a car
as it fell off of a cliff, and

00:00:08.240 --> 00:00:12.660
the equations were x as a
function of t was-- and I will

00:00:12.660 --> 00:00:16.480
write that-- x as a function of
t was equal to 10, I believe.

00:00:16.480 --> 00:00:18.180
I did this a couple of
hours ago, so I think

00:00:18.180 --> 00:00:18.670
that's what I said.

00:00:18.670 --> 00:00:21.570
10 plus 5t.

00:00:21.570 --> 00:00:30.360
And y as a function of t was
equal to 50 minus 5t squared.

00:00:30.360 --> 00:00:32.790
And the graph looks
something like this.

00:00:32.790 --> 00:00:33.950
Let me redraw it.

00:00:33.950 --> 00:00:36.200
It never hurts.

00:00:36.200 --> 00:00:38.180
That was the y-axis.

00:00:38.180 --> 00:00:41.470
That's the x-axis.

00:00:41.470 --> 00:00:45.480
And we saw that at t equals 0,
and we could try t equals 0

00:00:45.480 --> 00:00:48.690
here, and we'll get the
point 10 comma 50.

00:00:48.690 --> 00:00:51.300
So the point 10 comma
50 was right there.

00:00:51.300 --> 00:00:54.990
That was at t equal to 0,
and then we plotted a few

00:00:54.990 --> 00:00:56.880
points in the last video.

00:00:56.880 --> 00:00:59.010
I think t equals 1 was there.

00:00:59.010 --> 00:01:03.510
t equals 2 was, like, there,
and t equal 3 looked something

00:01:03.510 --> 00:01:08.910
like that, and this was the
path of the actual car as

00:01:08.910 --> 00:01:12.600
it hurtled to the ground.

00:01:12.600 --> 00:01:16.130
But this parametric equation
actually doesn't just describe

00:01:16.130 --> 00:01:18.470
this part of the curve.

00:01:18.470 --> 00:01:22.120
It describes a curve that goes
in both directions forever.

00:01:22.120 --> 00:01:25.190
So it describes a curve that
does something like this.

00:01:25.190 --> 00:01:30.900
If you actually plot t equals
minus 1, what do you get here?

00:01:30.900 --> 00:01:32.920
You get minus 5.

00:01:32.920 --> 00:01:35.140
So 10 minus 5 is 5.

00:01:35.140 --> 00:01:39.290
If you put a minus 1 here,
this becomes a plus 1.

00:01:39.290 --> 00:01:42.260
So minus 1, so you get 5, 45.

00:01:42.260 --> 00:01:43.510
So you get that
point right there.

00:01:43.510 --> 00:01:45.280
And if you did minus 2, you're
going to get a point that

00:01:45.280 --> 00:01:46.330
looks something like that.

00:01:46.330 --> 00:01:47.690
And minus 3, you're
going to get a point

00:01:47.690 --> 00:01:48.630
something like that.

00:01:48.630 --> 00:01:52.280
So the whole curve described by
this parametric equation will

00:01:52.280 --> 00:01:53.340
look something like this.

00:01:53.340 --> 00:01:56.140
Let me do it in a
different color.

00:01:56.140 --> 00:01:59.500
It looks something like this.

00:01:59.500 --> 00:02:02.990
And the direction as x
increases-- sorry, as

00:02:02.990 --> 00:02:05.260
t increases, looks
like this, right?

00:02:05.260 --> 00:02:11.920
This is t is equal to minus 3,
minus 2, minus 1, 0, 1, 2,

00:02:11.920 --> 00:02:13.510
and so forth and so on.

00:02:13.510 --> 00:02:17.480
So, in the last example, our
path was actually just a subset

00:02:17.480 --> 00:02:20.400
of the path described by
this parametric equation.

00:02:20.400 --> 00:02:22.840
And I'm saying all of this
because sometimes it's useful

00:02:22.840 --> 00:02:26.180
to just bound your parametric
equation and say this is a path

00:02:26.180 --> 00:02:28.410
only for certain values of t.

00:02:28.410 --> 00:02:31.870
And in our example, it was when
we leave this point, which is

00:02:31.870 --> 00:02:35.490
at t equals 0, all the way
to when we hit the ground.

