Part 2 of proof of Heron's formula

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:00.850
In

00:00:00.850 --> 00:00:04.750
the last video, I claimed that
this result we got for the area

00:00:04.750 --> 00:00:09.770
of a triangle that had sides
of length a, b, and c is

00:00:09.770 --> 00:00:11.810
equivalent to Heron's formula.

00:00:11.810 --> 00:00:14.150
And what I want to do in this
video is show you that this is

00:00:14.150 --> 00:00:16.780
equivalent to Heron's formula
by essentially just doing a

00:00:16.780 --> 00:00:18.990
bunch of algebraic
manipulation.

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So the first thing we want to
do-- let's just spring this

00:00:21.590 --> 00:00:23.590
1/2 c under the radical sign.

00:00:23.590 --> 00:00:28.170
So 1/2 c, that's the same
thing as the square root

00:00:28.170 --> 00:00:30.480
of c squared over 4.

00:00:30.480 --> 00:00:32.910
You take the square root
of that you get 1/2 c.

00:00:32.910 --> 00:00:36.270
So this whole expression is
equal to-- instead of drawing

00:00:36.270 --> 00:00:41.450
the radical, I'll just write
the square root of this,

00:00:41.450 --> 00:00:48.200
of c squared over 4
times all of this.

00:00:48.200 --> 00:00:49.530
I'll just copy and paste it.

00:00:53.040 --> 00:00:55.610
Copy and paste.

00:00:55.610 --> 00:00:57.160
So times all of that.

00:00:57.160 --> 00:01:01.160
And of course, it has
to be distributed.

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So c squared over 4
times all of that.

00:01:03.960 --> 00:01:06.390
And then we have to
close the square root.

00:01:08.990 --> 00:01:11.460
Let me just distribute
the c squared over 4.

00:01:11.460 --> 00:01:13.960
This is going to be equal
to the square root.

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This is going to be hairy,
but I think you'll find it

00:01:15.940 --> 00:01:18.620
satisfying to see how this
could turn into something as

00:01:18.620 --> 00:01:20.470
simple as Heron's formula.

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The square root of c squared
over 4 times a squared is c

00:01:24.660 --> 00:01:32.560
squared a squared over 4,
minus c squared over 4.

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I'm just distributing this.

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And I'm going to write it as
the numerator squared over

00:01:37.600 --> 00:01:39.060
the denominator squared.

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So times c squared plus
a squared minus b

00:01:44.090 --> 00:01:45.950
squared, squared.

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Over-- if I square the
denominator that's 4c squared.

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And we immediately see that c
squared and that c squared

00:01:54.840 --> 00:01:55.600
are going to cancel out.

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Let me close all of the
parentheses just like that.

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And, of course, this 4 times
that 4, that's going to

00:02:02.530 --> 00:02:04.520
result in-- well let
me write it this way.

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That's the same
thing as 4 squared.

00:02:06.490 --> 00:02:08.850
And I'm instead of writing
16, you'll see why

00:02:08.850 --> 00:02:09.890
I'm writing that.

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Now this I can rewrite.

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This is going to be equal to
the square root-- I'm

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arbitrarily switching colors--
of ca over 2 squared.

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This is the same thing as that.

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Right?

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I'm just writing it as
the whole thing squared.

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If I square that, that's the c
squared a squared over 2

00:02:30.360 --> 00:02:34.930
squared over 4, minus-- and I'm
going to write this whole thing

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as an expression squared.

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So that's c squared
plus a squared minus

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b squared, over 4.

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And we are squaring both the
numerator and the denominator.

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Now this might look a little
bit interesting to you.

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Let me make the parentheses in
a slightly different color.

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You might remember from
factoring polynomials that if I

00:03:00.775 --> 00:03:03.460
have something of the form x
squared minus y squared, that

00:03:03.460 --> 00:03:08.520
factors into x plus
y times x minus y.

00:03:08.520 --> 00:03:11.000
And we're going to be using
this over and over again.

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Now if you call ca over 2x, and
you call this whole big thing

00:03:15.590 --> 00:03:19.110
y, then we have x squared
minus y squared.

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So we can factor it.

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So this whole thing is going to
be equal to the square root of

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x plus y, or in this case it's
ca over 2 plus the y, which is

00:03:34.740 --> 00:03:40.960
c squared plus a squared
minus b squared over 4.

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Times x minus y.

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So this is our x.

00:03:45.570 --> 00:03:51.370
ca over 2, minus all of
this business over here.

00:03:51.370 --> 00:03:53.840
Or even better, let me just
say plus and then let me

00:03:53.840 --> 00:03:54.680
just write the negative.

00:03:54.680 --> 00:04:01.980
So plus minus c squared minus
a squared plus b squared.

00:04:01.980 --> 00:04:05.140
All of that over 4.

00:04:05.140 --> 00:04:10.180
So all I did here is I said
this is the same thing as this

00:04:10.180 --> 00:04:15.120
plus this, this plus this,
times this minus this, this

00:04:15.120 --> 00:04:18.610
minus-- I just said plus
the negative of this.

00:04:18.610 --> 00:04:21.770
So minus c squared minus a
squared plus b squared.

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All I did is that right there.

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Now let's see if we can
simplify this, or if we

00:04:26.610 --> 00:04:28.870
could add these fractions.

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Well, we can get a
common denominator.

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ca over 2, that's the same
thing as 2ca over 4.

00:04:35.650 --> 00:04:38.910
ca over 2, that's the same
thing as 2ca over 4, just

00:04:38.910 --> 00:04:41.160
multiplying the numerator
and the denominator by 2.

00:04:41.160 --> 00:04:44.420
And now we can add
the numerators.

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So our whole expression is now
going to be equal to the square

00:04:49.540 --> 00:04:55.645
root of this first expression,
will become-- and I'm going

00:04:55.645 --> 00:04:56.460
to write it this way.

00:04:56.460 --> 00:05:07.820
I'm going to write c squared
plus 2ca plus a squared minus b

00:05:07.820 --> 00:05:11.820
squared, all of that over 4.

00:05:11.820 --> 00:05:13.900
That's our first expression.

00:05:13.900 --> 00:05:18.010
And then our second expression
is going to become-- well,

00:05:18.010 --> 00:05:20.190
everything's going to be over
4, so I'll just write

00:05:20.190 --> 00:05:21.070
that right now.

00:05:21.070 --> 00:05:21.965
Everything over 4.

00:05:27.280 --> 00:05:36.030
And then we could write this
as b squared, minus c squared

00:05:36.030 --> 00:05:43.490
minus 2ca plus a squared.

00:05:43.490 --> 00:05:46.570
Just to make sure, I have
a minus a squared here.

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Plus times a minus, still
it's a minus a squared.

00:05:49.320 --> 00:05:51.420
I have a plus 2ca over here.

00:05:51.420 --> 00:05:54.080
Minus times a minus,
that's a plus 2ca.

00:05:54.080 --> 00:05:55.580
I have a minus c squared here.

00:05:55.580 --> 00:05:57.170
I have a minus c squared here.

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So these two things
are equivalent.

00:06:00.530 --> 00:06:04.630
Now the next thing we need to
recognize, or hopefully we can

00:06:04.630 --> 00:06:09.940
recognize, is that this over
here-- this might get a little

00:06:09.940 --> 00:06:13.690
bit messy-- that's the same
thing as c plus a squared.

00:06:13.690 --> 00:06:14.350
Let me write this.

00:06:14.350 --> 00:06:20.860
This is equal to the square
root, open parentheses, of this

00:06:20.860 --> 00:06:29.940
over here is c plus a squared
minus b squared, over 4.

00:06:29.940 --> 00:06:31.480
That's that first term.

00:06:31.480 --> 00:06:33.020
And then the second term.

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This over here is the same
thing as c minus a squared.

00:06:35.920 --> 00:06:39.120
So that whole thing will
simplify to b squared

00:06:39.120 --> 00:06:47.470
minus c minus a squared,
all of that over 4.

00:06:47.470 --> 00:06:48.910
So we're making some headway.

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As I told you, this
is a hairy problem.

00:06:51.830 --> 00:06:53.950
But we're seeing some neat
applications of factoring

00:06:53.950 --> 00:06:57.320
polynomials, and we're seeing
how a fairly bizarre looking

00:06:57.320 --> 00:07:00.160
equation can be transformed
into a simpler one.

00:07:00.160 --> 00:07:02.090
Now we can use this exact same
property-- we have that

00:07:02.090 --> 00:07:04.770
pattern-- something squared
minus something else squared.

00:07:07.310 --> 00:07:08.500
So we can factor it out.

00:07:08.500 --> 00:07:09.580
And I'll do it in
the same line.

00:07:09.580 --> 00:07:12.120
So this is going to be equal
to-- I'm going to write a

00:07:12.120 --> 00:07:14.040
little bit small, just so I
don't run out of space--

00:07:14.040 --> 00:07:15.310
the square root.

00:07:15.310 --> 00:07:20.000
This will factor into
this plus this.

00:07:20.000 --> 00:07:29.510
So c plus a plus b times
c plus a minus b.

00:07:29.510 --> 00:07:29.850
Right?

00:07:29.850 --> 00:07:32.030
It's the exact same pattern
that I did over here.

00:07:32.030 --> 00:07:34.470
This is x squared,
this is y squared.

00:07:34.470 --> 00:07:41.760
So times c plus a minus
b, all of that over 4.

00:07:41.760 --> 00:07:43.260
And then we have this one.

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This is going to be
b plus c minus a.

00:07:50.620 --> 00:07:53.180
Let me scroll down to
the right a little bit.

00:07:53.180 --> 00:07:59.030
Times b plus c minus a--
that's x plus y-- times

00:07:59.030 --> 00:08:02.640
b minus c minus a.

00:08:02.640 --> 00:08:09.020
Or that's the same thing
as b minus c plus a.

00:08:09.020 --> 00:08:13.110
This is the same thing
as b minus c minus a.

00:08:13.110 --> 00:08:14.140
Right?

00:08:14.140 --> 00:08:14.570
All right.

00:08:14.570 --> 00:08:20.370
And all of that over 4.

00:08:20.370 --> 00:08:23.910
Now, I can rewrite this
whole expression.

00:08:23.910 --> 00:08:25.580
I don't want to
run out of space.

00:08:25.580 --> 00:08:30.305
I can rewrite this whole
expression as, well 4 is

00:08:30.305 --> 00:08:32.955
the product of 2 times 2.

00:08:36.380 --> 00:08:40.620
So our whole area expression
has been, arguably, simplified

00:08:40.620 --> 00:08:44.780
to it equals the square root--
and this is really the home

00:08:44.780 --> 00:08:50.560
stretch-- of this right here,
which I can just write as

00:08:50.560 --> 00:08:55.780
a plus b plus c over 2.

00:08:55.780 --> 00:08:57.690
That's that term right there.

00:08:57.690 --> 00:09:00.640
Times this term.

00:09:00.640 --> 00:09:02.480
Times that term.

00:09:02.480 --> 00:09:05.340
And let me write a
simplification here. c plus a

00:09:05.340 --> 00:09:13.200
minus b, that's the same thing
as a plus b plus c minus 2b.

00:09:13.200 --> 00:09:14.490
These two things
are equivalent.

00:09:14.490 --> 00:09:14.700
Right?

00:09:14.700 --> 00:09:19.450
You have an a, you have a c,
and then b minus 2b is going

00:09:19.450 --> 00:09:22.510
to be equal to minus b.

00:09:22.510 --> 00:09:24.750
Right? b minus 2b,
that's minus b.

00:09:24.750 --> 00:09:29.690
So this next term is going
to be a plus b plus

00:09:29.690 --> 00:09:34.330
c minus 2b, over 2.

00:09:34.330 --> 00:09:36.240
Or instead of writing it like
that, let me write this

00:09:36.240 --> 00:09:40.570
over 2 minus this over 2.

00:09:40.570 --> 00:09:43.920
And then our next
term right here.

00:09:43.920 --> 00:09:46.180
Same exact logic.

00:09:46.180 --> 00:09:55.360
That's the same thing as a
plus b plus c minus 2a,

00:09:55.360 --> 00:09:56.500
all of that over 2.

00:09:56.500 --> 00:09:56.770
Right?

00:09:56.770 --> 00:09:59.960
If we add the minus 2a to
the a we get minus a.

00:09:59.960 --> 00:10:02.040
So we get b plus c minus a.

00:10:02.040 --> 00:10:03.820
These are identical things.

00:10:03.820 --> 00:10:06.950
So all this over 2, or we
can split the denominators

00:10:06.950 --> 00:10:09.130
just like that over 2.

00:10:09.130 --> 00:10:10.680
And then one last term.

00:10:10.680 --> 00:10:13.690
And you might already
recognize the rule of

00:10:13.690 --> 00:10:16.500
Heron's formula popping up.

00:10:16.500 --> 00:10:19.570
I was thinking not the rule
of Heron-- Heron's formula.

00:10:19.570 --> 00:10:23.050
That term right there is
the exact same thing as a

00:10:23.050 --> 00:10:27.570
plus b plus c minus 2c.

00:10:27.570 --> 00:10:27.860
Right?

00:10:27.860 --> 00:10:31.200
You take 2c away from the c you
get a minus c, and then you

00:10:31.200 --> 00:10:32.650
still have the a and the b.

00:10:32.650 --> 00:10:34.540
And then all of that over 2.

00:10:34.540 --> 00:10:37.640
You could write that over
2 minus that over 2.

00:10:37.640 --> 00:10:39.600
And, of course, we're
taking the square root

00:10:39.600 --> 00:10:41.540
of all of this stuff.

