Multiplying: using an area model

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.810
So I'm going to
multiply 16 times 27.

00:00:03.810 --> 00:00:06.520
I'm going to do it using
something called an area model.

00:00:06.520 --> 00:00:08.060
And the whole point
of an area model

00:00:08.060 --> 00:00:09.710
is to really understand
what's going on

00:00:09.710 --> 00:00:11.970
in the multiplication process.

00:00:11.970 --> 00:00:16.620
So 16, you can represent
literally as 10 plus 6.

00:00:16.620 --> 00:00:18.370
This 1 is in the tens place.

00:00:18.370 --> 00:00:20.250
It represents one 10.

00:00:20.250 --> 00:00:27.280
So we can represent
that as 10 plus 6.

00:00:27.280 --> 00:00:29.850
Let me do the 6 in
that same green color.

00:00:29.850 --> 00:00:32.640
10 plus 6.

00:00:32.640 --> 00:00:34.760
And let me mark
off 10 and 6 here.

00:00:34.760 --> 00:00:43.020
So this is 1, 2, 3,
4, 5, 6, 7, 8, 9, 10.

00:00:43.020 --> 00:00:45.770
So this part in
blue, that's the 10.

00:00:45.770 --> 00:00:48.150
So I'm representing--
I've gone 10 slashes,

00:00:48.150 --> 00:00:50.430
or I'm representing 10
boxes right over there.

00:00:50.430 --> 00:00:52.120
And then the 6,
which I want to do in

00:00:52.120 --> 00:00:56.200
that green color, the sixth,
let's mark off 6 boxes.

00:00:56.200 --> 00:01:01.390
1, 2, 3, 4, 5, 6.

00:01:01.390 --> 00:01:03.990
So 16 is this whole length.

00:01:03.990 --> 00:01:05.730
The blue part is the 10.

00:01:05.730 --> 00:01:09.280
The green part is the 6.

00:01:09.280 --> 00:01:11.020
The 10 comes from
the one in the tens.

00:01:11.020 --> 00:01:14.620
This is literally 1 10 and
this is literally 6 1s.

00:01:14.620 --> 00:01:17.640
Now let's think about 27.

00:01:17.640 --> 00:01:20.080
Well, we already know that
the 2 in the 10s place

00:01:20.080 --> 00:01:21.780
is representing 20.

00:01:21.780 --> 00:01:22.955
So let's count off 20.

00:01:22.955 --> 00:01:36.537
So it's 1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14 15, 16 17,

00:01:36.537 --> 00:01:38.290
18, 19.

00:01:38.290 --> 00:01:39.990
20.

00:01:39.990 --> 00:01:42.410
So up until that
point right over here,

00:01:42.410 --> 00:01:44.830
this line right over
here has a length of 20.

00:01:47.195 --> 00:01:48.695
But we're not just
talking about 20.

00:01:48.695 --> 00:01:50.130
We're talking about 27.

00:01:50.130 --> 00:01:54.150
So it's 20 plus 7.

00:01:54.150 --> 00:02:03.550
So let's count off 7
here, 1, 2, 3, 4, 5, 6, 7.

00:02:03.550 --> 00:02:05.720
Now if we look at
the total number.

00:02:05.720 --> 00:02:10.440
So if we have 16 times 27,
the product 16 times 27, what

00:02:10.440 --> 00:02:14.820
gives you the area of
a 16 by 27 rectangle.

00:02:14.820 --> 00:02:16.040
So let me draw that.

00:02:16.040 --> 00:02:18.040
So it's 16-- this
is a width of 16.

00:02:18.040 --> 00:02:23.120
Let's bring a little line down
right over here-- and then

00:02:23.120 --> 00:02:27.105
the height of my rectangle,
the way I've drawn it is 27.

00:02:33.670 --> 00:02:36.990
This is l, 2, 3, 4, 5, 6, 7.

00:02:36.990 --> 00:02:40.740
So the area of this rectangle
is going to be 16 times

00:02:40.740 --> 00:02:43.620
27, the area of this
entire rectangle.

00:02:43.620 --> 00:02:46.410
Now, we could do is break it
up by parts, because it'll

00:02:46.410 --> 00:02:50.740
be easier to compute, and we
can see what part of the area

00:02:50.740 --> 00:02:52.690
those different
products represent.

00:02:52.690 --> 00:02:54.610
So, for example, we
can think about what

00:02:54.610 --> 00:03:00.010
10-- so let me separate
out the sections.

00:03:00.010 --> 00:03:00.880
So let me draw it.

00:03:00.880 --> 00:03:04.440
So this section is 10 wide.

00:03:04.440 --> 00:03:06.580
And then let me draw
a section like this.

00:03:10.740 --> 00:03:13.970
So we could figure out the
areas of each of these sections,

00:03:13.970 --> 00:03:15.860
and then the area of
the entire rectangle,

00:03:15.860 --> 00:03:17.450
which is going to
be this product,

00:03:17.450 --> 00:03:20.124
is going to be the area of all
of these rectangles combined.

00:03:20.124 --> 00:03:21.790
So we could first
think about it-- well,

00:03:21.790 --> 00:03:24.730
let's think about
what 20 times 10 is.

00:03:24.730 --> 00:03:27.420
Well 20 times 10 is a fairly
straightforward thing.

00:03:27.420 --> 00:03:30.000
It's going to be 200.

00:03:30.000 --> 00:03:32.920
You could think of as 2 times
1, and you have two 0's there.

00:03:32.920 --> 00:03:35.100
Or you probably know
in your head 20 times

00:03:35.100 --> 00:03:37.350
10 is just going to be--
you're going to add a 0 here.

00:03:37.350 --> 00:03:38.820
So it's going to be 200.

00:03:38.820 --> 00:03:41.580
So 20 times 10 is 200.

00:03:41.580 --> 00:03:43.100
And let me highlight that in.

00:03:43.100 --> 00:03:46.010
So you have-- that's
not the color I wanted

00:03:46.010 --> 00:03:49.760
to use-- let me use
this blue right here.

00:03:49.760 --> 00:03:53.230
So that's the blue from the 10.

00:03:53.230 --> 00:03:56.140
Let me put some orange
in there from the 20

00:03:56.140 --> 00:03:58.280
to make it clear that this
is the product of both

00:03:58.280 --> 00:04:00.210
of those numbers.

00:04:00.210 --> 00:04:04.980
So 20 times 10 is 200.

00:04:04.980 --> 00:04:07.680
Now what's 20 times 6?

00:04:07.680 --> 00:04:12.430
Well 2 times 6 is 12,
plus you have this 0 here.

00:04:12.430 --> 00:04:15.610
So this right over here
is going to be 120.

00:04:15.610 --> 00:04:17.149
And it has the orange in it.

00:04:19.959 --> 00:04:21.700
So this is a 20.

00:04:21.700 --> 00:04:23.783
And then let me put
some green in for the 6.

00:04:26.330 --> 00:04:29.470
20 times the 6.

00:04:29.470 --> 00:04:31.670
Now what's the area of this
section right over here?

00:04:31.670 --> 00:04:34.770
Well, it's 7 high
and it is 10 wide.

00:04:34.770 --> 00:04:39.210
So it's going to be
7 times 10 or 70.

00:04:39.210 --> 00:04:41.980
I'll have an area
of 70 square units.

00:04:41.980 --> 00:04:46.685
So let me do it in
this purple, and I'll

00:04:46.685 --> 00:04:48.930
thrown some blue in there, too.

00:04:48.930 --> 00:04:50.730
It's kind of a fun art project.

00:04:50.730 --> 00:04:52.577
I'll throw some
blue in there too.

00:04:52.577 --> 00:04:54.910
And then finally, what's the
area of this little section

00:04:54.910 --> 00:04:55.840
right over here?

00:04:55.840 --> 00:04:58.830
It's 7 high and it's 6 wide.