00:02:35.490 --> 00:02:37.780
So if we want to know those
boundaries, well, we know the

00:02:37.780 --> 00:02:42.470
first boundary is t equals 0,
so this in the last example,

00:02:42.470 --> 00:02:45.600
for when we actually talked
about a car careening off of a

00:02:45.600 --> 00:02:49.410
cliff, it was between 0, so t
is greater than or

00:02:49.410 --> 00:02:49.850
equal to zero.

00:02:49.850 --> 00:02:53.940
And then we have to figure
out what t-value makes

00:02:53.940 --> 00:02:54.940
us hit the ground?

00:02:54.940 --> 00:02:56.670
Or when does y equal 0?

00:02:56.670 --> 00:02:59.980
Because when y equals 0-- y is
our altitude, so when y equals

00:02:59.980 --> 00:03:01.100
0, we've hit the ground.

00:03:01.100 --> 00:03:03.590
So if we say what t-value
makes y equal 0?

00:03:03.590 --> 00:03:07.090
We just say 0 equals
50 minus 5t squared.

00:03:07.090 --> 00:03:08.290
So let's do that.

00:03:08.290 --> 00:03:13.220
0 is equal to 50
minus 5t squared.

00:03:13.220 --> 00:03:15.930
I just took this and set y
equal to 0 because that's

00:03:15.930 --> 00:03:18.570
when we're going to
be hitting the floor.

00:03:18.570 --> 00:03:20.440
Add 5t squared to both sides.

00:03:20.440 --> 00:03:24.340
You get 5t squared
is equal to 50.

00:03:24.340 --> 00:03:27.500
Divide both sides by 5. t
squared is equal to 10.

00:03:27.500 --> 00:03:31.070
t is equal to the
square root of 3.

00:03:31.070 --> 00:03:33.110
And you can actually say plus
or minus square root of 3

00:03:33.110 --> 00:03:36.770
because this is-- we hit the
ground right at t is equal to

00:03:36.770 --> 00:03:38.660
square root of-- oh, sorry.

00:03:38.660 --> 00:03:41.010
t is equal to
square root of 10.

00:03:41.010 --> 00:03:42.470
t is equal to
square root of 10.

00:03:42.470 --> 00:03:44.975
My brain did 3 because I
immediately said, oh, square

00:03:44.975 --> 00:03:48.300
root of 10, that's 3 point
something, something.

00:03:48.300 --> 00:03:51.850
So this is t equals square root
of 10, which is approximately

00:03:51.850 --> 00:03:54.800
equal to 3 point--
I don't know.

00:03:54.800 --> 00:03:55.580
I don't know what it is.

00:03:55.580 --> 00:03:58.770
I should probably get my
calculator going, but that's

00:03:58.770 --> 00:03:59.780
not important for this.

00:03:59.780 --> 00:04:02.700
And if we actually took the
positive and negative square

00:04:02.700 --> 00:04:07.720
root, this would be t is
equal to the negative

00:04:07.720 --> 00:04:09.530
square root of 10.

00:04:09.530 --> 00:04:11.930
Out here there's a negative
sign right there, and if you

00:04:11.930 --> 00:04:14.500
could go backwards in time,
you would have kind of

00:04:14.500 --> 00:04:15.490
hit the ground there.

00:04:15.490 --> 00:04:17.420
But remember, in our
example, we weren't

00:04:17.420 --> 00:04:18.510
going backwards in time.

00:04:18.510 --> 00:04:21.440
If we went backwards in time at
t minus 1, we were actually on

00:04:21.440 --> 00:04:23.780
a plateau, and then there would
have been a different set of

00:04:23.780 --> 00:04:26.110
parametric equations that would
have described the car when it

00:04:26.110 --> 00:04:29.220
was just driving
on the plateau.

00:04:29.220 --> 00:04:31.840
So if we really want to
describe the path of the car,

00:04:31.840 --> 00:04:35.710
we should have bounded it and
said that between time is equal

00:04:35.710 --> 00:04:40.460
to 0 and time being less than
the square root of 10,

00:04:40.460 --> 00:04:42.990
this would have been our
parametric equations.

00:04:42.990 --> 00:04:44.580
Fair enough.