00:10:41.540 --> 00:10:52.230
Now, if we define an S to be
equal to a plus b plus c over

00:10:52.230 --> 00:10:55.560
2, then this equation
simplifies a good bit.

00:10:55.560 --> 00:10:57.800
This right here is S.

00:10:57.800 --> 00:11:00.130
That right there is S.

00:11:00.130 --> 00:11:01.705
That right there is S.

00:11:01.705 --> 00:11:03.940
And that right there is S.

00:11:03.940 --> 00:11:07.720
And these simplify
a good bit too.

00:11:07.720 --> 00:11:12.030
Minus 2b over 2, that's just
the same thing as minus b.

00:11:12.030 --> 00:11:14.880
Minus 2a over 2, that's the
same thing as minus a.

00:11:14.880 --> 00:11:17.100
Minus 2c over 2, that's the
same thing as minus c.

00:11:17.100 --> 00:11:23.590
So this whole equation for our
area now is equal to-- I'll

00:11:23.590 --> 00:11:24.620
rewrite the square root.

00:11:24.620 --> 00:11:30.670
The radical, the square root,
of S-- that's that right there.

00:11:33.550 --> 00:11:34.500
I'll do it in the same colors.

00:11:34.500 --> 00:11:46.890
Times S minus b, times this is
S minus a, times-- and we're

00:11:46.890 --> 00:11:49.555
at the last one-- S minus c.

00:11:52.390 --> 00:11:56.510
And we have proved Heron's
formula is the exact same thing

00:11:56.510 --> 00:11:59.410
as what we proved at the
end of the last video.

00:11:59.410 --> 00:12:02.250
So this was pretty neat.

00:12:02.250 --> 00:12:05.910
And we just had to do a
little bit of hairy algebra

00:12:05.910 --> 00:12:08.150
to actually prove it.


Part 1 of proof of Heron's formula

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.550
Let's say I've got a triangle.

00:00:03.550 --> 00:00:05.900
There is my triangle
right there.

00:00:05.900 --> 00:00:08.550
And I only know the lengths of
the sides of the triangle.

00:00:08.550 --> 00:00:12.490
This side has length a, this
side has length b, and

00:00:12.490 --> 00:00:14.360
that side has length c.

00:00:14.360 --> 00:00:17.480
And I'm asked to find the
area of that triangle.

00:00:17.480 --> 00:00:21.870
So far all I'm equipped with is
the idea that the area, the

00:00:21.870 --> 00:00:26.930
area of a triangle is equal
to 1/2 times the base of

00:00:26.930 --> 00:00:30.380
the triangle times the
height of the triangle.

00:00:30.380 --> 00:00:33.570
So the way I've drawn this
triangle, the base of this

00:00:33.570 --> 00:00:39.060
triangle, would be side c, but
the height we don't know.

00:00:39.060 --> 00:00:41.900
The height would be that h
right there and we don't

00:00:41.900 --> 00:00:43.750
even know what that h is.

00:00:43.750 --> 00:00:45.200
So this would be the h.

00:00:45.200 --> 00:00:48.360
So the question is how do
we figure out the area

00:00:48.360 --> 00:00:49.660
of this triangle?

00:00:49.660 --> 00:00:51.150
If you watched the last
video you know that you

00:00:51.150 --> 00:00:52.440
use Heron's formula.

00:00:52.440 --> 00:00:55.910
But the idea here is to try
to prove Heron's formula.

00:00:55.910 --> 00:00:59.530
So let's just try to figure
out h from just using

00:00:59.530 --> 00:01:00.970
the Pythagorean theorem.

00:01:00.970 --> 00:01:04.190
And from there, once we know h,
we can apply this formula and

00:01:04.190 --> 00:01:07.100
figure out the area
of this triangle.

00:01:07.100 --> 00:01:11.370
So we already
labeled this as h.

00:01:11.370 --> 00:01:13.120
Let me define another
variable here.

00:01:15.880 --> 00:01:19.270
This is a trick you'll see
pretty often in geometry.

00:01:19.270 --> 00:01:24.990
Let me define this is x, and if
this is x in magenta, then in

00:01:24.990 --> 00:01:29.990
this bluish-purplish color,
that would be c minus x, right?

00:01:29.990 --> 00:01:33.710
This whole length is c
-- the whole base is c.

00:01:33.710 --> 00:01:37.560
So if this part is x, then
this part is c minus x.

00:01:37.560 --> 00:01:41.060
What I could do now, since
these are both right angles,

00:01:41.060 --> 00:01:44.250
and I know that because this is
the height, I can set up two

00:01:44.250 --> 00:01:46.600
Pythagorean theorem equations.

00:01:46.600 --> 00:01:50.890
First, I could do this left
hand side and I can write that

00:01:50.890 --> 00:01:57.880
x squared plus h squared
is equal to a squared.

00:01:57.880 --> 00:02:00.690
That's what I get from
this left hand triangle.

00:02:00.690 --> 00:02:05.000
Then from this right hand
triangle, I get c minus x

00:02:05.000 --> 00:02:14.030
squared plus h squared
is equal to b squared.

00:02:14.030 --> 00:02:17.760
So I'm assuming I know a, b and
c, so I have two equations

00:02:17.760 --> 00:02:18.950
with two unknowns.

00:02:18.950 --> 00:02:22.290
The unknowns are x and h.

00:02:22.290 --> 00:02:24.220
And remember, h is what
we're trying to figure out

00:02:24.220 --> 00:02:25.270
because we already know c.

00:02:25.270 --> 00:02:27.540
If we know h, we can
apply the area formula.

00:02:27.540 --> 00:02:28.900
So how can we do that?

00:02:28.900 --> 00:02:32.200
Well, let's substitute
for h to figure out x.

00:02:32.200 --> 00:02:36.360
When I say that I mean let's
solve for h squared here.

00:02:36.360 --> 00:02:38.890
If we solve for h squared
here we just subtract x

00:02:38.890 --> 00:02:40.320
squared from both sides.

00:02:40.320 --> 00:02:44.540
We can write that x squared --
sorry, we could write that

00:02:44.540 --> 00:02:51.720
h squared is equal to a
squared minus x squared.

00:02:51.720 --> 00:02:53.770
Then we could take this
information and substitute

00:02:53.770 --> 00:02:56.640
it over here for h squared.

00:02:56.640 --> 00:03:02.030
So this bottom equation
becomes c minus x

00:03:02.030 --> 00:03:04.990
squared plus h squared.

00:03:04.990 --> 00:03:08.610
h squared we know from this
left hand side equation.

00:03:08.610 --> 00:03:11.620
h squared is going to be equal
to -- so plus, I'll do it in

00:03:11.620 --> 00:03:19.160
that color -- a squared minus x
squared is equal to b squared.

00:03:19.160 --> 00:03:21.650
I just substituted the
value of that in here, the

00:03:21.650 --> 00:03:23.280
value of that in there.

00:03:23.280 --> 00:03:25.860
Now let's expand this
expression out.

00:03:25.860 --> 00:03:29.750
c minus x squared, that
is c squared minus

00:03:29.750 --> 00:03:34.320
2cx plus x squared.

00:03:34.320 --> 00:03:38.200
Then we have the minus --
sorry, we have the plus a

00:03:38.200 --> 00:03:44.280
squared minus x squared
equals b squared.

00:03:47.660 --> 00:03:50.060
We have an x squared and
a minus x squared there,

00:03:50.060 --> 00:03:51.610
so those cancel out.

00:03:54.680 --> 00:03:58.790
Let's add the 2cx to both
sides of this equation.

00:03:58.790 --> 00:04:01.930
So now our equation
would become c squared

00:04:01.930 --> 00:04:04.720
plus a squared.

00:04:04.720 --> 00:04:06.490
I'm adding 2cx to both sides.

00:04:06.490 --> 00:04:10.440
So you add 2cx to this,
you get 0 is equal to

00:04:10.440 --> 00:04:13.580
b squared plus 2cx.

00:04:13.580 --> 00:04:16.370
All I did here is I canceled
out the x squared and then I

00:04:16.370 --> 00:04:19.600
added 2cx to both sides
of this equation.

00:04:19.600 --> 00:04:22.130
My goal here is to solve for x.

00:04:22.130 --> 00:04:24.580
Once I solve for x, then
I can solve for h and

00:04:24.580 --> 00:04:26.350
apply that formula.

00:04:26.350 --> 00:04:29.090
Now to solve for x, let's
subtract b squared

00:04:29.090 --> 00:04:30.040
from both sides.

00:04:30.040 --> 00:04:36.200
So we'll get c squared
plus a squared minus b

00:04:36.200 --> 00:04:41.020
squared is equal to 2cx.

00:04:41.020 --> 00:04:46.160
Then if we divide both sides by
2c, we get c squared plus a

00:04:46.160 --> 00:04:52.600
squared minus b squared
over 2c is equal to x.

00:04:52.600 --> 00:04:54.880
We've just solved for x here.

00:04:54.880 --> 00:04:57.290
Now, our goal is to solve
for the height, so that

00:04:57.290 --> 00:04:59.930
we can apply 1/2 times
base times height.

00:04:59.930 --> 00:05:04.120
So to do that, we go back to
this equation right here

00:05:04.120 --> 00:05:07.040
and solve for our height.

00:05:07.040 --> 00:05:10.800
Let me scroll down
a little bit.

00:05:10.800 --> 00:05:16.290
We know that our height
squared is equal to a

00:05:16.290 --> 00:05:20.520
squared minus x squared.

00:05:20.520 --> 00:05:23.330
Instead of just writing x
squared let's substitute here.

00:05:23.330 --> 00:05:27.430
So it's minus x squared -- x
is this thing right here.

00:05:27.430 --> 00:05:32.880
So c squared plus a
squared minus b squared

00:05:32.880 --> 00:05:36.670
over 2c, squared.

00:05:36.670 --> 00:05:39.320
This is the same
thing as x squared.

00:05:39.320 --> 00:05:41.090
We just solved for that.

00:05:41.090 --> 00:05:47.950
So h is going to be equal to
the square root of all this

00:05:47.950 --> 00:05:51.610
business in there -- I'll
switch the colors -- of a

00:05:51.610 --> 00:06:00.070
squared minus c squared plus
a squared minus b squared

00:06:00.070 --> 00:06:02.150
-- all of that squared.

00:06:02.150 --> 00:06:04.800
Let me make it a little bit
neater than that because

00:06:04.800 --> 00:06:06.720
I don't want to--.

00:06:06.720 --> 00:06:13.980
The square root -- make sure I
have enough space -- of a

00:06:13.980 --> 00:06:20.130
squared minus all of this stuff
squared -- we have c squared

00:06:20.130 --> 00:06:25.800
plus a squared minus b
squared, all of that over 2c.

00:06:25.800 --> 00:06:27.670
That is the height
of our triangle.

00:06:27.670 --> 00:06:30.310
The triangle that we
started off with up here.

00:06:30.310 --> 00:06:33.360
Let me copy and paste that
just so that we can remember

00:06:33.360 --> 00:06:36.070
what we're dealing with.

00:06:36.070 --> 00:06:41.600
Copy it and then let me
paste it down here.

00:06:41.600 --> 00:06:43.300
So I've pasted it down here.

00:06:43.300 --> 00:06:45.210
So we know what the height
is -- it's this big

00:06:45.210 --> 00:06:46.830
convoluted formula.

00:06:46.830 --> 00:06:51.180
The height in terms of a, b
and c is this right here.

00:06:51.180 --> 00:06:54.570
So if we wanted to figure out
the area -- the area of our

00:06:54.570 --> 00:06:58.270
triangle -- let me
do it in pink.

00:06:58.270 --> 00:07:03.772
The area of our triangle is
going to be 1/2 times our base

00:07:03.772 --> 00:07:09.850
-- our base is this entire
length, c -- times c times our

00:07:09.850 --> 00:07:13.260
height, which is this
expression right here.

00:07:13.260 --> 00:07:15.680
Let me just copy and
paste this instead of--.

00:07:15.680 --> 00:07:21.390
So let me copy and paste.

00:07:21.390 --> 00:07:24.450
So times the height.

00:07:24.450 --> 00:07:27.910
So this now is our
expression for the area.

00:07:27.910 --> 00:07:29.810
Now you're immediately saying
gee, that doesn't look a lot

00:07:29.810 --> 00:07:32.790
like Heron's formula,
and you're right.

00:07:32.790 --> 00:07:35.360
It does not look a lot like
Heron's formula, but what I'm

00:07:35.360 --> 00:07:37.820
going to show you in the next
video is that this essentially

00:07:37.820 --> 00:07:39.230
is Heron's formula.

00:07:39.230 --> 00:07:43.050
This is a harder to remember
version of Heron's formula.

00:07:43.050 --> 00:07:46.000
I'm going to apply a lot of
algebra to essentially simplify

00:07:46.000 --> 00:07:47.260
this to Heron's formula.

00:07:47.260 --> 00:07:49.430
But this will work.

00:07:49.430 --> 00:07:51.520
If you could memorize this,
I think Heron's a lot

00:07:51.520 --> 00:07:53.050
easier to memorize.

00:07:53.050 --> 00:07:56.300
But if you can memorize this
and you just know a, b and

00:07:56.300 --> 00:08:00.700
c, you apply this formula
right here and you will get

00:08:00.700 --> 00:08:04.940
the area of a triangle.

00:08:04.940 --> 00:08:07.290
Well, actually let's just apply
this just to show that this at

00:08:07.290 --> 00:08:09.710
least gives the same
number as Heron's.