00:04:58.830 --> 00:05:02.640
So it's going to be 7 times
6 or 42 square units of area.

00:05:05.320 --> 00:05:06.490
Let me color it in.

00:05:06.490 --> 00:05:09.440
So I got some magenta
in there, and then I

00:05:09.440 --> 00:05:13.350
got some green in
there right over there.

00:05:13.350 --> 00:05:15.870
So what's the area of this
entire thing going to be?

00:05:15.870 --> 00:05:23.880
Well, it's going to be the
200 plus the 120 plus--

00:05:23.880 --> 00:05:27.740
let me do it this way-- it's
going to be 200 plus 120

00:05:27.740 --> 00:05:32.660
plus 70 plus 42.

00:05:32.660 --> 00:05:35.180
When you add that up, you get,
let's see, in the ones place,

00:05:35.180 --> 00:05:36.210
you get a 2.

00:05:36.210 --> 00:05:40.410
Then you get 2 tens plus 7
tens is 9 tens, plus 4 tens,

00:05:40.410 --> 00:05:45.850
is 13 tens, which is the same
thing as 3 tens and 1 hundred.

00:05:45.850 --> 00:05:47.460
And then this is a 4.

00:05:47.460 --> 00:05:48.460
Did I add that up right?

00:05:48.460 --> 00:05:49.420
Let's see, this will be 11.

00:05:49.420 --> 00:05:50.336
Yup, that looks right.

00:05:50.336 --> 00:05:52.650
So this is 432.

00:05:52.650 --> 00:05:59.115
So this is going
to be equal to 432.

00:05:59.115 --> 00:06:00.990
And you might be saying,
hey, Sal, why did we

00:06:00.990 --> 00:06:02.510
go through all of this business?

00:06:02.510 --> 00:06:05.110
I've seen before
that if I take 16--

00:06:05.110 --> 00:06:08.630
I could take 16
times 27 like this.

00:06:08.630 --> 00:06:12.150
You probably learned
this type of a process.

00:06:12.150 --> 00:06:13.350
16 times 27.

00:06:13.350 --> 00:06:16.510
Then you say, OK, let's start
with the 7 in the ones place

00:06:16.510 --> 00:06:19.250
and you do 7 times 6 is 42.

00:06:22.540 --> 00:06:23.374
And you carry the 4.

00:06:23.374 --> 00:06:25.456
You're really just putting
the 4 in the tens place

00:06:25.456 --> 00:06:26.560
because it's a 40.

00:06:26.560 --> 00:06:29.420
But right when you
did that 7 times 6,

00:06:29.420 --> 00:06:32.880
we essentially calculated
this right over there.

00:06:32.880 --> 00:06:36.500
And then when you multiply
the 7 times the 1,

00:06:36.500 --> 00:06:38.480
you really multiplying
7 times 10,

00:06:38.480 --> 00:06:40.740
and then you're adding
the 4 from the 42.

00:06:40.740 --> 00:06:43.490
So when you do the
7 times this 1.

00:06:43.490 --> 00:06:46.160
This is you're actually
calculating this area.

00:06:46.160 --> 00:06:48.150
And then when you
carry it when you add

00:06:48.150 --> 00:06:50.514
this carried 4, and
put it right down here.

00:06:50.514 --> 00:06:52.930
You're essentially taking the
sum of both of these things,

00:06:52.930 --> 00:06:56.710
because you're multiplying 7
times the 16 to get this area.

00:06:59.167 --> 00:07:00.000
So let's just do it.

00:07:00.000 --> 00:07:05.120
7 times 1 is 7 plus 4 is 11.

00:07:05.120 --> 00:07:08.250
So 112, what you just
figured out right over here,

00:07:08.250 --> 00:07:11.020
is this area Right over here.

00:07:11.020 --> 00:07:18.360
So 7 times the 10 or the
70, plus 7 times 6, the 42.

00:07:18.360 --> 00:07:20.540
Now when you go
into the tens place

00:07:20.540 --> 00:07:22.450
you've probably always
said, oh, you know,

00:07:22.450 --> 00:07:25.270
I just throw a 0 down there.

00:07:25.270 --> 00:07:27.310
But why do you throw
a 0 down there?

00:07:27.310 --> 00:07:29.400
Because this 2 is
representing 10.

00:07:29.400 --> 00:07:32.850
So if you multiply 2 times
the 6, and you put a 12

00:07:32.850 --> 00:07:34.262
and put a 2 down
and carry the 1.

00:07:34.262 --> 00:07:35.220
That wouldn't be right.

00:07:35.220 --> 00:07:36.170
This is a 20.

00:07:36.170 --> 00:07:38.050
So that's why you
put the 0 there.

00:07:38.050 --> 00:07:40.300
So let's scratch this out
so we don't get confused.

00:07:40.300 --> 00:07:43.730
2 times 6 is 12.

00:07:43.730 --> 00:07:48.700
So we're used to carrying the
2 down here and carrying the 1.

00:07:48.700 --> 00:07:53.360
But notice we're really thinking
about 20 times 60 is 120.

00:07:53.360 --> 00:07:55.350
Just doing that we just
calculated that right

00:07:55.350 --> 00:07:56.360
over there.

00:07:56.360 --> 00:08:01.440
And then we do 20
times 10 is 200.

00:08:01.440 --> 00:08:03.740
But then we had that
one from the 120.

00:08:03.740 --> 00:08:08.480
So this is going to be 300.

00:08:08.480 --> 00:08:12.300
So what we just did when we
multiplied the 2 times the 16,

00:08:12.300 --> 00:08:15.680
we just calculated
this total area.

00:08:15.680 --> 00:08:18.810
We did the 20 times
the 6 to get that.

00:08:18.810 --> 00:08:21.010
And then we had the 20
times a 10 to get that.

00:08:21.010 --> 00:08:23.780
When we did the 20 times
the 6, we carried this 1

00:08:23.780 --> 00:08:25.150
into the hundreds place.

00:08:25.150 --> 00:08:28.110
And so we added it all
together and got 320.

00:08:28.110 --> 00:08:29.960
And then this step
right over here,

00:08:29.960 --> 00:08:32.770
you're literally just
finding the combined area--

00:08:32.770 --> 00:08:36.460
the area of this plus the
area of that the area.

00:08:36.460 --> 00:08:40.240
So that's going to get us to---
we deserve a drum roll now.

00:08:40.240 --> 00:08:41.039
2 plus 0 is 2.

00:08:41.039 --> 00:08:43.380
1 plus 2 is 3.

00:08:43.380 --> 00:08:45.872
1 plus 3 is 4.


Regrouping numbers intro various place values

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:05.500
We have 4,5000 equals 3
thousands plus how many

00:00:05.500 --> 00:00:07.600
hundreds, question
mark hundreds?

00:00:07.600 --> 00:00:09.240
So let's write this
left-hand side,

00:00:09.240 --> 00:00:10.823
but I'm going to
write it out in terms

00:00:10.823 --> 00:00:11.917
of thousands and hundreds.

00:00:11.917 --> 00:00:13.500
So I'll write the
thousands in orange.

00:00:13.500 --> 00:00:17.225
So this is equal to
4 thousands, which

00:00:17.225 --> 00:00:22.580
is the same thing as just 4,000,
plus 500, which you could also

00:00:22.580 --> 00:00:25.060
view as 5 hundreds.

00:00:25.060 --> 00:00:26.930
So this is the left-hand side.

00:00:26.930 --> 00:00:28.560
Now let's look at
the right-hand side.

00:00:28.560 --> 00:00:30.460
We have 3 thousands.

00:00:30.460 --> 00:00:33.470
So it's 3 thousands.

00:00:33.470 --> 00:00:35.220
Now let's not even
look at this right now.