00:04:44.580 --> 00:04:47.000
Now, the next thing that's
interesting is, you know, we

00:04:47.000 --> 00:04:49.215
took these parametric
equations, and we generalized

00:04:49.215 --> 00:04:51.270
it, and we just said what
it would look like if

00:04:51.270 --> 00:04:52.110
we didn't bound it.

00:04:52.110 --> 00:04:54.490
If this wasn't the case, then
the parametric equation would

00:04:54.490 --> 00:04:56.860
just keep going like that, and
it would have this kind

00:04:56.860 --> 00:04:58.780
of direction, right?

00:04:58.780 --> 00:05:01.420
It would move in
that direction.

00:05:01.420 --> 00:05:06.020
But can we express the set of
parametric equations as just a

00:05:06.020 --> 00:05:11.180
normal equation where y is
expressed as a function of x,

00:05:11.180 --> 00:05:13.490
or x is expressed as
a function of y?

00:05:13.490 --> 00:05:15.930
And let's try.

00:05:15.930 --> 00:05:20.010
So the easiest thing to do is
let's just rewrite them, just

00:05:20.010 --> 00:05:22.120
so we clean up our board.

00:05:22.120 --> 00:05:30.760
So if you have x is equal to 10
plus 5t and y is equal to 50

00:05:30.760 --> 00:05:38.020
minus 5t squared, if we want to
write this system as just an

00:05:38.020 --> 00:05:40.460
equation with just y and x,
what we can do is we could just

00:05:40.460 --> 00:05:44.810
solve for t in terms of either
x or y and then substitute

00:05:44.810 --> 00:05:46.860
back into the other equation.

00:05:46.860 --> 00:05:49.630
So the easier one to solve is
this top equation because

00:05:49.630 --> 00:05:50.570
it's a linear equation.

00:05:50.570 --> 00:05:51.820
This is a little
more complicated.

00:05:51.820 --> 00:05:54.720
We'll have, you know, t is
equal to the square root of y,

00:05:54.720 --> 00:05:56.350
and it becomes a lot more
complicated, so let's

00:05:56.350 --> 00:05:58.680
just use this.

00:05:58.680 --> 00:06:00.680
So if we want to solve
for t, let's subtract

00:06:00.680 --> 00:06:01.960
10 from both sides.

00:06:01.960 --> 00:06:06.700
You get x minus 10
is equal to 5t.

00:06:06.700 --> 00:06:08.400
Divide both sides by 5.

00:06:08.400 --> 00:06:11.650
You get x over 5
minus 2, right?

00:06:11.650 --> 00:06:14.240
10 divided by 5 is equal to t.

00:06:14.240 --> 00:06:16.860
And now, we can take
this and substitute it.

00:06:16.860 --> 00:06:18.190
This is t, right?

00:06:18.190 --> 00:06:21.720
So we can take this and
substitute it back here

00:06:21.720 --> 00:06:24.700
for t because we say
this is equal to t.

00:06:24.700 --> 00:06:33.710
So our equation becomes y is
equal to 50 minus 5 times x

00:06:33.710 --> 00:06:39.230
minus x over 5 minus 2 squared.

00:06:39.230 --> 00:06:40.260
Now, let's simplify this.

00:06:40.260 --> 00:06:43.750
This is equal to--
let's switch colors.

00:06:43.750 --> 00:06:50.780
50 minus 5 times x squared
over 25 minus-- let's see,

00:06:50.780 --> 00:06:58.310
2/5, so it's 4/5x plus 4.

00:06:58.310 --> 00:07:00.870
And then if you multiply--
and this is a y out here.

00:07:00.870 --> 00:07:03.970
So you get y is equal to 50.

00:07:03.970 --> 00:07:08.610
And let's see, minus 5 times--
minus 5 over 25 is minus 1/5,

00:07:08.610 --> 00:07:11.820
so minus x squared over 5.

00:07:11.820 --> 00:07:17.020
Minus 5 times minus
4/5, that's plus 4x.

00:07:17.020 --> 00:07:21.490
And then minus 5
times 4 is minus 20.

00:07:21.490 --> 00:07:22.250
We're almost there.