00:08:09.710 --> 00:08:15.920
So in the last video we had a
triangle that had sides 9, 11

00:08:15.920 --> 00:08:22.350
and 16, and its area using
Heron's was equal to 18

00:08:22.350 --> 00:08:26.290
times the square root of 7.

00:08:26.290 --> 00:08:29.780
Let's see what we get when we
applied this formula here.

00:08:29.780 --> 00:08:36.260
So we get the area is equal
to 1/2 times 16 times the

00:08:36.260 --> 00:08:40.300
square root of a squared.

00:08:40.300 --> 00:08:49.330
That is 81 minus -- let's see,
c squared is 16, so that's 256.

00:08:49.330 --> 00:08:58.020
256 plus a squared, that's
at 81 minus b squared,

00:08:58.020 --> 00:09:02.250
so minus 121.

00:09:02.250 --> 00:09:04.120
All of this stuff is squared.

00:09:04.120 --> 00:09:09.530
All of that over 2 times c
-- all of that over 32.

00:09:09.530 --> 00:09:12.150
So let's see if we can
simplify this a little bit.

00:09:12.150 --> 00:09:15.770
81 minus 121, that is minus 40.

00:09:15.770 --> 00:09:18.790
So this becomes 216 over 32.

00:09:18.790 --> 00:09:22.470
So area is equal to
1/2 times 8 is 8.

00:09:22.470 --> 00:09:24.530
Let me switch colors.

00:09:24.530 --> 00:09:38.820
1/2 times 16 is 8 times the
square root of 81 minus 256.

00:09:38.820 --> 00:09:41.370
81 minus 121, that's minus 40.

00:09:41.370 --> 00:09:43.370
256 minus 40 is 216.

00:09:43.370 --> 00:09:48.270
216 over 32 squared.

00:09:48.270 --> 00:09:50.630
Now, this is a lot of
math to do so let me

00:09:50.630 --> 00:09:51.870
get out a calculator.

00:09:51.870 --> 00:09:54.140
I'm really just trying to show
you that these two numbers

00:09:54.140 --> 00:09:57.440
should give us our same number.

00:09:57.440 --> 00:10:01.290
So if we turn on
our calculator--.

00:10:01.290 --> 00:10:02.440
First of all, let's just
figure out what 18

00:10:02.440 --> 00:10:03.420
square root of 7 are.

00:10:03.420 --> 00:10:07.590
18 times the square root
of 7 -- this is what

00:10:07.590 --> 00:10:08.580
we got using Heron's.

00:10:08.580 --> 00:10:11.100
We got 47.62.

00:10:11.100 --> 00:10:13.160
Let's see if this is 47.62.

00:10:13.160 --> 00:10:26.700
So we have 8 times the square
root of 81 minus 216 divided

00:10:26.700 --> 00:10:35.140
by 32 squared, and then we
close our square roots.

00:10:35.140 --> 00:10:37.990
And we get the
exact same number.

00:10:37.990 --> 00:10:39.890
I was worried -- I actually
didn't do this calculation

00:10:39.890 --> 00:10:41.580
ahead of time so I might have
made a careless mistake.

00:10:41.580 --> 00:10:43.310
But there you go, you get
the exact same number.

00:10:43.310 --> 00:10:47.170
So our formula just now gave
us the exact same value

00:10:47.170 --> 00:10:48.350
as Heron's formula.

00:10:48.350 --> 00:10:54.030
But what I'm going to do in the
next video is prove to you that

00:10:54.030 --> 00:10:57.690
this can actually be reduced
algebraically to Heron's.


Heron's formula

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:03.240
I think it's pretty common
knowledge how to find the area

00:00:03.240 --> 00:00:06.030
of the triangle if we know the
length of its base

00:00:06.030 --> 00:00:07.250
and its height.

00:00:07.250 --> 00:00:10.540
So, for example, if that's my
triangle, and this length right

00:00:10.540 --> 00:00:14.910
here-- this base-- is of length
b and the height right here is

00:00:14.910 --> 00:00:19.080
of length h, it's pretty common
knowledge that the area of this

00:00:19.080 --> 00:00:23.170
triangle is going to be equal
to 1/2 times the base

00:00:23.170 --> 00:00:24.440
times the height.

00:00:24.440 --> 00:00:30.240
So, for example, if the base
was equal to 5 and the height

00:00:30.240 --> 00:00:37.180
was equal to 6, then our area
would be 1/2 times 5 times 6,

00:00:37.180 --> 00:00:41.770
which is 1/2 times 30--
which is equal to 15.

00:00:41.770 --> 00:00:45.120
Now what is less well-known is
how to figure out the area of a

00:00:45.120 --> 00:00:48.250
triangle when you're only given
the sides of the triangle.

00:00:48.250 --> 00:00:49.740
When you aren't
given the height.

00:00:49.740 --> 00:00:53.470
So, for example, how do
you figure out a triangle

00:00:53.470 --> 00:00:55.570
where I just give you the
lengths of the sides.

00:00:55.570 --> 00:01:00.530
Let's say that's side a, side
b, and side c. a, b, and c are

00:01:00.530 --> 00:01:01.640
the lengths of these sides.

00:01:01.640 --> 00:01:03.360
How do you figure that out?

00:01:03.360 --> 00:01:05.270
And to do that we're
going to apply something

00:01:05.270 --> 00:01:06.430
called Heron's Formula.

00:01:12.210 --> 00:01:13.790
And I'm not going to
prove it in this video.

00:01:13.790 --> 00:01:15.200
I'm going to prove it
in a future video.

00:01:15.200 --> 00:01:17.400
And really to prove it you
already probably have

00:01:17.400 --> 00:01:18.720
the tools necessary.

00:01:18.720 --> 00:01:20.480
It's really just the
Pythagorean theorem and

00:01:20.480 --> 00:01:22.220
a lot of hairy algebra.

00:01:22.220 --> 00:01:24.230
But I'm just going to show you
the formula now and how to

00:01:24.230 --> 00:01:26.760
apply it, and then you'll
hopefully appreciate that it's

00:01:26.760 --> 00:01:28.590
pretty simple and pretty
easy to remember.

00:01:28.590 --> 00:01:31.660
And it can be a nice trick
to impress people with.

00:01:31.660 --> 00:01:36.320
So Heron's Formula says first
figure out this third variable

00:01:36.320 --> 00:01:38.640
S, which is essentially
the perimeter of this

00:01:38.640 --> 00:01:40.660
triangle divided by 2.

00:01:40.660 --> 00:01:45.810
a plus b plus c, divided by 2.

00:01:45.810 --> 00:01:49.480
Then once you figure out S, the
area of your triangle-- of this

00:01:49.480 --> 00:01:55.840
triangle right there-- is going
to be equal to the square root

00:01:55.840 --> 00:01:59.710
of S-- this variable S right
here that you just calculated--

00:01:59.710 --> 00:02:10.540
times S minus a, times S
minus b, times S minus c.

00:02:10.540 --> 00:02:12.480
That's Heron's
Formula right there.

00:02:12.480 --> 00:02:13.830
This combination.

00:02:13.830 --> 00:02:16.130
Let me square it off for you.

00:02:16.130 --> 00:02:18.700
So that right there
is Heron's Formula.

00:02:18.700 --> 00:02:21.610
And if that looks a little bit
daunting-- it is a little bit

00:02:21.610 --> 00:02:24.290
more daunting, clearly, than
just 1/2 times base

00:02:24.290 --> 00:02:25.290
times height.

00:02:25.290 --> 00:02:28.040
Let's do it with an actual
example or two, and actually

00:02:28.040 --> 00:02:31.350
see this is actually
not so bad.

00:02:31.350 --> 00:02:33.320
So let's say I have a triangle.

00:02:33.320 --> 00:02:35.300
I'll leave the
formula up there.

00:02:35.300 --> 00:02:37.460
So let's say I have a
triangle that has sides

00:02:37.460 --> 00:02:44.920
of length 9, 11, and 16.

00:02:44.920 --> 00:02:47.040
So let's apply Heron's Formula.

00:02:47.040 --> 00:02:51.190
S in this situation is going to
be the perimeter divided by 2.

00:02:51.190 --> 00:02:56.630
So 9 plus 11 plus
16, divided by 2.

00:02:56.630 --> 00:03:00.430
Which is equal to 9 plus
11-- is 20-- plus 16 is

00:03:00.430 --> 00:03:04.660
36, divided by 2 is 18.

00:03:04.660 --> 00:03:09.430
And then the area by Heron's
Formula is going to be equal to

00:03:09.430 --> 00:03:19.380
the square root of S-- 18--
times S minus a-- S minus 9.

00:03:19.380 --> 00:03:27.790
18 minus 9, times 18 minus
11, times 18 minus 16.

00:03:31.490 --> 00:03:38.200
And then this is equal to
the square root of 18

00:03:38.200 --> 00:03:44.730
times 9 times 7 times 2.

00:03:44.730 --> 00:03:47.340
Which is equal to-- let's
see, 2 times 18 is 36.

00:03:47.340 --> 00:03:48.900
So I'll just
rearrange it a bit.

00:03:48.900 --> 00:03:56.700
This is equal to the square
root of 36 times 9 times 7,

00:03:56.700 --> 00:04:05.540
which is equal to the square
root of 36 times the square

00:04:05.540 --> 00:04:09.330
root of 9 times the
square root of 7.

00:04:09.330 --> 00:04:14.130
The square root
of 36 is just 6.

00:04:14.130 --> 00:04:16.040
This is just 3.

00:04:16.040 --> 00:04:17.750
And we don't deal with the
negative square roots,

00:04:17.750 --> 00:04:19.920
because you can't have
negative side lengths.

00:04:19.920 --> 00:04:23.460
And so this is going to
be equal to 18 times

00:04:23.460 --> 00:04:26.120
the square root of 7.

00:04:26.120 --> 00:04:28.060
So just like that, you saw it,
it only took a couple of

00:04:28.060 --> 00:04:30.760
minutes to apply Heron's
Formula, or even less than

00:04:30.760 --> 00:04:33.420
that, to figure out that the
area of this triangle

00:04:33.420 --> 00:04:38.710
right here is equal to 18
square root of seven.

00:04:38.710 --> 00:04:42.040
Anyway, hopefully you
found that pretty neat.


Amplitude, period, frequency and wavelength of periodic waves

WEBVTT
Kind: captions
Language: en

00:00:00.430 --> 00:00:03.360
In the last video we talked
about the idea that if I start

00:00:03.360 --> 00:00:06.570
with some type of a string
there, and if I were to take

00:00:06.570 --> 00:00:08.870
the left end of the string-- I
could just have equally have

00:00:08.870 --> 00:00:11.370
done the right, but if I take
the left end of the string and

00:00:11.370 --> 00:00:14.280
jerk it up, then all the way
down, and then back to its

00:00:14.280 --> 00:00:16.890
resting position, it'll generate
this disturbance in

00:00:16.890 --> 00:00:17.460
the string.

00:00:17.460 --> 00:00:20.190
And the disturbance might
initially look like this after

00:00:20.190 --> 00:00:23.270
I've done that jerking
up and down once.

00:00:23.270 --> 00:00:25.960
And that disturbance is going to
propagate down the string.

00:00:25.960 --> 00:00:28.330
It's going to move down
the string like that.

00:00:28.330 --> 00:00:31.090
Let me color this in black.

00:00:31.090 --> 00:00:33.950
So this is right after I do that
first cycle-- that first

00:00:33.950 --> 00:00:35.870
jerking up and down.

00:00:35.870 --> 00:00:38.300
The string might look
something like that.

00:00:38.300 --> 00:00:40.710
And if we wait a little while,
the string might look

00:00:40.710 --> 00:00:43.420
something like this, assuming
that I only did that once.

00:00:43.420 --> 00:00:46.980
The string might look something
like this, where

00:00:46.980 --> 00:00:50.630
that pulse has actually
propagated down the string.

00:00:50.630 --> 00:00:52.570
That pulse has propagated
down the string.

00:00:52.570 --> 00:00:54.675
And in the last video, we said,
hey, this disturbance

00:00:54.675 --> 00:00:57.520
that's propagating down the
string, or propagating down

00:00:57.520 --> 00:00:59.580
this medium-- although it
doesn't necessarily have to

00:00:59.580 --> 00:01:05.180
have a medium- we called this
a wave. And in particular,

00:01:05.180 --> 00:01:07.690
this wave right here,
this is a pulse.

00:01:07.690 --> 00:01:11.260
This is a pulse wave because we
only have, essentially, one

00:01:11.260 --> 00:01:13.140
perturbation of the string.

00:01:13.140 --> 00:01:16.010
Now if I kept doing that-- if I
kept going up and down, and

00:01:16.010 --> 00:01:19.290
up and down, essentially, if I
periodically did it at regular

00:01:19.290 --> 00:01:21.430
intervals, then my
string would look

00:01:21.430 --> 00:01:22.680
something like this.

00:01:27.660 --> 00:01:30.070
Doing my best to
draw it neatly.

00:01:30.070 --> 00:01:33.550
It might look something like
this, where once again, the

00:01:33.550 --> 00:01:36.280
perturbations are going
towards-- the disturbances are

00:01:36.280 --> 00:01:38.830
going to move to-- the right.

00:01:38.830 --> 00:01:41.870
They're going to move to the
right with some velocity.