00:00:35.220 --> 00:00:36.594
Let's just think
about what do we

00:00:36.594 --> 00:00:38.597
have to add to this
right-hand side

00:00:38.597 --> 00:00:40.180
in order to get the
same thing that we

00:00:40.180 --> 00:00:42.210
have on the left-hand side?

00:00:42.210 --> 00:00:44.480
Well, if you compare
the 3,000 and the 4,000,

00:00:44.480 --> 00:00:46.310
you see you have an
extra 1,000 over here.

00:00:46.310 --> 00:00:48.770
So let's add an extra
1,000 on the right.

00:00:48.770 --> 00:00:51.480
So we're going to
add one extra 1,000.

00:00:51.480 --> 00:00:53.710
And now we just have
3,000 plus 1,000.

00:00:53.710 --> 00:00:56.870
This makes it the 4,000.

00:00:56.870 --> 00:00:59.030
But then, of course, we
also need another 500.

00:00:59.030 --> 00:01:04.170
So we're going to need
plus a 500 right over here.

00:01:04.170 --> 00:01:06.280
So in order for these
two things to be equal,

00:01:06.280 --> 00:01:24.520
we need to say 4,000 plus 500
is equal to 3,000 plus 1,500.

00:01:24.520 --> 00:01:27.280
Now, the way they've set
this up, we need to express--

00:01:27.280 --> 00:01:28.960
so it almost looks the same.

00:01:28.960 --> 00:01:31.040
On the left-hand
side, this is 4,500.

00:01:31.040 --> 00:01:35.850
So this right over here, this
is the same thing as 4,500.

00:01:35.850 --> 00:01:37.980
This is this right over here.

00:01:37.980 --> 00:01:40.420
And on the right-hand
side, we have 3 thousands.

00:01:40.420 --> 00:01:42.860
So that's this right over here.

00:01:42.860 --> 00:01:44.450
That's the 3 thousands.

00:01:44.450 --> 00:01:47.130
And then we just need to
express this as hundreds.

00:01:47.130 --> 00:01:52.464
So 1,500, this is the
same thing as 15 hundreds.

00:01:52.464 --> 00:01:53.630
So let's rewrite everything.

00:01:53.630 --> 00:01:57.087
We can rewrite this
as saying 4,500,

00:01:57.087 --> 00:01:59.670
just to get the exact same form
that they wrote it over there.

00:01:59.670 --> 00:02:11.300
So we could write 4,500 is
equal to 3 thousands plus-- now

00:02:11.300 --> 00:02:13.310
we have to just write
this in terms of hundreds.

00:02:13.310 --> 00:02:13.810
Well,

00:02:13.810 --> 00:02:15.540
This is 15 hundreds.

00:02:15.540 --> 00:02:18.720
Literally, if you
took 15 times 100,

00:02:18.720 --> 00:02:21.270
it's going to be equal to 1,500.

00:02:21.270 --> 00:02:28.250
So this could be viewed as 15
hundreds, so plus 15 hundreds.

00:02:28.250 --> 00:02:33.331
So in this situation, the
question mark is equal to 15.


Comparing whole number place values

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.750
Compare the expressions on both
sides of the question marks.

00:00:03.750 --> 00:00:12.180
So on the left-hand side of the
question mark, I have 7,907.

00:00:12.180 --> 00:00:18.500
So, it's 7,000 plus
900 plus 7, which

00:00:18.500 --> 00:00:23.035
is going to be the
same thing as 7,907.

00:00:23.035 --> 00:00:24.410
Now let's compare
that to what we

00:00:24.410 --> 00:00:27.230
have on the right-hand
side of the question mark.

00:00:27.230 --> 00:00:35.530
On the right-hand side, we have
7,000 plus 970, which is 7,970.

00:00:35.530 --> 00:00:37.240
Or if we wanted
to expand it out,

00:00:37.240 --> 00:00:43.290
we could write that as
7,000 plus 900 plus 70,

00:00:43.290 --> 00:00:45.490
and then we have 0 ones.

00:00:45.490 --> 00:00:47.670
So if we compare place
value by place value,

00:00:47.670 --> 00:00:50.270
we have 7,000 in either case.

00:00:50.270 --> 00:00:51.841
So the thousands
place is the same.

00:00:51.841 --> 00:00:53.340
We have the same
number of hundreds.

00:00:53.340 --> 00:00:54.980
We have 9 hundreds.

00:00:54.980 --> 00:00:57.310
But here we have no
tens, and we have a 7.

00:00:57.310 --> 00:00:59.270
And here we have no
ones, and we have 70.

00:00:59.270 --> 00:01:02.610
And clearly, 70
is greater than 7.

00:01:02.610 --> 00:01:04.730
So the number on
the right is greater

00:01:04.730 --> 00:01:07.370
than the number on the left,
or the number on the left

00:01:07.370 --> 00:01:12.000
is less than the
number on the right.

00:01:12.000 --> 00:01:13.650
I always remember
the less than symbol

00:01:13.650 --> 00:01:16.610
because it points to the
smaller number, or either

00:01:16.610 --> 00:01:19.050
of these symbols point
to the smaller number.

00:01:19.050 --> 00:01:21.090
So this right over here
would mean greater than.

00:01:21.090 --> 00:01:22.400
You have the larger
number on the left.

00:01:22.400 --> 00:01:23.370
This is less than.

00:01:23.370 --> 00:01:25.330
You have the smaller
number on the left.

00:01:25.330 --> 00:01:28.210
So we would go with less than.

00:01:28.210 --> 00:01:30.820
This is less than that.


Creating the largest possible number

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:04.700
Arrange the digits
2, 6, 0, and 1

00:00:04.700 --> 00:00:09.362
so that you create the highest
possible four-digit number.

00:00:09.362 --> 00:00:10.820
So the way I like
to think about it

00:00:10.820 --> 00:00:14.910
is, if I'm trying to create as
large of a number as possible,

00:00:14.910 --> 00:00:19.980
I want to put the largest
numbers in the largest place

00:00:19.980 --> 00:00:20.740
value.

00:00:20.740 --> 00:00:24.630
So if it's a four-digit number--
so it's one, two, three,

00:00:24.630 --> 00:00:27.340
four-digit number--
whatever I put here,

00:00:27.340 --> 00:00:29.970
this is going to
represent thousands.

00:00:29.970 --> 00:00:33.260
Whatever I put here is
going to represent hundreds.

00:00:33.260 --> 00:00:36.190
Whatever I put here
represents tens.

00:00:36.190 --> 00:00:38.870
And whatever I put
here represents ones.

00:00:38.870 --> 00:00:41.810
So I want to maximize the
number of thousands I have.

00:00:41.810 --> 00:00:45.070
For example, the largest
number here is 6.

00:00:45.070 --> 00:00:46.500
I could make it 6,000.

00:00:46.500 --> 00:00:47.560
I could make it 600.

00:00:47.560 --> 00:00:50.200
I could make it 60,
or I could make it 6.

00:00:50.200 --> 00:00:53.060
Well if I want as large
a number as possible,

00:00:53.060 --> 00:00:55.960
I'm going to make it 6,000.

00:00:55.960 --> 00:00:58.290
Notice, if I put any other
number in that place value--

00:00:58.290 --> 00:01:00.206
if I put a 0 there, I
would have no thousands.

00:01:00.206 --> 00:01:02.270
If I put 2 there,
I'd only have 2,000.

00:01:02.270 --> 00:01:04.190
If I put 1 there,
I'd only have 1,000.

00:01:04.190 --> 00:01:05.890
6,000 is definitely
going to be bigger

00:01:05.890 --> 00:01:07.931
than any of the numbers
that could be constructed

00:01:07.931 --> 00:01:11.460
with the 2, 0, or 1 in
the thousands place.

00:01:11.460 --> 00:01:14.350
Now the exact same logic, we
want the next largest number

00:01:14.350 --> 00:01:15.750
in the hundreds place.