00:07:22.250 --> 00:07:28.740
We can add the 50 to the
negative 20, and we get y--

00:07:28.740 --> 00:07:34.430
that's too dark-- y is equal
to minus x squared over

00:07:34.430 --> 00:07:38.440
5 plus 4x plus 30.

00:07:38.440 --> 00:07:39.250
And we're done.

00:07:39.250 --> 00:07:43.630
We just, I guess we could
say, simplified this set of

00:07:43.630 --> 00:07:49.340
parametric equations to this
one equation, where y

00:07:49.340 --> 00:07:50.670
is a function of x.

00:07:50.670 --> 00:07:52.690
And so you might be saying,
Sal, why didn't we just

00:07:52.690 --> 00:07:54.200
always have it this way?

00:07:54.200 --> 00:07:57.170
This is so much more simple,
and we can just graph this, and

00:07:57.170 --> 00:08:00.460
you know how to deal with it,
and it's just one equation

00:08:00.460 --> 00:08:01.470
with just y and x.

00:08:01.470 --> 00:08:03.550
We don't have this
third variable t.

00:08:03.550 --> 00:08:06.180
And that might be OK if
you just want to know

00:08:06.180 --> 00:08:08.140
the shape of the path.

00:08:08.140 --> 00:08:11.900
So if you took this equation,
and if I told you to graph it,

00:08:11.900 --> 00:08:14.700
you would indeed get a
parabola, kind of a

00:08:14.700 --> 00:08:18.320
downward-opening parabola that
looks something like that.

00:08:18.320 --> 00:08:22.730
But when you go from these two
equations to this one equation,

00:08:22.730 --> 00:08:24.520
you lose information.

00:08:24.520 --> 00:08:27.470
If I told you that this
describes the path of an

00:08:27.470 --> 00:08:29.650
object, some object, it
doesn't have to be the car.

00:08:29.650 --> 00:08:31.510
We've progressed beyond that.

00:08:31.510 --> 00:08:33.630
It describes a path
of some object.

00:08:33.630 --> 00:08:37.970
Just with this equation alone,
you don't know if the object

00:08:37.970 --> 00:08:40.500
is-- is the object
going this way?

00:08:40.500 --> 00:08:41.830
Is it going like that?

00:08:41.830 --> 00:08:43.850
Or is the object going
in the other direction

00:08:43.850 --> 00:08:44.580
like our car did.

00:08:44.580 --> 00:08:45.790
Is the object going that way?

00:08:45.790 --> 00:08:48.600
We don't know because we
don't know how it moves

00:08:48.600 --> 00:08:50.140
according to time.

00:08:50.140 --> 00:08:53.920
Likewise, if I told you this
was the path of the object and

00:08:53.920 --> 00:08:58.610
then I asked you where is the
object after 5 seconds,

00:08:58.610 --> 00:08:59.670
you don't know.

00:08:59.670 --> 00:09:01.890
Because this just tells you
what the shape-- where x is

00:09:01.890 --> 00:09:04.580
relative to y or where
y is relative to x.

00:09:04.580 --> 00:09:07.040
It doesn't tell you where you
are after 5 seconds, so we've

00:09:07.040 --> 00:09:08.890
lost all of that information.

00:09:08.890 --> 00:09:12.500
We've lost information that
this is t equals 0, this is t

00:09:12.500 --> 00:09:16.520
equals 1, this is t equals 2,
this is t equals 3, this is t

00:09:16.520 --> 00:09:17.880
equals the square root of 10.

00:09:17.880 --> 00:09:20.350
And that's why parametric
equations are useful.

00:09:20.350 --> 00:09:23.840
But anyway, I wanted to do this
example mainly to show you that

00:09:23.840 --> 00:09:27.850
you can express the same shape
this way and also to show you

00:09:27.850 --> 00:09:32.820
how to express a parametric
equation in terms of a y and an

00:09:32.820 --> 00:09:35.230
x, but also for you to
appreciate that the parametric

00:09:35.230 --> 00:09:39.210
equation is useful and that it
does contain information

00:09:39.210 --> 00:09:42.430
that this just single
equation won't contain.

00:09:42.430 --> 00:09:45.020
Anyway, see you in
the next video.