00:01:41.870 --> 00:01:44.510
And what I want to do in this
video is really focus on this

00:01:44.510 --> 00:01:49.390
type of wave. This type of wave
right here, which you can

00:01:49.390 --> 00:01:53.310
imagine, since I'm periodically
moving this left

00:01:53.310 --> 00:01:57.200
side up and down, up and down,
and creating these periodic

00:01:57.200 --> 00:02:01.370
movements in the wave, we call
this a periodic wave. This is

00:02:01.370 --> 00:02:08.979
a periodic wave. The motion is
repeated over and over again.

00:02:08.979 --> 00:02:11.980
So what I want to talk about is
some of the properties of a

00:02:11.980 --> 00:02:15.190
periodic wave. Now, the first
thing you might say is, hey,

00:02:15.190 --> 00:02:17.990
how far are you jerking
it up and down?

00:02:17.990 --> 00:02:20.630
How far are these movements
from rest?

00:02:20.630 --> 00:02:24.130
So if this is the resting
position right there, how far

00:02:24.130 --> 00:02:28.590
are these movements above the
resting position and below the

00:02:28.590 --> 00:02:29.640
resting position?

00:02:29.640 --> 00:02:33.430
And we call that the amplitude
of the wave. So that distance

00:02:33.430 --> 00:02:37.170
right there- I'll do it in
magenta-- that distance right

00:02:37.170 --> 00:02:38.420
there is the amplitude.

00:02:44.340 --> 00:02:47.660
Sometimes mariners will have
an idea of wave height.

00:02:47.660 --> 00:02:50.660
Wave height, they normally refer
to from the bottom--

00:02:50.660 --> 00:02:53.560
from the trough-- of
a wave to its peak.

00:02:53.560 --> 00:02:55.730
Amplitude, we're talking
about from the resting

00:02:55.730 --> 00:02:57.480
position to it's peak.

00:02:57.480 --> 00:02:58.950
So let me label peak.

00:02:58.950 --> 00:03:00.160
I think you know what
peak means.

00:03:00.160 --> 00:03:03.300
Peak is the highest point on
the wave. That's the peak.

00:03:03.300 --> 00:03:05.580
And that's the trough.

00:03:05.580 --> 00:03:08.510
If you're in a fishing boat and
you wanted to see how big

00:03:08.510 --> 00:03:12.280
a wave is, you'd probably care
about the wave height-- not so

00:03:12.280 --> 00:03:14.920
much if your boat's sitting down
here, you have to care

00:03:14.920 --> 00:03:15.760
about this whole distance.

00:03:15.760 --> 00:03:17.780
But anyway, we won't talk
too much about that.

00:03:17.780 --> 00:03:20.600
So that's the first interesting
idea behind a

00:03:20.600 --> 00:03:24.170
wave. And not all waves are
being generated by Sal

00:03:24.170 --> 00:03:25.980
jiggling a string on
the left-hand side.

00:03:25.980 --> 00:03:28.430
But I think you get the idea
that these waves can represent

00:03:28.430 --> 00:03:32.680
many different-- this graph can
represent many different

00:03:32.680 --> 00:03:36.290
types of wave forms. And this,
essentially, displacement,

00:03:36.290 --> 00:03:40.290
from the resting position, or
from the zero position, that

00:03:40.290 --> 00:03:42.330
is your amplitude.

00:03:42.330 --> 00:03:45.210
Now the next question you might
ask is, OK, I know how

00:03:45.210 --> 00:03:49.890
far you're jiggling this string
up and down, but how

00:03:49.890 --> 00:03:51.850
quickly are you doing it?

00:03:51.850 --> 00:03:55.730
So how long does it take for you
to go all the way up, all

00:03:55.730 --> 00:03:58.050
the way down, and back again?

00:03:58.050 --> 00:04:03.630
So how long for each cycle?

00:04:03.630 --> 00:04:06.070
Cycle is me going up,
down, back again.

00:04:06.070 --> 00:04:10.190
How long for each cycle?

00:04:10.190 --> 00:04:13.640
Or you might say how long
for each period?

00:04:13.640 --> 00:04:14.760
We're saying this is periodic.

00:04:14.760 --> 00:04:19.350
Each period is each repetition
of the wave. So this idea of

00:04:19.350 --> 00:04:22.205
how long for each cycle, we
call that the period.

00:04:25.590 --> 00:04:27.560
And this is going to
be a unit of time.

00:04:27.560 --> 00:04:30.420
Maybe I'm doing it every
two seconds.

00:04:30.420 --> 00:04:32.570
It takes me two seconds to
go up, down, back again.

00:04:32.570 --> 00:04:33.610
Up, down, back again.

00:04:33.610 --> 00:04:36.080
That's going to be
two seconds.

00:04:36.080 --> 00:04:38.570
A very related term is,
how many cycles

00:04:38.570 --> 00:04:39.820
am I doing per second?

00:04:53.730 --> 00:04:57.500
So in other words, you
could say, how many

00:04:57.500 --> 00:04:58.840
seconds for each cycle?

00:04:58.840 --> 00:05:00.090
We could even write that.

00:05:06.870 --> 00:05:12.200
So for example, a period might
look like something like 5

00:05:12.200 --> 00:05:16.260
seconds per cycle.

00:05:16.260 --> 00:05:18.240
Or maybe it's 2 seconds
per cycle.

00:05:18.240 --> 00:05:20.870
But what if we're asked how
many cycles per second?

00:05:20.870 --> 00:05:22.600
So we're asking the
opposite question.

00:05:22.600 --> 00:05:25.480
It's not how long, how many
seconds does it take for me to

00:05:25.480 --> 00:05:27.270
go up, down, and back again.

00:05:27.270 --> 00:05:30.200
We're saying in each second, how
many times am I going up,

00:05:30.200 --> 00:05:31.440
down, back again?

00:05:31.440 --> 00:05:33.080
So how many cycles per second?

00:05:33.080 --> 00:05:34.510
That's the inverse of period.

00:05:34.510 --> 00:05:37.600
So period, the notation
is normally a big

00:05:37.600 --> 00:05:40.160
capital T for period.

00:05:40.160 --> 00:05:41.410
This is frequency.

00:05:49.210 --> 00:05:52.430
It's normally denoted by an F.

00:05:52.430 --> 00:05:54.770
And this, you're going to
say cycles per second.

00:05:54.770 --> 00:05:58.240
So if you're going 5 seconds per
cycle, that means you're

00:05:58.240 --> 00:06:08.550
doing 1/5 of a cycle or, 1/5
of a cycle per second.

00:06:08.550 --> 00:06:10.510
All I did is invert
this right there.

00:06:10.510 --> 00:06:11.560
And that make sense.

00:06:11.560 --> 00:06:14.490
Because the period and the
frequency are inversions of

00:06:14.490 --> 00:06:14.860
each other.

00:06:14.860 --> 00:06:17.160
This is how many seconds
per cycle.

00:06:17.160 --> 00:06:20.550
How long does one up, down,
back again take?

00:06:20.550 --> 00:06:22.710
And this is how many up,
down, back agains

00:06:22.710 --> 00:06:23.670
are there in a second?

00:06:23.670 --> 00:06:25.040
So they are inverses
of each other.

00:06:25.040 --> 00:06:30.090
So we could say that frequency
is equal to 1 over the period.

00:06:30.090 --> 00:06:32.180
Or you could say that
period is equal

00:06:32.180 --> 00:06:34.580
to 1 over the frequency.

00:06:34.580 --> 00:06:39.860
So if I told you that I'm
vibrating the left end of this

00:06:39.860 --> 00:06:49.060
rope at 10 cycles per second--
and, by the way, the unit of

00:06:49.060 --> 00:06:52.300
cycles per second, this is a
hertz, so I could have also

00:06:52.300 --> 00:06:54.120
written this down as 10
hertz, which you've

00:06:54.120 --> 00:06:56.000
probably heard before.

00:06:56.000 --> 00:06:59.360
10 hertz just means 10
cycles per second.

00:06:59.360 --> 00:07:03.770
If my frequency is 10 cycles per
second, my period is going

00:07:03.770 --> 00:07:05.540
to be 1 over that.

00:07:05.540 --> 00:07:13.580
So 1 over 10 seconds per cycle,
which makes sense.

00:07:13.580 --> 00:07:16.680
If in 10 times, I can go up and
down, a whole up, down,

00:07:16.680 --> 00:07:19.560
back again, if I can do that
10 times in a second, it's

00:07:19.560 --> 00:07:23.690
going to take me 1/10 of a
second to do it each time.

00:07:23.690 --> 00:07:28.460
Now another question we might
ask ourselves is, how quickly

00:07:28.460 --> 00:07:31.170
is this wave moving, in this
case, to the right?

00:07:31.170 --> 00:07:33.010
Since I'm jiggling the left
end of the string.

00:07:33.010 --> 00:07:37.700
How quickly is it moving
to the right?

00:07:37.700 --> 00:07:40.170
So the velocity.

00:07:40.170 --> 00:07:42.770
So to do that, we need to figure
out how far did the

00:07:42.770 --> 00:07:45.200
wave go after one cycle?

00:07:45.200 --> 00:07:47.270
Or after one period?

00:07:47.270 --> 00:07:51.750
So after I jiggled this once,
how far did the wave go?

00:07:51.750 --> 00:07:54.520
What is this distance from this
resting point to this

00:07:54.520 --> 00:07:56.010
resting point there?

00:07:56.010 --> 00:07:57.370
And we call that a wavelength.

00:08:01.760 --> 00:08:04.480
And there's a lot of different
ways you can define a

00:08:04.480 --> 00:08:05.060
wavelength.

00:08:05.060 --> 00:08:10.160
You could view a wavelength as
how far the initial pulse went

00:08:10.160 --> 00:08:13.070
after completing exactly
one cycle.

00:08:13.070 --> 00:08:16.390
Or you could view it as the
distance from one peak to

00:08:16.390 --> 00:08:17.360
another peak.

00:08:17.360 --> 00:08:19.360
That is also going to
be the wavelength.

00:08:19.360 --> 00:08:23.070
Or you could view it as a
distance from one trough to

00:08:23.070 --> 00:08:24.100
the other trough.

00:08:24.100 --> 00:08:25.390
That's also the wavelength.

00:08:25.390 --> 00:08:30.240
Or in general, you could view
the wavelength as one exactly

00:08:30.240 --> 00:08:32.880
equal point on the
wave. From that

00:08:32.880 --> 00:08:34.700
distance to that distance.

00:08:34.700 --> 00:08:37.970
That is also one wavelength.

00:08:37.970 --> 00:08:39.940
Where you're completing, between
that point and that

00:08:39.940 --> 00:08:43.150
point, you're completing one
entire cycle to get exactly

00:08:43.150 --> 00:08:45.090
back to that same point.

00:08:45.090 --> 00:08:47.500
And when I say exactly back to
that same point, this point

00:08:47.500 --> 00:08:48.450
doesn't count.

00:08:48.450 --> 00:08:51.220
Because this point, although
we're in the same position,

00:08:51.220 --> 00:08:52.670
we're now going down.

00:08:52.670 --> 00:08:53.940
We want to go to
the point where

00:08:53.940 --> 00:08:55.330
we're in the same position.

00:08:55.330 --> 00:08:57.390
And notice over here,
we're going up.

00:08:57.390 --> 00:08:58.890
We want to be going up again.

00:08:58.890 --> 00:09:01.590
So distance is not
one wavelength.

00:09:01.590 --> 00:09:02.530
To go one wavelength,
we have to go

00:09:02.530 --> 00:09:03.780
back to the same position.

00:09:03.780 --> 00:09:06.340
And we're moving in the
same direction.

00:09:06.340 --> 00:09:08.920
So this is also one
wavelength.

00:09:08.920 --> 00:09:12.630
So if we know how far we've
travelled after one period--

00:09:12.630 --> 00:09:22.220
let me write it this way;
wavelength is equal to how far

00:09:22.220 --> 00:09:31.625
the wave has traveled
after one period.

00:09:34.590 --> 00:09:37.170
Or you could say after
one cycle.

00:09:41.130 --> 00:09:44.020
Because remember, a period is
how long does it take to

00:09:44.020 --> 00:09:45.540
complete one cycle.

00:09:45.540 --> 00:09:48.340
One to complete up, down,
and back again notion.

00:09:48.340 --> 00:09:51.600
So if we know how far we've
traveled, and we know how long

00:09:51.600 --> 00:09:54.750
it took us, it took us one
period, how can we figure out

00:09:54.750 --> 00:09:56.350
the velocity?

00:09:56.350 --> 00:10:02.175
Well, the velocity is equal to
distance divided by time.

00:10:09.450 --> 00:10:12.340
For a wave, your velocity--
and I could write it as a

00:10:12.340 --> 00:10:14.260
vector, but I think you
get the general idea.

00:10:14.260 --> 00:10:15.800
Your velocity-- what's
the distance

00:10:15.800 --> 00:10:17.650
you travel in a period?

00:10:17.650 --> 00:10:23.390
Well, the distance you travel in
a period is your wavelength

00:10:23.390 --> 00:10:27.640
after one up, down,
back again.

00:10:27.640 --> 00:10:30.210
The wave pulse would have
traveled exactly that far.

00:10:30.210 --> 00:10:31.910
That would be my wavelength.

00:10:31.910 --> 00:10:34.830
So I've traveled the distance of
a wavelength, and how long

00:10:34.830 --> 00:10:36.850
did it take me to travel
that distance?

00:10:36.850 --> 00:10:39.180
Well, it took me a period
to travel that distance.

00:10:39.180 --> 00:10:42.170
So it's wavelength divided
by period.

00:10:46.320 --> 00:10:50.790
Now I just said that 1 over the
period is the same thing

00:10:50.790 --> 00:10:51.410
as the frequency.