00:01:15.750 --> 00:01:18.120
So the next largest
number here is a 2.

00:01:18.120 --> 00:01:19.980
So I'm going to put
a 2 right over here.

00:01:19.980 --> 00:01:22.850
I'd rather have two hundreds
than zero hundreds or one

00:01:22.850 --> 00:01:23.350
hundreds.

00:01:23.350 --> 00:01:25.080
That's going to make it bigger.

00:01:25.080 --> 00:01:27.760
Then, same exact idea--
we want the next largest

00:01:27.760 --> 00:01:29.220
number in the tens place.

00:01:29.220 --> 00:01:32.710
I'd rather have one
10 than zero 10.

00:01:32.710 --> 00:01:34.920
And then we're just
left with the 0.

00:01:34.920 --> 00:01:37.880
So we're just going to
put the 0 right over here.

00:01:37.880 --> 00:01:41.860
So we can make 6,210.

00:01:41.860 --> 00:01:45.120
If I wanted to make the smallest
possible four-digit number,

00:01:45.120 --> 00:01:47.020
then I would
rearrange this, so I

00:01:47.020 --> 00:01:49.340
have the smallest possible
number in the thousands

00:01:49.340 --> 00:01:52.030
and the largest possible
number in the ones.

00:01:52.030 --> 00:01:54.090
So the smallest possible
number I could create

00:01:54.090 --> 00:02:05.297
is-- I'm going to be
careful-- 0, 1, 2, 6.

00:02:05.297 --> 00:02:07.880
So the smallest possible number,
if I just switch this around,

00:02:07.880 --> 00:02:12.257
would be 126 that I could
construct out of these digits.

00:02:12.257 --> 00:02:13.590
But this is what they asked for.

00:02:13.590 --> 00:02:15.560
So we'll give 6,210.


Finding y intercept given slope and point

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:04.630
The following line passes
through the point 5 comma 8,

00:00:04.630 --> 00:00:11.140
and the equation of the line
is y is equal to 17/13x plus b.

00:00:11.140 --> 00:00:14.110
What is the value of
the y-intercept b?

00:00:14.110 --> 00:00:17.570
So we know that this
point, this x and y value

00:00:17.570 --> 00:00:19.760
must satisfy this
equation, so we

00:00:19.760 --> 00:00:24.750
know that when x is equal
to 5, y is equal to 8.

00:00:24.750 --> 00:00:31.850
So we can say-- so when x is
equal to 5, y is equal to 8.

00:00:31.850 --> 00:00:43.040
So we can say 8 must be equal
to 17/13 times x times 5 plus b,

00:00:43.040 --> 00:00:45.520
and then we can solve for b.

00:00:45.520 --> 00:00:48.140
So if we simplify
this a little bit,

00:00:48.140 --> 00:00:53.850
we get 8 is equal to--
let's see, 5 times 17 is 50,

00:00:53.850 --> 00:01:00.230
plus 35 is 85-- is 85/13 plus b.

00:01:00.230 --> 00:01:04.849
Then to solve for b, we just
subtract 85/13 from both sides.

00:01:04.849 --> 00:01:11.240
So we have 8 minus 85/13.

00:01:11.240 --> 00:01:16.080
85/13 Is equal to b.

00:01:16.080 --> 00:01:19.750
And now we just have to
subtract these two numbers.

00:01:19.750 --> 00:01:21.350
So 8 is the same
thing as-- let's see.

00:01:21.350 --> 00:01:29.650
80 plus 24 is 104,
so it is 104/13--

00:01:29.650 --> 00:01:35.830
this is the same thing
as 8-- minus 85/13,

00:01:35.830 --> 00:01:37.620
which is going to
be-- let's see.

00:01:37.620 --> 00:01:39.777
This is 19/13.

00:01:39.777 --> 00:01:40.360
Is that right?

00:01:40.360 --> 00:01:42.930
Yes, if this was 105,
then it would be 20/13.

00:01:42.930 --> 00:01:48.810
So this is 19/13, so it's equal
to 19/13, which is equal to b.

00:01:48.810 --> 00:01:51.190
So the equation of this
line is going to be y

00:01:51.190 --> 00:01:57.636
is equal to 17/13x plus 19/13.


Slope-intercept form from table

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:02.850
A line goes through
the following points,

00:00:02.850 --> 00:00:06.890
and the equation of that line
is written in y equals mx plus b

00:00:06.890 --> 00:00:07.390
form.

00:00:07.390 --> 00:00:09.890
Also known as
slope-intercept form.

00:00:09.890 --> 00:00:11.880
What is the equation
of the line?

00:00:11.880 --> 00:00:13.630
So the first thing we
want to think about,

00:00:13.630 --> 00:00:14.990
what is the slope of this line?

00:00:14.990 --> 00:00:16.530
What is m here?

00:00:16.530 --> 00:00:20.020
So what is our change in
y for given change in x?

00:00:20.020 --> 00:00:21.915
So this is an
interesting example here.

00:00:21.915 --> 00:00:24.540
And I encourage you to pause the
video and try it out yourself.

00:00:24.540 --> 00:00:28.150
Because no matter how much we
change x, y is not changing.

00:00:28.150 --> 00:00:30.350
y is a constant, 2.

00:00:30.350 --> 00:00:33.476
So your change in y between any
two points is going to be 0.

00:00:33.476 --> 00:00:35.100
It doesn't matter
what your change in x

00:00:35.100 --> 00:00:38.690
is, your change in x could be
1, your change in x could be 4,

00:00:38.690 --> 00:00:40.740
your change in y is always 0.

00:00:40.740 --> 00:00:43.690
So y is not changing
as you change x.

00:00:43.690 --> 00:00:48.660
So your slope for this
relationship is actually 0.

00:00:48.660 --> 00:00:52.700
Y is equal to 0x
plus-- and then, you

00:00:52.700 --> 00:00:55.740
could just realize that
the equation of this

00:00:55.740 --> 00:00:58.030
is just that y is
always equal to 2.

00:00:58.030 --> 00:01:02.730
So it's 0x plus 2, which is the
same thing as y is equal to 2.

00:01:02.730 --> 00:01:04.120
You could substitute back in.

00:01:04.120 --> 00:01:08.130
You could say OK, well, if
y is equal to 0x plus b,

00:01:08.130 --> 00:01:11.100
that means that y is equal to b.

00:01:11.100 --> 00:01:16.000
Well, y is always equal to 2,
no matter what thing you pick,

00:01:16.000 --> 00:01:17.660
so b is equal to 2.

00:01:17.660 --> 00:01:20.400
So either way, this
just boils down to y

00:01:20.400 --> 00:01:24.340
is equal to 0x plus 2,
or y is just equal to 2.

00:01:24.340 --> 00:01:26.980
Let's do another one of these.

00:01:26.980 --> 00:01:28.890
Maybe one where the y
is actually changing.

00:01:28.890 --> 00:01:33.180
So here, the y is clearly
actually changing.

00:01:33.180 --> 00:01:34.745
So let me copy and paste this.

00:01:39.030 --> 00:01:40.440
I want to put on my scratch pad.

00:01:40.440 --> 00:01:42.340
We can work it out.

00:01:42.340 --> 00:01:45.400
So we'll stick it
right over here.

00:01:45.400 --> 00:01:50.460
And then we are told a line
goes through the-- OK, so

00:01:50.460 --> 00:01:51.004
same thing.

00:01:51.004 --> 00:01:52.420
The line goes
through these points

00:01:52.420 --> 00:01:53.984
with the equation of a line.

00:01:53.984 --> 00:01:55.400
So the main idea
here is, you only

00:01:55.400 --> 00:01:57.360
need 2 points for
an equation of line.

00:01:57.360 --> 00:01:59.110
They've given us
more than necessary.