00:10:51.410 --> 00:10:54.910
So I could rewrite this
as wavelength.

00:10:54.910 --> 00:10:57.570
And actually, I should
be clear here.

00:10:57.570 --> 00:11:00.540
The notation for wavelength
tends to be the

00:11:00.540 --> 00:11:01.810
Greek letter lambda.

00:11:01.810 --> 00:11:06.550
So we could say velocity
is equal to

00:11:06.550 --> 00:11:09.510
wavelength over period.

00:11:09.510 --> 00:11:11.170
Which is the same thing
as wavelength

00:11:11.170 --> 00:11:14.110
times 1 over my period.

00:11:14.110 --> 00:11:17.050
And we just said that 1 over the
period, this is the same

00:11:17.050 --> 00:11:18.570
thing is your frequency.

00:11:18.570 --> 00:11:21.650
So velocity is equal to

00:11:21.650 --> 00:11:25.600
wavelength times your frequency.

00:11:25.600 --> 00:11:28.090
And if you know this, you can
pretty much solve all of the

00:11:28.090 --> 00:11:31.470
basic problems that you might
encounter in waves.

00:11:31.470 --> 00:11:34.400
So for example, if someone
tells you that I have a

00:11:34.400 --> 00:11:42.520
velocity of-- I don't know-- 100
meters per second to the

00:11:42.520 --> 00:11:45.940
right, so in that direction--
velocity you have to give a

00:11:45.940 --> 00:11:54.410
direction-- and they were to
tell you that my frequency is

00:11:54.410 --> 00:12:00.770
equal to-- let's say my
frequency is 20 cycles per

00:12:00.770 --> 00:12:04.370
second, which is the same
thing as 20 hertz.

00:12:04.370 --> 00:12:08.790
So literally, if you had a
little window where you're

00:12:08.790 --> 00:12:13.410
only able to observe this part
of your wave, you'd only

00:12:13.410 --> 00:12:15.160
observe that part
of my string.

00:12:15.160 --> 00:12:18.020
If we're talking about 20 hertz,
then in 1 second, you

00:12:18.020 --> 00:12:19.960
would see this go up and
down twenty times.

00:12:19.960 --> 00:12:23.020
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17,

00:12:23.020 --> 00:12:24.410
18, 19, 20.

00:12:24.410 --> 00:12:26.280
In exactly one second,
you would see this go

00:12:26.280 --> 00:12:27.510
up and down 20 times.

00:12:27.510 --> 00:12:30.960
That's what we mean by the
frequency being 20 hertz, or

00:12:30.960 --> 00:12:32.590
20 cycles per second.

00:12:32.590 --> 00:12:33.495
So, they gave you
the velocity.

00:12:33.495 --> 00:12:34.470
They gave you the frequency.

00:12:34.470 --> 00:12:36.380
What's the wavelength here?

00:12:36.380 --> 00:12:39.380
So the wavelength, in this
situation-- you would say the

00:12:39.380 --> 00:12:45.130
velocity-- your velocity is
equal to your wavelength times

00:12:45.130 --> 00:12:47.780
your frequency.

00:12:47.780 --> 00:12:49.380
Divide both sides by 20.

00:12:49.380 --> 00:12:52.470
And actually, let me make sure
I get the units right.

00:12:52.470 --> 00:12:58.110
So this is meters per second,
is equal to lambda times 20

00:12:58.110 --> 00:13:02.450
cycles per second.

00:13:02.450 --> 00:13:05.610
So if you divide both sides by
20 cycles per second, you get

00:13:05.610 --> 00:13:14.010
100 meters per second times
1/20 seconds per cycle.

00:13:14.010 --> 00:13:16.490
And then this becomes a 5.

00:13:16.490 --> 00:13:17.370
This becomes a 1.

00:13:17.370 --> 00:13:21.760
So you get 5, and then the
seconds cancel out.

00:13:21.760 --> 00:13:23.680
So you get 5 meters per cycle.

00:13:23.680 --> 00:13:30.830
So this is equal to 5 meters per
cycle, which would be your

00:13:30.830 --> 00:13:32.150
wavelength in this situation.

00:13:32.150 --> 00:13:34.350
So it's 5 meters.

00:13:34.350 --> 00:13:37.130
You could say 5 meters per
cycle, but wavelength implies

00:13:37.130 --> 00:13:39.810
that you're talking about
the distance per cycle.

00:13:39.810 --> 00:13:44.760
So in this situation, if this is
moving to the right at 100

00:13:44.760 --> 00:13:47.440
meters per second and this
frequency-- I see this moving

00:13:47.440 --> 00:13:50.830
up and down 20 times in a
second-- then this distance,

00:13:50.830 --> 00:13:53.700
right here, must be 5 meters.

00:13:53.700 --> 00:13:57.050
Likewise, we can figure out
the period very easily.

00:13:57.050 --> 00:13:59.710
The period here is just going
to be 1 over the frequency.

00:13:59.710 --> 00:14:06.470
It's going to be 1/20
seconds per cycle.

00:14:06.470 --> 00:14:10.190
So using these formulas-- and I
don't want you to memorize a

00:14:10.190 --> 00:14:13.350
formula, it should be
intuitive for you.

00:14:13.350 --> 00:14:15.470
And hopefully, this video made
it a little bit intuitive.

00:14:15.470 --> 00:14:18.380
But using this, you can really
answer almost any question if

00:14:18.380 --> 00:14:22.770
you're given two of these
variables and you need to

00:14:22.770 --> 00:14:23.990
solve for the third.

00:14:23.990 --> 00:14:26.190
Anyway, hopefully you
found that helpful.


Interesting Polynomial Coefficient Problem

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.950
I've been sent is pretty
interesting algebra problem,

00:00:02.950 --> 00:00:06.760
and I think the question isn't
properly formed, or it might be

00:00:06.760 --> 00:00:08.400
missing some piece
of information.

00:00:08.400 --> 00:00:10.050
But it's an interesting
question nonetheless so I

00:00:10.050 --> 00:00:12.340
thought I'd work it out,
and we can talk about why

00:00:12.340 --> 00:00:14.150
it's not properly formed.

00:00:14.150 --> 00:00:16.520
So they give us a
function, f of x.

00:00:16.520 --> 00:00:18.910
They say it's a third degree
polynomial of the form

00:00:18.910 --> 00:00:28.020
ax to the third plus bx
squared plus cx plus d.

00:00:28.020 --> 00:00:32.280
And they tell us a couple of
the 0's of this polynomial.

00:00:32.280 --> 00:00:40.330
They say that we have 0's at
the point minus 1, 0 and

00:00:40.330 --> 00:00:42.150
the point 2 comma 0.

00:00:42.150 --> 00:00:46.360
And they tell us we have a
y-intercept, y-intercept,

00:00:46.360 --> 00:00:50.760
at the point 0, minus 2.

00:00:50.760 --> 00:00:58.320
And what they ask us is, what
is a plus b plus c plus d?

00:00:58.320 --> 00:01:01.330
So let's see if we can make
some headway on this.

00:01:01.330 --> 00:01:03.830
So the first thing is to think
about, is what would a third

00:01:03.830 --> 00:01:06.010
degree polynomial look like and
what are we even talking

00:01:06.010 --> 00:01:07.070
about when we say 0's.

00:01:07.070 --> 00:01:09.430
So let me just draw a
little bit of a graph.

00:01:09.430 --> 00:01:12.280
And I don't know exactly what
this third degree polynomial

00:01:12.280 --> 00:01:14.150
looks like just yet.

00:01:14.150 --> 00:01:16.840
Let me draw my axis.

00:01:16.840 --> 00:01:20.880
Now a third degree polynomial
can have as many as three 0's.

00:01:20.880 --> 00:01:23.350
And 0's are just the
points where the

00:01:23.350 --> 00:01:24.750
function is equal to 0.

00:01:24.750 --> 00:01:27.730
So, we have a 0 at minus 1, 0.

00:01:27.730 --> 00:01:30.320
So that's minus 1, 0,
that's right there.

00:01:30.320 --> 00:01:32.640
We have a zero 2, 0.

00:01:32.640 --> 00:01:34.110
1, 2, 0.

00:01:34.110 --> 00:01:36.530
So those are the two 0's
that they've given us.

00:01:36.530 --> 00:01:40.290
They also told us the
y-intercept at 0, minus 2.

00:01:40.290 --> 00:01:42.720
So it intersects the
y-axis right there.

00:01:42.720 --> 00:01:46.200
But this guy can have
as many as three 0's.

00:01:46.200 --> 00:01:49.270
Now some of them
might be complex.

00:01:49.270 --> 00:01:52.750
But complex 0's come in
pairs of two, the two

00:01:52.750 --> 00:01:53.980
complex conjugates.

00:01:53.980 --> 00:01:55.710
I won't go into too
much detail here.

00:01:55.710 --> 00:01:59.630
So this one must have a third
real 0, because if the third

00:01:59.630 --> 00:02:02.060
one was complex, you'd need
another complex 0, and you

00:02:02.060 --> 00:02:06.920
can't have four 0's for a
third degree polynomial.

00:02:06.920 --> 00:02:09.760
So that third root, it could
be sitting some place out

00:02:09.760 --> 00:02:11.460
here, maybe over here.

00:02:11.460 --> 00:02:13.910
Or it could even be a
repeat of one of these

00:02:13.910 --> 00:02:14.850
two roots over here.

00:02:14.850 --> 00:02:16.890
But let's just assume that
there's some third root, we

00:02:16.890 --> 00:02:18.260
don't know what it looks like.

00:02:18.260 --> 00:02:20.610
Let's say it looks something
like-- let's say that third

00:02:20.610 --> 00:02:22.410
root is sitting out
here some place.

00:02:22.410 --> 00:02:25.550
And then a potential graph, and
I could be completely wrong,

00:02:25.550 --> 00:02:27.750
I'm just guessing, but just to
get a sense of what third

00:02:27.750 --> 00:02:30.980
degree polynomials look like,
and how can a graph intersect

00:02:30.980 --> 00:02:34.160
the x-axis three times, a
potential graph might look

00:02:34.160 --> 00:02:35.880
something like this.

00:02:35.880 --> 00:02:39.305
Curves and bam, hits the other
0, hits the y-intercept, and

00:02:39.305 --> 00:02:40.530
then goes back up like that.

00:02:40.530 --> 00:02:43.040
It might go like that, it might
go the other way, go like that.

00:02:43.040 --> 00:02:45.120
Up, and then down like that.

00:02:45.120 --> 00:02:47.920
Well, there's many ways you
could draw something that

00:02:47.920 --> 00:02:51.580
essentially curves twice to
intersect these three points

00:02:51.580 --> 00:02:54.870
and also that y-intercept,
but I'm not going to go into

00:02:54.870 --> 00:02:55.720
all of those right now.

00:02:55.720 --> 00:02:59.190
But let's see if we can figure
out the coefficients here.

00:02:59.190 --> 00:03:01.590
So the key point is me
telling you that there

00:03:01.590 --> 00:03:03.250
must be a third 0 here.

00:03:03.250 --> 00:03:06.600
Let's call that third 0, it's
at the point, it's at the

00:03:06.600 --> 00:03:10.470
point-- let's call
it --r3 comma 0.

00:03:10.470 --> 00:03:13.390
And I'm using the
letter r for roots.

00:03:13.390 --> 00:03:16.890
Roots are the x
values of the 0's.

00:03:16.890 --> 00:03:23.950
So if we say f of minus 1 is
equal to 0, we say that x is

00:03:23.950 --> 00:03:26.720
equal to minus 1 is a root.

00:03:26.720 --> 00:03:32.360
Similarly, we know that f of
2, f of 2, is equal to 0.

00:03:32.360 --> 00:03:36.780
Or we could say x equals 2
is a root, these are roots.

00:03:36.780 --> 00:03:40.160
So when someone tells you a 0
they're also essentially

00:03:40.160 --> 00:03:43.360
telling you the roots, and then
we know there must be a third

00:03:43.360 --> 00:03:46.300
root at x is equal to r3.

00:03:46.300 --> 00:03:49.340
So x is equal to r3.

00:03:49.340 --> 00:03:53.980
Now, if we have a third degree
polynomial where these three x

00:03:53.980 --> 00:03:57.570
values make it 0, we can
rewrite this third degree

00:03:57.570 --> 00:04:01.270
polynomial as-- we can rewrite
it as-- I'll do it in a

00:04:01.270 --> 00:04:07.820
slightly different color --f of
x is equal to x plus 1-- you'll

00:04:07.820 --> 00:04:10.300
see why I'm doing x plus 1
instead of x minus 1 in a

00:04:10.300 --> 00:04:20.230
second --plus 1 times x
minus 2 times x minus r3.

00:04:20.230 --> 00:04:24.015
And we want to put an a out
front, and I'll tell you in

00:04:24.015 --> 00:04:26.550
a second why I'm putting
this a out front.

00:04:26.550 --> 00:04:30.400
Because if you were to multiply
just the x terms here, when

00:04:30.400 --> 00:04:33.370
actually do your distributed
property and multiply out these

00:04:33.370 --> 00:04:36.260
binomials, you'll just get x to
the third, but we had

00:04:36.260 --> 00:04:37.390
an ax to the third.

00:04:37.390 --> 00:04:40.360
So you just needed a constant
a there to make this

00:04:40.360 --> 00:04:42.100
an ax to the third.

00:04:42.100 --> 00:04:44.490
Now, why did I write
out like this?

00:04:44.490 --> 00:04:47.520
Well, if you put a minus 1 in
here what's going to happen?

00:04:47.520 --> 00:04:49.660
This term is going to
be minus 1 plus 1.

00:04:49.660 --> 00:04:50.850
It's going to be 0.

00:04:50.850 --> 00:04:53.210
Who cares what these are
or what a is, the whole

00:04:53.210 --> 00:04:54.380
thing's going to be 0.

00:04:54.380 --> 00:04:57.170
If we put x is equal
to 2, this term right

00:04:57.170 --> 00:04:58.610
here is going to be 0.

00:04:58.610 --> 00:05:01.190
Who cares what the other
terms or the a is, f

00:05:01.190 --> 00:05:03.070
of 2 is going to be 0.

00:05:03.070 --> 00:05:06.360
Similarly, when x is equal to
r3, this last term is 0, who

00:05:06.360 --> 00:05:08.170
cares about anything over here.

00:05:08.170 --> 00:05:11.550
So we know that this guy
up here can be rewritten

00:05:11.550 --> 00:05:13.520
in this form over here.

00:05:13.520 --> 00:05:17.490
And so if we can solve for r3
we can just multiply this thing

00:05:17.490 --> 00:05:20.370
out and try to do a little bit
of pattern matching to figure

00:05:20.370 --> 00:05:23.760
out what these coefficients
are going to be.

00:05:23.760 --> 00:05:28.100
Now, the other big clue they
gave us is a y-intercept.

00:05:28.100 --> 00:05:31.640
That point right there,
the point 0, minus 2.

00:05:31.640 --> 00:05:36.160
They're telling us that f
of 0 is equal to minus 2.

00:05:36.160 --> 00:05:37.940
Well what's f of 0?

00:05:37.940 --> 00:05:42.730
f of 0-- I'll write it here --f
of 0-- If you put 0 in here,

00:05:42.730 --> 00:05:46.420
this term becomes 0, this term
becomes 0, that term becomes 0,

00:05:46.420 --> 00:05:47.840
you are just left with the d.

00:05:47.840 --> 00:05:49.940
So f of 0 is equal to d.

00:05:49.940 --> 00:05:53.760
Or we could say that d
must be equal to minus 2.

00:05:53.760 --> 00:05:57.530
d is equal to minus 2, so
we solved at least one

00:05:57.530 --> 00:05:59.090
of the call efficient.

00:05:59.090 --> 00:06:02.450
The constant term. d
is equal to minus 2.

00:06:02.450 --> 00:06:05.970
Now, if d is equal to minus 2,
that means all of these

00:06:05.970 --> 00:06:08.740
constant terms, when you were
to multiply out this

00:06:08.740 --> 00:06:12.340
expression, and I would include
the a, because the a is scaling

00:06:12.340 --> 00:06:15.620
everything, that must
be equal to minus 2.

00:06:15.620 --> 00:06:16.730
Let me be clear here.

00:06:16.730 --> 00:06:19.910
So let me rewrite
this expression.

00:06:19.910 --> 00:06:22.970
This is equal to, I'll just
multiply this a times

00:06:22.970 --> 00:06:23.740
this last term.

00:06:23.740 --> 00:06:27.400
We do it in any order you want,
so this is the same thing as x

00:06:27.400 --> 00:06:36.450
plus 1 times x minus 2 to times
a a minus r, let me

00:06:36.450 --> 00:06:40.870
write, minus ar3.

00:06:40.870 --> 00:06:43.400
This is the exact same thing as
this thing up here, I'll just

00:06:43.400 --> 00:06:46.860
write this is with f of x is
equal to, and this is the exact

00:06:46.860 --> 00:06:48.460
same thing is that up there.

00:06:48.460 --> 00:06:53.560
Now, to solve for an r3, or
attempt to solve for an r3, you

00:06:53.560 --> 00:06:56.540
just have to realize that when
you multiply this thing out,

00:06:56.540 --> 00:06:59.690
the way to get the constant
term over here, or the d over

00:06:59.690 --> 00:07:02.840
here, that's going to be a
product of the constant

00:07:02.840 --> 00:07:04.310
terms in our expressions.

00:07:04.310 --> 00:07:06.920
Because if I introduced any
products with any x terms,

00:07:06.920 --> 00:07:09.870
you're going to get one of
these other terms over here.

00:07:09.870 --> 00:07:13.300
The ax cubed term is generated
from that term, that term,

00:07:13.300 --> 00:07:15.320
and that term, being
multiplied together.

00:07:15.320 --> 00:07:17.680
and then the constant term
is multiplied by the

00:07:17.680 --> 00:07:19.790
constant terms to be
multiplied together.

00:07:19.790 --> 00:07:22.080
And then the two in the middle
are multiplied by different

00:07:22.080 --> 00:07:24.260
combinations of constant terms.

00:07:24.260 --> 00:07:26.460
And you'll see that if
you actually want to

00:07:26.460 --> 00:07:28.260
multiply this out.

00:07:28.260 --> 00:07:35.000
But if you take my word for it,
that times that times that

00:07:35.000 --> 00:07:37.340
times, actually let me
write it this way.

00:07:37.340 --> 00:07:43.820
That times that times that,
have got to be equal to date we

00:07:43.820 --> 00:07:47.550
can then attempt to find some
relationship between a and d,

00:07:47.550 --> 00:07:49.500
which is mine is minus 2.

00:07:49.500 --> 00:07:57.890
So we could say 1 times minus
2 times minus ar3, these two

00:07:57.890 --> 00:08:00.050
minuses will cancel out.

00:08:00.050 --> 00:08:02.170
It is going to be
equal to our d terms.

00:08:02.170 --> 00:08:07.110
It's going to be equal to minus
2, and so if we simplify that a

00:08:07.110 --> 00:08:14.720
little bit we get 2ar3 is equal
to minus 2, divide both sides

00:08:14.720 --> 00:08:18.755
by 2a and you get r3 is
equal to minus 1/a.

00:08:21.320 --> 00:08:24.010
So we haven't solved for the
third root, we've just gotten

00:08:24.010 --> 00:08:27.990
it in terms of, in terms
of our coefficient a.

00:08:27.990 --> 00:08:33.060
But letls see if we can still
use that in some useful way.

00:08:33.060 --> 00:08:37.710
So if r3 is minus 1/a, this
term right here becomes what?

00:08:37.710 --> 00:08:39.210
Let me rewrite it.

00:08:39.210 --> 00:08:41.260
Let me rewrite it.

00:08:41.260 --> 00:08:43.030
All right, I'll do it in blue.

00:08:43.030 --> 00:08:50.700
f of x is going to be equal to
x plus 1 times x minus 2 times

00:08:50.700 --> 00:08:54.610
ax-- now minus a times r3.

00:08:54.610 --> 00:08:55.250
Let me do it over here.

00:08:55.250 --> 00:08:59.780
Minus a times r3
three is minus 1/a.

00:08:59.780 --> 00:09:01.340
So that's a minus
times a minus.

00:09:01.340 --> 00:09:02.420
That's a plus.

00:09:02.420 --> 00:09:03.420
The a's cancel out.

00:09:03.420 --> 00:09:04.210
You just get a 1.

00:09:04.210 --> 00:09:07.300
So this term now if we
substitute r3 with this will

00:09:07.300 --> 00:09:11.140
just turn into a this will
just turn into a plus 1.

00:09:11.140 --> 00:09:15.640
Minus a times minus
1/a that's plus 1.

00:09:15.640 --> 00:09:16.450
Plus 1.

00:09:16.450 --> 00:09:20.160
We just substituted that for
that to get this right here.

00:09:20.160 --> 00:09:21.790
Now let's multiply
this thing out.

00:09:21.790 --> 00:09:24.120
Let's actually do the
algebraic multiplication

00:09:24.120 --> 00:09:26.130
and see what we get.

00:09:26.130 --> 00:09:27.980
So first I'm going to
multiply these two terms.

00:09:27.980 --> 00:09:30.580
I'm going to do it in
a different color.

00:09:30.580 --> 00:09:34.050
Let me multiply those
two terms right there.

00:09:34.050 --> 00:09:39.740
So this is going to be equal to
x squared, x squared plus 1x

00:09:39.740 --> 00:09:43.400
minus 2x So that's
minus x, right?

00:09:43.400 --> 00:09:48.235
1 times x is x, minus 2
times x is x minus 2x.

00:09:48.235 --> 00:09:53.110
So it's minus x, and then minus
2, just multiplied these two

00:09:53.110 --> 00:09:54.940
binomials out right here.

00:09:54.940 --> 00:09:59.000
And then I'm going to have to
multiply that times ax plus 1,

00:09:59.000 --> 00:10:00.880
so I'm going to write the
ax plus 1 down here

00:10:00.880 --> 00:10:02.720
just to save space.

00:10:02.720 --> 00:10:05.810
ax plus 1.

00:10:05.810 --> 00:10:09.600
And now we just do a little bit
of algebraic multiplication.

00:10:09.600 --> 00:10:14.140
1 times minus 2 is-- I'll do
it in magenta --minus 2.

00:10:14.140 --> 00:10:17.150
1 times minus x is minus x.

00:10:17.150 --> 00:10:20.690
1 times x squared is x squared.

00:10:20.690 --> 00:10:26.820
Now ax times minus
2 is minus 2ax.

00:10:26.820 --> 00:10:29.910
Minus 2ax.

00:10:29.910 --> 00:10:37.680
ax times minus x is minus
ax squared, right?

00:10:37.680 --> 00:10:41.160
Minus ax squared and
then ax times x squared

00:10:41.160 --> 00:10:43.020
is ax to the third.

00:10:43.020 --> 00:10:44.750
ax to the third.

00:10:44.750 --> 00:10:48.220
Then we add everything
together, and we get our f of x

00:10:48.220 --> 00:10:52.380
is going to be equal to ax to
the third-- let me scroll to

00:10:52.380 --> 00:10:54.710
the left a little bit
--ax to the third.

00:10:54.710 --> 00:10:57.920
x squared minus ax squared.

00:10:57.920 --> 00:11:04.620
So that's plus 1 minus
a, 1 minus a x squared.

00:11:04.620 --> 00:11:08.360
And then we have these two
added together so this is minus

00:11:08.360 --> 00:11:17.660
1 minus 2a or we could call
this equal to-- so minus 1

00:11:17.660 --> 00:11:25.890
minus 2a --or we could say
minus 1 plus 2ax and then we

00:11:25.890 --> 00:11:30.090
have our minus 2, which makes
sense because that is our d.

00:11:30.090 --> 00:11:31.300
That's our d.

00:11:31.300 --> 00:11:33.030
So this is really the solution.

00:11:33.030 --> 00:11:36.410
This is the form that are
equation is going to take and

00:11:36.410 --> 00:11:38.590
the reason why I said in the
beginning of the problem that

00:11:38.590 --> 00:11:42.270
the equation or the problem
wasn't well defined is that I

00:11:42.270 --> 00:11:48.430
can pick any real number a
and this equation will

00:11:48.430 --> 00:11:51.280
satisfy these conditions
that we were given.

00:11:51.280 --> 00:11:54.270
So I suspect that I wasn't
given all of the constraints

00:11:54.270 --> 00:11:54.940
on this problem.

00:11:54.940 --> 00:11:56.930
Maybe there was a third
constraint that a is equal to

00:11:56.930 --> 00:11:59.990
1, or b is equal to something
else, or a plus b is

00:11:59.990 --> 00:12:00.860
equal to something else.

00:12:00.860 --> 00:12:04.060
But just using these
constraints, just using these

00:12:04.060 --> 00:12:07.050
constraints, the equation
will take this form.

00:12:07.050 --> 00:12:11.690
So, if it was in this form, of
course this is a, this is b,

00:12:11.690 --> 00:12:15.800
which would be 1 minus a,
that's b, that's a, and then

00:12:15.800 --> 00:12:18.540
this right here is equal to c.

00:12:18.540 --> 00:12:21.730
So we could pick, you know, we
could pick one particular

00:12:21.730 --> 00:12:24.320
choice if we just want to get
an answer to their problem, but

00:12:24.320 --> 00:12:26.210
it might not be the answer they
were looking for because they

00:12:26.210 --> 00:12:27.670
might have picked
a different a.

00:12:27.670 --> 00:12:30.290
But if you just pick a simple
a is equal to 1, than

00:12:30.290 --> 00:12:31.690
what's b equal to?

00:12:31.690 --> 00:12:34.750
Then b is equal to 1 minus
a, which is equal to 0.

00:12:34.750 --> 00:12:38.730
Then b is equal to 0, and then
c is equal-- actually c would

00:12:38.730 --> 00:12:42.530
include this minus sign right
there --c would be one plus 2a.

00:12:42.530 --> 00:12:46.470
So 2a is 2, 1 plus 2 is 3.

00:12:46.470 --> 00:12:47.790
Put a minus sign.

00:12:47.790 --> 00:12:49.920
So c would be equal to minus 3.

00:12:49.920 --> 00:12:52.280
And of course no matter
what we do, d is going

00:12:52.280 --> 00:12:55.480
to be equal to minus 2.

00:12:55.480 --> 00:13:00.230
So if we pick a is equal to 1,
the answer here to what is a

00:13:00.230 --> 00:13:07.970
plus b plus c plus d, would
be equal to 1 plus minus 3.

00:13:07.970 --> 00:13:11.410
1 plus minus 3 is minus 2.

00:13:11.410 --> 00:13:14.530
Minus another minus
2 is minus 4.