00:01:59.110 --> 00:02:01.000
So I'd like to pick
the two points that

00:02:01.000 --> 00:02:03.080
make things a
little bit simpler.

00:02:03.080 --> 00:02:08.179
So I'll pick the
point 4, 2 and 7, 0.

00:02:08.179 --> 00:02:09.970
I just picked those
two points because they

00:02:09.970 --> 00:02:12.970
have nice, clean numbers
associated with it.

00:02:12.970 --> 00:02:15.310
So what is our change in x here?

00:02:15.310 --> 00:02:18.950
So our change in x here, if we
go from 4 to 7, our change in x

00:02:18.950 --> 00:02:19.575
is equal to 3.

00:02:22.170 --> 00:02:24.510
And what's our change in y here?

00:02:24.510 --> 00:02:26.620
So we went up from 4 to 7.

00:02:26.620 --> 00:02:28.100
We increased by 3.

00:02:28.100 --> 00:02:30.470
Our y decreased by 2.

00:02:30.470 --> 00:02:32.800
Change in y is
equal to negative 2.

00:02:32.800 --> 00:02:37.580
So our slope, which is equal to
change in y over change in x,

00:02:37.580 --> 00:02:40.851
is equal to negative 2/3.

00:02:40.851 --> 00:02:42.850
And if you wanted to
relate that to the formulas

00:02:42.850 --> 00:02:44.260
that you normally
see for slope, you're

00:02:44.260 --> 00:02:45.551
just looking at your end point.

00:02:45.551 --> 00:02:49.370
So this is y2 minus
y1, which is negative 2

00:02:49.370 --> 00:02:52.780
over x2 minus x1,
which is 7 minus 4.

00:02:52.780 --> 00:02:55.520
But that just boils
down to negative 2/3.

00:02:55.520 --> 00:02:59.540
And so our equation is going
to be y is equal to negative

00:02:59.540 --> 00:03:03.920
2/3 x plus b.

00:03:03.920 --> 00:03:06.210
So let's substitute one
of these points in here,

00:03:06.210 --> 00:03:08.630
to figure out what
our b must be.

00:03:08.630 --> 00:03:12.030
And once again, I
want to figure out

00:03:12.030 --> 00:03:14.900
something where this is going
to become nice and clean.

00:03:14.900 --> 00:03:17.520
But this isn't going to
be really clean for any

00:03:17.520 --> 00:03:19.240
of these numbers
right over here.

00:03:19.240 --> 00:03:23.022
If we had a 3 for x, or a
6 for x, or a 0 for x, then

00:03:23.022 --> 00:03:24.230
things would work out nicely.

00:03:24.230 --> 00:03:26.460
But they don't give
us any of those.

00:03:26.460 --> 00:03:30.510
So let's just try
the 7 and the 0.

00:03:30.510 --> 00:03:34.450
So when x is equal to 0--
sorry, when x is equal to 7-- y

00:03:34.450 --> 00:03:36.000
is equal to 0.

00:03:36.000 --> 00:03:40.130
So when x is equal to 7, I'll
just do it in the same color,

00:03:40.130 --> 00:03:41.070
y is equal to 0.

00:03:41.070 --> 00:03:47.360
So 0 is equal to negative
2/3 times 7 plus b, or 0

00:03:47.360 --> 00:03:51.590
is equal to negative
14/3 plus b.

00:03:51.590 --> 00:03:56.790
Add 14/3 to both sides,
you get 14/3 is equal to b.

00:03:56.790 --> 00:03:59.930
So this is going to be
y is equal to negative--

00:03:59.930 --> 00:04:02.670
I'm going to go back
to the other screen--

00:04:02.670 --> 00:04:08.690
so y is equal to
negative 2/3 x plus 14/3.

00:04:08.690 --> 00:04:09.850
So let me do that.

00:04:09.850 --> 00:04:19.170
So y is equal to
negative 2/3 x plus 14/3.

00:04:22.280 --> 00:04:23.530
Let's check our answer.

00:04:23.530 --> 00:04:25.500
We got it right.


Using algebra to find measures of angles formed from transversal

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:02.590
So I've got two parallel lines.

00:00:02.590 --> 00:00:05.240
So that's the first
line right over there

00:00:05.240 --> 00:00:08.200
and then the second
line right over here.

00:00:08.200 --> 00:00:10.730
Let me denote that
these are parallel.

00:00:10.730 --> 00:00:12.960
These are parallel lines.

00:00:12.960 --> 00:00:15.780
Actually, I can do that
a little bit neater.

00:00:15.780 --> 00:00:21.510
And let me draw a transversal,
so a line that intersects both

00:00:21.510 --> 00:00:25.030
of these parallel lines,
so something like that.

00:00:25.030 --> 00:00:29.120
And now let's say
that we are told

00:00:29.120 --> 00:00:36.150
that this angle right
over here is 9x plus 88.

00:00:36.150 --> 00:00:37.720
And this is in degrees.

00:00:37.720 --> 00:00:42.680
And we're also told that
this angle right over here

00:00:42.680 --> 00:00:48.850
is 6x plus 182, once
again, in degrees.

00:00:48.850 --> 00:00:52.840
So my goal here and my question
for you is, can we figure out

00:00:52.840 --> 00:00:54.549
what these angles
actually are, given

00:00:54.549 --> 00:00:57.090
that these are parallel lines
and this is a transversal line?

00:00:57.090 --> 00:00:59.170
And I encourage you
to pause this video

00:00:59.170 --> 00:01:02.330
to try this on your own.

00:01:02.330 --> 00:01:07.840
Well, the key here to realize
is that these right over here

00:01:07.840 --> 00:01:09.730
are related by the
fact that they're

00:01:09.730 --> 00:01:13.070
formed from a transversal
intersecting parallel lines.

00:01:13.070 --> 00:01:16.310
And we know, for example,
that this angle corresponds

00:01:16.310 --> 00:01:17.920
to this angle right over here.

00:01:17.920 --> 00:01:19.920
They're going to be
congruent angles.

00:01:19.920 --> 00:01:22.230
And so this is 6x plus 182.

00:01:22.230 --> 00:01:27.270
This is also going
to be 6x plus 182.

00:01:27.270 --> 00:01:30.190
And then that helps us
realize that this blue angle

00:01:30.190 --> 00:01:31.850
and this orange
angle are actually

00:01:31.850 --> 00:01:32.980
going to be supplementary.

00:01:32.980 --> 00:01:35.920
They're going to add up to 180
degrees, because put together,

00:01:35.920 --> 00:01:38.290
when you make them
adjacent, their outer rays

00:01:38.290 --> 00:01:40.190
form a line right over here.

00:01:40.190 --> 00:01:49.460
So we know that 6x plus
182 plus 9x plus 88

00:01:49.460 --> 00:01:53.480
is going to be equal
to 180 degrees.

00:01:56.770 --> 00:01:58.940
And now we just have
to simplify this thing.

00:01:58.940 --> 00:02:03.380
So 6x plus 9x is
going to give us 15x.

00:02:03.380 --> 00:02:06.510
And then we have 182 plus 88.

00:02:06.510 --> 00:02:10.030
Let's see, 182 plus 8,
would get us to 190.

00:02:10.030 --> 00:02:11.750
And then we add another 80.

00:02:11.750 --> 00:02:18.700
It gets us to 270-- plus
270-- is equal to 180.

00:02:18.700 --> 00:02:21.740
If we subtract 270
from both sides,

00:02:21.740 --> 00:02:26.300
we get 15x is equal
to negative 90.

00:02:29.380 --> 00:02:31.650
And now we can divide
both sides by 15.

00:02:34.840 --> 00:02:37.460
And we get x is equal
to-- what is this?

00:02:37.460 --> 00:02:42.370
Let's see, 6 times 15
is 60 plus 30 is 90.

00:02:42.370 --> 00:02:46.380
So x is going to be
equal to negative 6.