00:13:14.530 --> 00:13:18.110
That's another possibility, but
we just as easily could have

00:13:18.110 --> 00:13:21.310
picked, you know a could have
been something, it could be a

00:13:21.310 --> 00:13:22.630
fraction, it could be anything.

00:13:22.630 --> 00:13:26.310
It could be, you know
a could be minus 1.

00:13:26.310 --> 00:13:29.990
a could be equal to minus 1, if
a is equal to minus 1, then b

00:13:29.990 --> 00:13:33.930
is equal to 2, then b
is equal to 2, right?

00:13:33.930 --> 00:13:35.400
1 minus minus 1.

00:13:35.400 --> 00:13:38.790
And then c is equal to 1 minus
2, which is minus 1, but then

00:13:38.790 --> 00:13:40.400
you have a minus out front.

00:13:40.400 --> 00:13:44.040
So then c is equal to 1,
and then d is always going

00:13:44.040 --> 00:13:46.230
to be equal to minus 2.

00:13:46.230 --> 00:13:47.930
d is equal to minus 2.

00:13:47.930 --> 00:13:53.020
And then you have a plus b plus
c plus d would be equal to.,

00:13:53.020 --> 00:13:57.350
well these would cancel out,
you'd get them equal to 0.

00:13:57.350 --> 00:14:01.110
So, we don't know exactly what
answer the problem writer

00:14:01.110 --> 00:14:03.500
was looking for, but it is a
pretty interesting problem.

00:14:03.500 --> 00:14:06.530
And then you know, just out of
curiosity, it-- let's take this

00:14:06.530 --> 00:14:09.750
first example that we took
--where a is equal to 1, we

00:14:09.750 --> 00:14:12.280
could then look at what
our root must be.

00:14:12.280 --> 00:14:17.470
If a is equal to 1, then our
third root is that r3 would be

00:14:17.470 --> 00:14:20.790
equal-- that's just for this
situation right here --than r3

00:14:20.790 --> 00:14:23.330
would be equal to negative 1
divided by 1 would be equal to

00:14:23.330 --> 00:14:27.410
minus 1, which means that we
would have a repeated root, we

00:14:27.410 --> 00:14:30.940
would have a repeated
root, right there.

00:14:30.940 --> 00:14:34.830
We wouldn't necessarily have
a distinct root out there.

00:14:34.830 --> 00:14:39.730
And if we're curious about what
that would look like, let's

00:14:39.730 --> 00:14:43.130
look at the situation
where a is equal to 1.

00:14:43.130 --> 00:14:44.280
Actually maybe we wil
look at both of them.

00:14:44.280 --> 00:14:48.710
When a is equal to 1, our
equation is x to the third--

00:14:48.710 --> 00:14:54.140
where is the x button --x to
the third, there's no x

00:14:54.140 --> 00:15:03.040
squared term, minus 3x minus
3x, minus 3x, minus 2.

00:15:03.040 --> 00:15:05.880
So let's grab both of
them simultaneously.

00:15:05.880 --> 00:15:08.160
So that's this one, where
we picked this choice

00:15:08.160 --> 00:15:10.210
of coefficients.

00:15:10.210 --> 00:15:13.760
And then let's do this choice
of coefficients, if we have, if

00:15:13.760 --> 00:15:23.270
we have minus x to the third,
right? a is minus 1 plus 2 x

00:15:23.270 --> 00:15:31.350
squared, and then we have, plus
x, c is 1, plus x minus 2.

00:15:31.350 --> 00:15:32.520
So those are 2 graps.

00:15:32.520 --> 00:15:34.370
Let's see what they look like.

00:15:34.370 --> 00:15:36.640
Let's graph them.

00:15:36.640 --> 00:15:39.500
So our first one, there we
go, we hit that point,

00:15:39.500 --> 00:15:41.500
that's our double root.

00:15:41.500 --> 00:15:44.200
Notice, both of these
two graphs meet our

00:15:44.200 --> 00:15:45.480
two constraints.

00:15:45.480 --> 00:15:47.790
They both have a root
there at minus 1.

00:15:47.790 --> 00:15:51.310
They both have a y-intercept
at y is equal to minus 2.

00:15:51.310 --> 00:15:54.780
And they both have a
root at x is equal to

00:15:54.780 --> 00:15:56.420
positive 2 right there.

00:15:56.420 --> 00:15:59.480
Now the second one right here
had a separate distinct third

00:15:59.480 --> 00:16:02.960
root, while the first one,
here, has a double root

00:16:02.960 --> 00:16:04.000
right over there.

00:16:04.000 --> 00:16:06.230
Anyway, I actually think this
problem was somewhat more

00:16:06.230 --> 00:16:08.850
interesting by the fact we
had to actually look at the

00:16:08.850 --> 00:16:10.240
different solutions to it.

00:16:10.240 --> 00:16:12.030
And there's actually an
infinite number of solutions

00:16:12.030 --> 00:16:15.180
to this based on what
you choose for your a.


Introduction to waves

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.170
Let's say I've got a rope.

00:00:03.170 --> 00:00:04.580
That's my rope.

00:00:04.580 --> 00:00:07.220
And what I'm going to do is, I'm
going to take the left end

00:00:07.220 --> 00:00:08.910
of the rope, and I'm going
to jerk it up,

00:00:08.910 --> 00:00:10.070
and then back down.

00:00:10.070 --> 00:00:13.190
And we're going to talk about
what happens or what possibly

00:00:13.190 --> 00:00:14.460
gets formed.

00:00:14.460 --> 00:00:19.140
So if I take it up over here,
it's going to, obviously, take

00:00:19.140 --> 00:00:21.300
the string to the right
of it up with it.

00:00:21.300 --> 00:00:24.170
And the string is going to
look something like this.

00:00:24.170 --> 00:00:26.030
It's going to look something
like that.

00:00:26.030 --> 00:00:29.060
Now I'm going to immediately
jerk it back down.

00:00:29.060 --> 00:00:31.470
And as it passes, let's see what
the rope will look like

00:00:31.470 --> 00:00:35.160
when the left-hand point is at
its original position again.

00:00:35.160 --> 00:00:37.690
So the left-hand point-- I've
pulled it back down.

00:00:37.690 --> 00:00:41.800
But in the last time period,
this part of the rope had some

00:00:41.800 --> 00:00:43.320
type of an upward velocity.

00:00:43.320 --> 00:00:44.940
You could imagine that way.

00:00:44.940 --> 00:00:48.340
And even after that point, even
though this left-hand

00:00:48.340 --> 00:00:53.720
point starts getting pulled
down, this point right here

00:00:53.720 --> 00:00:55.080
still has some upward
momentum.

00:00:55.080 --> 00:00:57.430
So it's still going to keep
moving up, maybe at a slower

00:00:57.430 --> 00:00:59.710
pace because it's starting
to be tugged down by

00:00:59.710 --> 00:01:00.550
the rope on its left.

00:01:00.550 --> 00:01:02.130
So it's going to look
something like that.

00:01:02.130 --> 00:01:05.099
And it's going to bring the
rope to its right with it.

00:01:05.099 --> 00:01:08.032
So the rope will look
something like this.

00:01:08.032 --> 00:01:10.480
The rope might look something
like that.

00:01:10.480 --> 00:01:13.600
And then I'm going to take this
guy-- this was just an

00:01:13.600 --> 00:01:16.160
intermediate position on the
way to being pulled all the

00:01:16.160 --> 00:01:17.410
way down here.

00:01:24.150 --> 00:01:26.560
So what's the rope going
to look like now?

00:01:26.560 --> 00:01:27.160
Well.

00:01:27.160 --> 00:01:30.820
This guy, he had some momentum
that got him there.

00:01:30.820 --> 00:01:33.490
But then all of that velocity
will essentially go to zero

00:01:33.490 --> 00:01:35.850
because he's being tugged
by the rope to the left.

00:01:35.850 --> 00:01:37.580
And now, he's going to
switch directions.

00:01:37.580 --> 00:01:40.820
And he will have gotten
here, at that point.

00:01:40.820 --> 00:01:45.770
The point on the line that was
here-- on the purple period of

00:01:45.770 --> 00:01:47.780
time-- it had some
upward momentum.

00:01:47.780 --> 00:01:50.000
So it's just going to keep
going, on maybe a slower pace.

00:01:50.000 --> 00:01:52.400
It'll be there and it will bring
the rest of the rope to

00:01:52.400 --> 00:01:53.290
the right of it with it.

00:01:53.290 --> 00:01:55.650
So now my rope is going to
look something like this.

00:01:58.790 --> 00:02:01.410
And then finally, where I'm
going to jerk the rope back to

00:02:01.410 --> 00:02:03.710
its original position-- so this
left-hand point is going

00:02:03.710 --> 00:02:06.180
to be there.

00:02:06.180 --> 00:02:09.580
This guy, in the previous
time period,

00:02:09.580 --> 00:02:11.060
was moving down rapidly.

00:02:11.060 --> 00:02:13.840
So he might get there ready to
switch directions again.

00:02:13.840 --> 00:02:15.780
This guy will start
moving down.

00:02:15.780 --> 00:02:19.080
This guy, right here, he had
some upward momentum.

00:02:19.080 --> 00:02:21.590
So he's going to be up
in this position now.

00:02:21.590 --> 00:02:23.390
And he's going to be ready
to switch directions.

00:02:23.390 --> 00:02:25.540
So finally, when I've done this
whole cycle, when I've

00:02:25.540 --> 00:02:29.610
moved up, down and back there
again, my rope might look

00:02:29.610 --> 00:02:32.650
exactly like this.

00:02:32.650 --> 00:02:33.940
And I could let go
of the rope.

00:02:33.940 --> 00:02:38.210
I could just leave this little
left point right there.

00:02:38.210 --> 00:02:41.230
And this the lump is going to
propagate along the rope.

00:02:41.230 --> 00:02:42.760
Because in the next moment
of time, what's it

00:02:42.760 --> 00:02:44.080
going to look like?

00:02:44.080 --> 00:02:47.590
This guy is going to be pulled
up by this left-hand point.

00:02:47.590 --> 00:02:50.550
So he'll go back to his
resting position.

00:02:50.550 --> 00:02:53.770
This guy's being pulled down
right here by the part of the

00:02:53.770 --> 00:02:56.050
rope to the left of him, so he's
going to be pulled down.

00:02:56.050 --> 00:02:58.730
This guy's being pulled down.

00:02:58.730 --> 00:03:01.550
But this guy had some upward
momentum in the time period

00:03:01.550 --> 00:03:03.530
before, so he will
have moved up.

00:03:03.530 --> 00:03:06.860
And so, the very next time
period, my rope is going to

00:03:06.860 --> 00:03:08.110
look something like this.

00:03:13.300 --> 00:03:16.600
And this disturbance in the
rope, if I do nothing else,

00:03:16.600 --> 00:03:19.480
and if I don't lose energy to
heat and friction and all

00:03:19.480 --> 00:03:22.610
that, it'll just continue
moving down the rope.

00:03:22.610 --> 00:03:25.890
If I look at the rope at some
future period in time, maybe

00:03:25.890 --> 00:03:29.410
not that far down, the rope will
look something like this.

00:03:29.410 --> 00:03:32.590
And if I were to keep watching
it, I'll see this disturbance.

00:03:32.590 --> 00:03:34.090
I keep using the word
disturbance, because there's

00:03:34.090 --> 00:03:37.010
really no better word
to use for it.

00:03:37.010 --> 00:03:40.110
I'll see this disturbance or
perturbation, or whatever you

00:03:40.110 --> 00:03:43.950
want to call it, moving
along the rope.

00:03:43.950 --> 00:03:47.700
When we think about what a wave
is, we essentially-- I

00:03:47.700 --> 00:03:49.620
kind of jumped the gun-- I
keep calling this is a

00:03:49.620 --> 00:03:51.890
disturbance, because I didn't
want to use the word, wave. I

00:03:51.890 --> 00:03:53.940
want to say, well what
really is a wave?

00:03:53.940 --> 00:03:57.890
And a wave really is just this
disturbance that's propagating

00:03:57.890 --> 00:03:59.190
down the rope.

00:03:59.190 --> 00:04:02.978
So this is a good time to
actually define a wave. A

00:04:02.978 --> 00:04:05.820
wave. Because once I define it,
I can start calling this a

00:04:05.820 --> 00:04:07.450
wave, as opposed to
a disturbance

00:04:07.450 --> 00:04:09.320
propagating down the rope.

00:04:09.320 --> 00:04:15.630
So a wave is a disturbance
propagating through space.

00:04:24.340 --> 00:04:27.755
And you might see other
definitions of a wave. One of

00:04:27.755 --> 00:04:32.020
the most typical ones is
energy or a disturbance

00:04:32.020 --> 00:04:35.010
propagating energy
through a medium.

00:04:35.010 --> 00:04:36.420
And when they say medium,
it's what is

00:04:36.420 --> 00:04:37.380
the wave going through?

00:04:37.380 --> 00:04:42.810
So in this example, the rope
would be our medium.

00:04:42.810 --> 00:04:46.560
But the reason why I don't want
to use that definition of

00:04:46.560 --> 00:04:49.210
a wave is because in future
videos, we'll learn about

00:04:49.210 --> 00:04:52.920
electromagnetic waves and those
don't propagate through

00:04:52.920 --> 00:04:53.680
any medium.

00:04:53.680 --> 00:04:55.440
They propagate through
a vacuum.

00:04:55.440 --> 00:04:58.380
So to keep things as general as
possible, we'll just call

00:04:58.380 --> 00:05:01.050
it a disturbance that propagates
through space.