00:02:46.380 --> 00:02:47.960
So far, we've made
a lot of progress.

00:02:47.960 --> 00:02:53.020
We figured out what x is equal
to. x is equal to negative 6,

00:02:53.020 --> 00:02:55.990
but we still haven't figured out
what these angles are equal to.

00:02:55.990 --> 00:02:59.050
So this angle right
over here, 9x plus 88,

00:02:59.050 --> 00:03:06.950
this is going to be equal to
9 times negative 6 plus 88.

00:03:06.950 --> 00:03:10.033
9 times negative
6 is negative 54.

00:03:13.332 --> 00:03:15.290
Let me write this down
before I make a mistake.

00:03:15.290 --> 00:03:19.230
Negative 54 plus
88 is going to be--

00:03:19.230 --> 00:03:25.550
let's see, to go from 88 minus
54 will give us 34 degrees.

00:03:25.550 --> 00:03:28.380
So this is equal to 34,
and it's in degrees.

00:03:28.380 --> 00:03:31.910
So this orange angle
right here is 34 degrees.

00:03:31.910 --> 00:03:35.770
The blue angle is going
to be 180 minus that.

00:03:35.770 --> 00:03:38.480
But we can verify that by
actually evaluating 6x plus

00:03:38.480 --> 00:03:39.720
182.

00:03:39.720 --> 00:03:42.780
So this is going to be
equal to 6 times negative 6

00:03:42.780 --> 00:03:46.490
is negative 36 plus 182.

00:03:46.490 --> 00:03:50.440
So this is going to
be equal to-- let's

00:03:50.440 --> 00:03:55.560
see, if I subtract the
6 first, I get to 176.

00:03:55.560 --> 00:03:59.000
So this gets us to 146 degrees.

00:03:59.000 --> 00:04:03.205
And you can verify-- 146 plus
34 is equal to 180 degrees.

00:04:08.917 --> 00:04:11.250
Now, we could also figure out
the other angles from this

00:04:11.250 --> 00:04:12.080
as well.

00:04:12.080 --> 00:04:15.590
We know that if
this is 34 degrees,

00:04:15.590 --> 00:04:17.490
then this must be
34 degrees as well.

00:04:17.490 --> 00:04:19.190
Those are opposite angles.

00:04:19.190 --> 00:04:21.209
This angle also
corresponds to this angle

00:04:21.209 --> 00:04:23.340
so it must also be
34 degrees, which

00:04:23.340 --> 00:04:26.950
is opposite to this angle,
which is going to be 34 degrees.

00:04:26.950 --> 00:04:30.660
Similarly, if this one right
over here is 146 degrees,

00:04:30.660 --> 00:04:33.160
we already know that this
one is going to be 146.

00:04:33.160 --> 00:04:35.880
This one's going to be
146 since it's opposite.

00:04:35.880 --> 00:04:39.856
And that's going to be
146 degrees as well.


Figuring out angles between transversal and parallel lines

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.210
Let's say that we have
two parallel lines.

00:00:03.210 --> 00:00:05.980
So that's one line
right over there,

00:00:05.980 --> 00:00:07.830
and then this is
the other line that

00:00:07.830 --> 00:00:09.650
is parallel to the first one.

00:00:09.650 --> 00:00:11.630
I'll draw it as
parallel as I can.

00:00:11.630 --> 00:00:13.460
So these two lines are parallel.

00:00:13.460 --> 00:00:15.000
This is the symbol
right over here

00:00:15.000 --> 00:00:17.740
to show that these two
lines are parallel.

00:00:17.740 --> 00:00:19.920
And then let me draw
a transversal here.

00:00:19.920 --> 00:00:21.920
So let me draw a transversal.

00:00:21.920 --> 00:00:24.630
This is also a line.

00:00:24.630 --> 00:00:29.550
Now, let's say that we know
that this angle right over here

00:00:29.550 --> 00:00:32.740
is 110 degrees.

00:00:32.740 --> 00:00:36.780
What other angles can
we figure out here?

00:00:36.780 --> 00:00:38.750
Well, the first thing
that we might realize

00:00:38.750 --> 00:00:41.440
is that, look, corresponding
angles are equivalent.

00:00:41.440 --> 00:00:43.632
This angle, the angle
between this parallel line

00:00:43.632 --> 00:00:45.090
and the transversal,
is going to be

00:00:45.090 --> 00:00:47.370
the same as the angle
between this parallel line

00:00:47.370 --> 00:00:48.730
and the transversal.

00:00:48.730 --> 00:00:52.640
So this right over here is
also going to be 110 degrees.

00:00:52.640 --> 00:00:55.660
Now, we also know that
vertical angles are equivalent.

00:00:55.660 --> 00:00:58.040
So if this is 110
degrees, then this angle

00:00:58.040 --> 00:01:00.580
right over here on the opposite
side of the intersection

00:01:00.580 --> 00:01:02.681
is also going to be 110 degrees.

00:01:02.681 --> 00:01:04.680
And we could use that
same logic right over here

00:01:04.680 --> 00:01:06.340
to say that if this
is 110 degrees,

00:01:06.340 --> 00:01:08.576
then this is also 110 degrees.

00:01:08.576 --> 00:01:09.950
We could've also
said that, look,

00:01:09.950 --> 00:01:12.910
this angle right over here
corresponds to this angle

00:01:12.910 --> 00:01:16.920
right over here so that they
also will have to be the same.

00:01:16.920 --> 00:01:19.360
Now, what about
these other angles?

00:01:19.360 --> 00:01:23.190
So this angle right over
here, its outside ray,

00:01:23.190 --> 00:01:25.277
I guess you could
say, forms a line

00:01:25.277 --> 00:01:26.610
with this angle right over here.

00:01:26.610 --> 00:01:31.110
This pink angle is supplementary
to this 110 degree angle.

00:01:31.110 --> 00:01:35.580
So this pink angle plus 110
is going to be equal to 180.

00:01:35.580 --> 00:01:39.990
Or we know that this pink angle
is going to be 70 degrees.

00:01:39.990 --> 00:01:43.430
And then we know that it's a
vertical angle with this angle

00:01:43.430 --> 00:01:46.320
right over here, so
this is also 70 degrees.

00:01:46.320 --> 00:01:50.620
This angle that's kind of
right below this parallel line

00:01:50.620 --> 00:01:53.910
with the transversal, the bottom
left, I guess you could say,

00:01:53.910 --> 00:01:56.520
corresponds to this bottom
left angle right over here.

00:01:56.520 --> 00:01:58.141
So this is also 70 degrees.

00:01:58.141 --> 00:02:00.140
And we could've also
figured that out by saying,

00:02:00.140 --> 00:02:03.992
hey, this angle is supplementary
to this angle right over here.

00:02:03.992 --> 00:02:05.700
And then we could use
multiple arguments.

00:02:05.700 --> 00:02:08.289
The vertical angle argument,
the supplementary argument two

00:02:08.289 --> 00:02:11.350
ways, or the corresponding
angle argument to say that,

00:02:11.350 --> 00:02:15.940
hey, this must be
70 degrees as well.


Find measure of vertical angles

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.120
Let's say we have two
intersecting lines.

00:00:03.120 --> 00:00:06.080
So that's one of the
lines right over there.

00:00:06.080 --> 00:00:10.490
And then I have another
line right over here.

00:00:10.490 --> 00:00:12.980
So those are my two
intersecting lines.

00:00:12.980 --> 00:00:15.240
And let's say we know that
the measure of this angle

00:00:15.240 --> 00:00:19.840
right over here is
equal to 7x plus 182.

00:00:19.840 --> 00:00:21.550
And this is being
given in degrees,

00:00:21.550 --> 00:00:24.160
so it's 7x plus 182 degrees.

00:00:24.160 --> 00:00:28.090
And we know that the measure
of this angle right over here

00:00:28.090 --> 00:00:33.740
is 9x plus 194 degrees.

00:00:33.740 --> 00:00:36.360
So my question to
you is, what is

00:00:36.360 --> 00:00:39.710
the measure of each
of these angles?

00:00:39.710 --> 00:00:43.300
And I encourage you to pause
the video and to think about it.

00:00:43.300 --> 00:00:45.050
Well, the thing that
might jump out at you

00:00:45.050 --> 00:00:47.870
is that these two things
are vertical angles.

00:00:47.870 --> 00:00:50.880
They're the opposite angles
when we have these intersecting

00:00:50.880 --> 00:00:52.220
lines right over here.

00:00:52.220 --> 00:00:55.200
And vertical angles are
equal to each other.

00:00:55.200 --> 00:00:57.790
So we know, because these
are vertical angles,

00:00:57.790 --> 00:01:08.680
that 9x plus 194 degrees must
be equal to 7x plus 182 degrees.

00:01:08.680 --> 00:01:11.080
And now we just
have to solve for x.

00:01:11.080 --> 00:01:13.340
So if we want all the x-terms
on the left-hand side,

00:01:13.340 --> 00:01:15.237
we could subtract 7x from here.

00:01:15.237 --> 00:01:17.070
We've got to do it to
both sides, of course,

00:01:17.070 --> 00:01:18.832
in order to maintain
the equality.

00:01:18.832 --> 00:01:20.790
And then we could put
all of our constant terms

00:01:20.790 --> 00:01:21.920
on the right-hand side.

00:01:21.920 --> 00:01:24.420
So we can subtract
194 from the left.

00:01:24.420 --> 00:01:26.610
We have to subtract
194 from the right

00:01:26.610 --> 00:01:29.020
in order to maintain
the inequality.

00:01:29.020 --> 00:01:33.830
And on the left, what
we're left with is just 2x.

00:01:33.830 --> 00:01:37.220
And on the right, what
we're left with-- let's see.

00:01:37.220 --> 00:01:39.710
182 minus 194.

00:01:39.710 --> 00:01:42.480
So if it was 194 minus 182,
it would be positive 12.

00:01:42.480 --> 00:01:44.024
But now it's going
to be negative 12.

00:01:44.024 --> 00:01:45.940
We're subtracting the
larger from the smaller,

00:01:45.940 --> 00:01:48.420
so it's equal to negative 12.

00:01:48.420 --> 00:01:50.970
And then divide both sides by 2.

00:01:50.970 --> 00:01:55.632
And we get x is
equal to negative 6.

00:01:55.632 --> 00:01:57.090
And now we can use
that information

00:01:57.090 --> 00:01:59.380
to find out the measure of
either one of these angles,

00:01:59.380 --> 00:02:01.060
which is the same
as the other one.

00:02:01.060 --> 00:02:07.260
So we can see here that if we
take 7 times negative 6 plus

00:02:07.260 --> 00:02:13.940
182, so 7 times negative
6 is negative 42,

00:02:13.940 --> 00:02:18.960
plus 182 is going to be
equal to 140 degrees.

00:02:18.960 --> 00:02:20.980
And you'll see the
same thing over here.

00:02:20.980 --> 00:02:28.600
If we say 9 times negative 6,
which is negative 54, plus 194,

00:02:28.600 --> 00:02:32.541
this also equals 140 degrees.


Determining probability

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:02.570
We were given two dice to roll.

00:00:02.570 --> 00:00:04.930
One is black with six sides.

00:00:04.930 --> 00:00:07.050
The other is white
with four sides.

00:00:07.050 --> 00:00:10.490
So a four-sided die is kind
of a pyramid-looking thing.

00:00:10.490 --> 00:00:13.420
It has exactly four sides.

00:00:13.420 --> 00:00:15.370
It would be a pyramid
with a triangular base.

00:00:15.370 --> 00:00:17.078
For a given roll, what
is the probability

00:00:17.078 --> 00:00:19.900
that the dice add up to 10?

00:00:19.900 --> 00:00:23.340
If it helps, you may select
the matching outcomes below.

00:00:23.340 --> 00:00:25.610
Your selections aren't
checked with your answer.

00:00:25.610 --> 00:00:27.230
So this might look a little
bit strange at first.

00:00:27.230 --> 00:00:28.688
You're like, what
is this triangle?

00:00:28.688 --> 00:00:32.940
This triangle is the face
of the four-sided die

00:00:32.940 --> 00:00:34.570
that is facing you.

00:00:34.570 --> 00:00:37.740
So this is the roll
of the four-sided die.

00:00:37.740 --> 00:00:39.380
So this is a roll of 1 and 1.

00:00:39.380 --> 00:00:41.089
So these clearly are
not adding up to 10.

00:00:41.089 --> 00:00:43.254
So let's think about all
the ones that add up to 10.

00:00:43.254 --> 00:00:44.316
So this adds up to 2.

00:00:44.316 --> 00:00:46.510
This adds up to 3.

00:00:46.510 --> 00:00:48.580
This adds up to 4.

00:00:48.580 --> 00:00:50.137
This adds up to 5.

00:00:50.137 --> 00:00:51.720
Let's see what
actually adds up to 10.

00:00:51.720 --> 00:00:55.560
So if we go all the way-- so
clearly, if we get a 4 and a 6,

00:00:55.560 --> 00:00:58.450
that's going to add up to 10.

00:00:58.450 --> 00:01:00.010
And are there any
other possibilities

00:01:00.010 --> 00:01:01.860
that add up to 10?

00:01:01.860 --> 00:01:05.440
Well, the highest possible roll
I can get on a four-sided die

00:01:05.440 --> 00:01:06.790
is a 4.

00:01:06.790 --> 00:01:09.010
So that's my best scenario.

00:01:09.010 --> 00:01:10.660
That's the highest
possible roll.

00:01:10.660 --> 00:01:13.270
And even if I get that, I still
need the highest possible roll

00:01:13.270 --> 00:01:14.650
on the six-sided die.

00:01:14.650 --> 00:01:16.650
So this is the only
scenario that adds up to 10.

00:01:16.650 --> 00:01:18.316
You can look at any
of these other ones.

00:01:18.316 --> 00:01:20.250
They'll add up to
something less than 10.

00:01:20.250 --> 00:01:24.130
So there's one possibility
that satisfies our constraints.

00:01:24.130 --> 00:01:25.960
The dice add up to 10.

00:01:25.960 --> 00:01:28.750
And how many total equally
likely possibilities

00:01:28.750 --> 00:01:29.530
were there?

00:01:29.530 --> 00:01:30.030
Let's see.

00:01:30.030 --> 00:01:32.930
This grid shows all the
equally likely possibilities.

00:01:32.930 --> 00:01:37.120
This is 1, 2, 3, 4
times 1, 2, 3, 4, 5, 6.

00:01:37.120 --> 00:01:39.970
There are 24 equally
likely possibilities,

00:01:39.970 --> 00:01:43.320
and that comes from you
have 6 possibilities

00:01:43.320 --> 00:01:45.376
from the six-sided die
times 4 possibilities

00:01:45.376 --> 00:01:46.750
from the four-sided
die gives you

00:01:46.750 --> 00:01:49.440
24 equally likely possibilities.

00:01:49.440 --> 00:01:54.440
Only one satisfies
the constraint

00:01:54.440 --> 00:01:56.520
that the dice add up to 10.

00:01:56.520 --> 00:01:59.390
Let's do a few more of these.

00:01:59.390 --> 00:02:02.280
So you're given two
six-sided dice to roll.

00:02:02.280 --> 00:02:03.990
For a given roll, what
is the probability

00:02:03.990 --> 00:02:06.500
that the dice add to 6?

00:02:06.500 --> 00:02:10.340
So here we have the
tops of both of my dice.