00:05:01.050 --> 00:05:02.835
And it usually transfers
energy.

00:05:17.030 --> 00:05:19.590
What do I mean by transferring
energy?

00:05:19.590 --> 00:05:21.660
On this left hand part
of the rope, I gave a

00:05:21.660 --> 00:05:22.540
little energy the rope.

00:05:22.540 --> 00:05:25.420
I moved it up, down, and
then back again.

00:05:25.420 --> 00:05:28.420
And then after I did that, that
up, down, back again is

00:05:28.420 --> 00:05:30.780
happening successively
to every point to the

00:05:30.780 --> 00:05:31.750
right on the rope.

00:05:31.750 --> 00:05:34.570
So if I waited long enough, at
this point on the rope right

00:05:34.570 --> 00:05:38.810
here, it's going to move up,
down, and then back again.

00:05:38.810 --> 00:05:41.040
Exactly what I did over here
is going to happen to this

00:05:41.040 --> 00:05:41.730
point on the rope.

00:05:41.730 --> 00:05:43.740
And then later on, it's going to
happen to some other future

00:05:43.740 --> 00:05:44.620
point on the rope.

00:05:44.620 --> 00:05:46.720
So that energy that I originally
put on the

00:05:46.720 --> 00:05:48.390
left-hand side of the
rope is being

00:05:48.390 --> 00:05:49.830
transferred down the rope.

00:05:49.830 --> 00:05:54.890
If I had some type of object
here sitting on the rope,

00:05:54.890 --> 00:05:59.440
maybe when the wave-- when the
disturbance- passes by it,

00:05:59.440 --> 00:06:01.710
this thing could get flipped
into the air; it might get

00:06:01.710 --> 00:06:03.800
pushed into the air,
and go into a

00:06:03.800 --> 00:06:05.530
higher potential energy.

00:06:05.530 --> 00:06:08.410
So this disturbance
is transferring

00:06:08.410 --> 00:06:10.320
energy in this case.

00:06:10.320 --> 00:06:13.440
Now, what I've drawn here isn't
the only type of wave

00:06:13.440 --> 00:06:16.620
you can have. I mean my
definition is fairly general.

00:06:16.620 --> 00:06:19.660
But the definition is more
general than just what I've

00:06:19.660 --> 00:06:20.270
drawn here.

00:06:20.270 --> 00:06:28.770
For example, you could have a
sound wave. If you just look

00:06:28.770 --> 00:06:33.110
at all of the molecules of the
air, they have some density

00:06:33.110 --> 00:06:35.850
that looks something
like that.

00:06:35.850 --> 00:06:38.620
And now let's say I had some
type of a membrane-- maybe

00:06:38.620 --> 00:06:41.480
it's a speaker--
that jolts this

00:06:41.480 --> 00:06:42.680
left-hand side of the air.

00:06:42.680 --> 00:06:46.820
So it just pushes-- so let me
see if I can draw this.

00:06:46.820 --> 00:06:52.560
Let's say I had some type of
surface here that just really

00:06:52.560 --> 00:06:55.440
quickly jolts-- that just moves
it in that direction,

00:06:55.440 --> 00:06:56.590
and then just comes back.

00:06:56.590 --> 00:06:58.630
So similar to what I did
here, I go up and down.

00:06:58.630 --> 00:07:01.320
But instead of doing that, it
just pushes the air and then

00:07:01.320 --> 00:07:02.010
pushes back.

00:07:02.010 --> 00:07:03.440
So what's going to happen?

00:07:03.440 --> 00:07:06.220
Right after it pushes it, the
air molecules that it pushes

00:07:06.220 --> 00:07:08.940
up against are going
to jam together.

00:07:08.940 --> 00:07:11.120
They're going to
get compressed.

00:07:11.120 --> 00:07:13.330
Right here, all these air
molecules that were right on

00:07:13.330 --> 00:07:15.490
the surface, they're going to
get pushed next to all these

00:07:15.490 --> 00:07:17.550
air molecules that
are right there.

00:07:17.550 --> 00:07:20.760
And then when it pushes back,
or when the membrane goes

00:07:20.760 --> 00:07:23.280
back, you're going to have
fewer air molecules here,

00:07:23.280 --> 00:07:25.050
because you're going to have
a low density here.

00:07:25.050 --> 00:07:28.030
And then these guys, that are
all scrunched up together,

00:07:28.030 --> 00:07:29.800
they're going to want to get
away from each other.

00:07:29.800 --> 00:07:31.750
They might even run
into each other.

00:07:31.750 --> 00:07:34.520
And so these guys are going to
run into those guys, who are

00:07:34.520 --> 00:07:35.950
going to run into
the next guys.

00:07:35.950 --> 00:07:37.060
And so on and so forth.

00:07:37.060 --> 00:07:39.680
And after these guys bump into
those guys, those guys are

00:07:39.680 --> 00:07:41.170
going to go back to
where they were.

00:07:41.170 --> 00:07:43.750
So essentially, you're going
to have this disturbance

00:07:43.750 --> 00:07:48.950
that's going to be a set of
molecules compressing, or

00:07:48.950 --> 00:07:50.620
bumping into, its neighboring
molecules.

00:07:50.620 --> 00:07:54.360
So if you look at this at some
future period in time, all of

00:07:54.360 --> 00:07:57.040
a sudden, this area
might look normal.

00:07:57.040 --> 00:08:01.330
Let me clear it and draw it
just the way it I started.

00:08:01.330 --> 00:08:02.835
So this area might
look normal.

00:08:06.620 --> 00:08:09.690
But that compression of the
particles might have reached

00:08:09.690 --> 00:08:10.940
right over here.

00:08:15.440 --> 00:08:17.930
And not only that, we saw that
right after the compression,

00:08:17.930 --> 00:08:20.960
you usually have this area
of low pressure.

00:08:20.960 --> 00:08:24.450
So if I were to really draw this
wave, and actually, if

00:08:24.450 --> 00:08:27.210
this membrane were to keep doing
it over, and over, and

00:08:27.210 --> 00:08:30.340
over again-- so it kept going
forward and back, forward and

00:08:30.340 --> 00:08:32.520
back, or right and left, right
and left-- what you would have

00:08:32.520 --> 00:08:34.650
is a series of compressions.

00:08:34.650 --> 00:08:36.630
The air would just have a
series of compressions.

00:08:36.630 --> 00:08:38.120
So that's one compression.

00:08:38.120 --> 00:08:41.780
You'd have another compression
right there.

00:08:41.780 --> 00:08:44.250
Another compression
right there.

00:08:44.250 --> 00:08:45.700
And then, in between
the compressions

00:08:45.700 --> 00:08:47.780
the air is less dense.

00:08:47.780 --> 00:08:50.550
The air is less dense
like this.

00:08:50.550 --> 00:08:53.790
And what we've essentially just
generated is a sound wave

00:08:53.790 --> 00:08:55.570
travelling through air.

00:08:55.570 --> 00:09:02.060
So this right here is a sound
wave. And this type of wave,

00:09:02.060 --> 00:09:06.500
where the direction of the
disturbance is the same, or

00:09:06.500 --> 00:09:09.570
along the same axis as the
direction in which the wave is

00:09:09.570 --> 00:09:12.150
travelling-- the wave is
travelling in that direction--

00:09:12.150 --> 00:09:16.570
this is call a longitudinal
wave. So sound waves sound

00:09:16.570 --> 00:09:18.410
through air, they're
longitudinal waves.

00:09:23.400 --> 00:09:28.090
Sometimes called a compression
wave. Same thing.

00:09:28.090 --> 00:09:31.720
Compression wave. Because it's
caused by compression.

00:09:31.720 --> 00:09:35.110
Our example of the string, this
is called a transverse

00:09:35.110 --> 00:09:44.100
wave. Because the disturbance,
the movement of the medium, is

00:09:44.100 --> 00:09:48.040
going in a direction transverse
to-- or at an axis

00:09:48.040 --> 00:09:50.250
that's transverse to-- the
direction of our movement.

00:09:50.250 --> 00:09:53.660
We're moving in that direction,
to the right.

00:09:53.660 --> 00:09:56.190
Actually, our wave is moving to
the right, but the actual

00:09:56.190 --> 00:09:58.310
medium is moving up and down.

00:09:58.310 --> 00:10:00.670
Our medium is moving
up and down.

00:10:00.670 --> 00:10:02.500
That's why this is called
transverse.

00:10:02.500 --> 00:10:05.160
While here the medium is moving
left and right while

00:10:05.160 --> 00:10:06.980
the wave moves to the right.

00:10:06.980 --> 00:10:08.310
So it's along the same axis.

00:10:08.310 --> 00:10:11.710
So we're dealing with the
compressional or longitudinal.

00:10:11.710 --> 00:10:14.980
Now in this first example,
I just did one cycle.

00:10:14.980 --> 00:10:18.280
I just jerked up, down,
and back again.

00:10:18.280 --> 00:10:20.510
And I created this
one disturbance.

00:10:20.510 --> 00:10:23.205
And we can call this, when you
just do it once, you can view

00:10:23.205 --> 00:10:24.455
this as a wave pulse.

00:10:26.860 --> 00:10:29.710
If I kept doing that-- if I
just went up, down, back

00:10:29.710 --> 00:10:31.470
again, up, down, back again,
and I kept doing it

00:10:31.470 --> 00:10:34.760
periodically, over and over
again, then I would generate a

00:10:34.760 --> 00:10:37.080
periodic wave. And my
string would look

00:10:37.080 --> 00:10:38.330
something like this.

00:10:43.610 --> 00:10:50.200
Well actually, it would look
something that right there,

00:10:50.200 --> 00:10:54.930
where that's the disturbance
generated from our first time

00:10:54.930 --> 00:10:59.600
that we moved this left-hand
part of our string.

00:10:59.600 --> 00:11:03.840
So this right here is
a periodic wave.

00:11:03.840 --> 00:11:05.790
In the next video, we're going
to talk about a lot of the

00:11:05.790 --> 00:11:09.800
properties of a periodic wave.
How the wavelength, and its

00:11:09.800 --> 00:11:12.330
frequency, and its period relate
to its velocity, and

00:11:12.330 --> 00:11:13.100
all of that.

00:11:13.100 --> 00:11:14.870
But I'll leave that alone
in this video.

00:11:14.870 --> 00:11:17.910
But I just wanted you to
appreciate what I think is a

00:11:17.910 --> 00:11:20.420
concept that we use
in everyday life.

00:11:20.420 --> 00:11:22.290
So it's a wave, it's a sound
wave, and all that.

00:11:22.290 --> 00:11:24.610
But it's a fairly abstract
notion where we talk about a

00:11:24.610 --> 00:11:27.590
wave, we're really just pointing
to a disturbance

00:11:27.590 --> 00:11:30.980
that's moving, usually along
a medium, at least when we

00:11:30.980 --> 00:11:32.720
visualize it, but not always.

00:11:32.720 --> 00:11:34.210
But we're just pointing
to this disturbance.

00:11:34.210 --> 00:11:37.000
And this disturbance could take
many forms. It could be a

00:11:37.000 --> 00:11:39.940
transverse disturbance if we're
dealing with a string.

00:11:39.940 --> 00:11:44.370
It could be a disturbance in
terms of the density of air

00:11:44.370 --> 00:11:47.440
molecules in terms of a sound
wave. And there is a relation.

00:11:47.440 --> 00:11:53.120
So if you wanted to just plot
the density here by position--

00:11:53.120 --> 00:11:55.920
if I were to mathematically
represent this compression

00:11:55.920 --> 00:11:59.300
wave right here-- let's say that
this line represents just

00:11:59.300 --> 00:12:02.380
resting before the sound
wave hits, that's

00:12:02.380 --> 00:12:03.700
just your normal density.

00:12:03.700 --> 00:12:05.510
If we were to plot the
density it might look

00:12:05.510 --> 00:12:07.410
something like this.

00:12:07.410 --> 00:12:08.870
Over here, we have very
high density.

00:12:08.870 --> 00:12:10.240
Over here, we have
very low density.

00:12:10.240 --> 00:12:11.610
Over here, we have very
high density.

00:12:11.610 --> 00:12:15.930
And if you were to plot it, it
would look a lot like that

00:12:15.930 --> 00:12:21.090
transverse wave that I did with
the rope at the beginning

00:12:21.090 --> 00:12:21.950
of this video.

00:12:21.950 --> 00:12:23.760
And that's why they're even
grouped together.

00:12:23.760 --> 00:12:26.460
Because mathematically, even
though a compression wave

00:12:26.460 --> 00:12:29.180
looks very different, or
you might visualize or

00:12:29.180 --> 00:12:31.340
conceptualize it very different
than a transverse

00:12:31.340 --> 00:12:34.190
wave, mathematically, they're
essentially the same thing.

00:12:34.190 --> 00:12:35.610
You have this quantity.

00:12:35.610 --> 00:12:39.170
In this case, it's the density
of the air varying over time.

00:12:39.170 --> 00:12:43.400
In this case, it's the height,
or the position, or how much

00:12:43.400 --> 00:12:45.860
your displacement from your
resting position-- that's the

00:12:45.860 --> 00:12:50.530
quantity varying through time,
that disruption is travelling

00:12:50.530 --> 00:12:51.700
over the course of the medium.

00:12:51.700 --> 00:12:54.600
That's why we call both
of these things waves.

00:12:54.600 --> 00:12:56.560
Anyway, I'll let you go here.

00:12:56.560 --> 00:12:58.500
And in the next video, we'll
talk a little bit more of the

00:12:58.500 --> 00:13:01.700
properties of periodic waves.