00:02:10.340 --> 00:02:12.786
And now what are all the
scenarios where the dice add up

00:02:12.786 --> 00:02:13.285
to 6?

00:02:16.680 --> 00:02:17.950
Well, let's think about this.

00:02:17.950 --> 00:02:22.640
If I have a 1 and a 5,
that's going to add up to 6.

00:02:22.640 --> 00:02:24.650
In this entire column,
that's the only way

00:02:24.650 --> 00:02:26.320
that I add up to 6.

00:02:26.320 --> 00:02:28.260
Right over here, let's see.

00:02:28.260 --> 00:02:31.550
If I have a 2 on the black
die, then I'm going to need a 4

00:02:31.550 --> 00:02:34.410
on the white die to add up to 6.

00:02:34.410 --> 00:02:36.620
If I have a 3 on the black
die, I'm going to need a 3

00:02:36.620 --> 00:02:38.572
on the white die to add up to 6.

00:02:38.572 --> 00:02:40.780
If I have a 4 on the black
die, I'm going to need a 2

00:02:40.780 --> 00:02:41.990
on the white die.

00:02:41.990 --> 00:02:44.550
If I have a 5 on the black
die, I'm going to need a 1

00:02:44.550 --> 00:02:45.620
on the white die.

00:02:45.620 --> 00:02:47.370
And then if I have 6
six on the black die,

00:02:47.370 --> 00:02:50.190
it's actually impossible that
the sum will be exactly 6.

00:02:50.190 --> 00:02:51.790
So there's no zero
on the white die.

00:02:51.790 --> 00:02:54.510
So none of these
meet the constraint.

00:02:54.510 --> 00:02:58.790
So here I have one, two,
three, four, five possibilities

00:02:58.790 --> 00:03:01.060
where my dice add up to 6.

00:03:01.060 --> 00:03:03.620
So I have five
possibilities out of a total

00:03:03.620 --> 00:03:06.070
of 36 equally likely ones.

00:03:06.070 --> 00:03:07.950
This is a six by
six grid, and it

00:03:07.950 --> 00:03:09.490
comes from six
possibilities for one

00:03:09.490 --> 00:03:12.120
die times six possibilities
for the other die.

00:03:12.120 --> 00:03:14.690
So 5 possibilities
satisfy my constraint

00:03:14.690 --> 00:03:18.371
out of a total of 36.

00:03:18.371 --> 00:03:19.120
Let's do one more.

00:03:19.120 --> 00:03:22.410
This is actually a lot of fun.

00:03:22.410 --> 00:03:26.692
You're given two
six-sided dice to roll.

00:03:26.692 --> 00:03:28.400
For a given roll, what
is the probability

00:03:28.400 --> 00:03:33.720
at least one die is a six?

00:03:33.720 --> 00:03:34.540
So let's see.

00:03:34.540 --> 00:03:36.700
So we just have to look
for all of the situations

00:03:36.700 --> 00:03:39.500
where at least one
of the dice is a six.

00:03:39.500 --> 00:03:41.870
So that one, that one.

00:03:41.870 --> 00:03:44.492
Every time I see a six, I
should just click on that.

00:03:44.492 --> 00:03:45.436
That one.

00:03:45.436 --> 00:03:46.130
That one.

00:03:46.130 --> 00:03:49.554
And then obviously all of these
scenarios right over here.

00:03:49.554 --> 00:03:51.720
So these are all of the
scenarios where at least one

00:03:51.720 --> 00:03:52.990
of the dice is a six.

00:03:52.990 --> 00:03:55.390
Here, of course, is where
both of the dice are a six.

00:03:55.390 --> 00:03:57.000
So how many scenarios are there?

00:03:57.000 --> 00:04:03.900
1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11.

00:04:03.900 --> 00:04:06.640
So there are 11
possibilities out

00:04:06.640 --> 00:04:09.220
of a total of 36
equally likely ones.

00:04:09.220 --> 00:04:10.560
Remember, 6 times 6.

00:04:10.560 --> 00:04:13.730
There's 36 equally
likely outcomes.

00:04:13.730 --> 00:04:15.400
So let's check our answer.

00:04:15.400 --> 00:04:17.404
And we got it right.


Find measure of supplementary angles

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:04.910
We're told that the measure
of angle QPR-- so that's

00:00:04.910 --> 00:00:08.860
this angle right over
here-- is 2x plus 122.

00:00:08.860 --> 00:00:11.220
And I'll assume that
these are in degrees.

00:00:11.220 --> 00:00:15.870
So it's 2x plus 122 degrees.

00:00:15.870 --> 00:00:20.800
And the measure of
angle RPS-- so that's

00:00:20.800 --> 00:00:26.840
this angle right over here--
is 2x plus 22 degrees.

00:00:26.840 --> 00:00:29.840
And they ask us to find
the measure of angle RPS.

00:00:29.840 --> 00:00:32.617
So we need to figure out
this right over here.

00:00:32.617 --> 00:00:34.200
So we would be able
to figure that out

00:00:34.200 --> 00:00:36.010
if we just knew what x is.

00:00:36.010 --> 00:00:38.390
And lucky for us, we
can use the information

00:00:38.390 --> 00:00:39.960
given to solve for
x and then figure

00:00:39.960 --> 00:00:42.620
out what 2 times x plus 22 is.

00:00:42.620 --> 00:00:46.200
And the main big idea here,
the thing that pops out here,

00:00:46.200 --> 00:00:48.720
is that the outside rays
for both of these angle

00:00:48.720 --> 00:00:49.930
form a line.

00:00:49.930 --> 00:00:52.210
These two angles form a line.

00:00:52.210 --> 00:00:55.006
You could say that
they are supplementary.

00:00:55.006 --> 00:00:56.630
Both of these angles
are supplementary.

00:00:56.630 --> 00:01:02.150
2x plus 22 plus another 2x plus
122 is going to add up to 180.

00:01:02.150 --> 00:01:08.350
We know that this entire angle
right over here is 180 degrees.

00:01:08.350 --> 00:01:11.200
So we can say that the measure
of angle QPR, this angle

00:01:11.200 --> 00:01:17.090
right over here, 2x plus
122, plus the green angle,

00:01:17.090 --> 00:01:22.970
plus angle RPS-- so
plus 2x plus 22-- is

00:01:22.970 --> 00:01:25.225
going to be equal
to 180 degrees.

00:01:28.420 --> 00:01:31.690
And now we can start
simplifying this.

00:01:31.690 --> 00:01:32.510
We have two x's.

00:01:32.510 --> 00:01:34.100
We have another two x's.

00:01:34.100 --> 00:01:36.560
So those are going
to add up to be 4x.

00:01:36.560 --> 00:01:41.000
And then we have 122 plus 22.

00:01:41.000 --> 00:01:43.400
So that's going to be 144.

00:01:43.400 --> 00:01:45.710
And the sum of
those two are going

00:01:45.710 --> 00:01:47.900
to be equal to 180 degrees.

00:01:47.900 --> 00:01:50.660
We can subtract 144
from both sides.

00:01:54.140 --> 00:01:56.140
On the left-hand
side, we're just

00:01:56.140 --> 00:01:59.870
going to be left with a
4x, this 4x right here.

00:01:59.870 --> 00:02:01.530
And on the right-hand
side, we're

00:02:01.530 --> 00:02:04.910
going to have-- let's see,
if we were subtracting 140,

00:02:04.910 --> 00:02:06.005
we would have 40 left.

00:02:06.005 --> 00:02:07.630
And then we have to
subtract another 4,

00:02:07.630 --> 00:02:09.410
so it's going to be 36.

00:02:09.410 --> 00:02:16.955
Divide both sides by 4,
and we get x is equal to 9.

00:02:16.955 --> 00:02:18.330
Now remember,
we're not done yet.

00:02:18.330 --> 00:02:19.890
They didn't say solve for x.

00:02:19.890 --> 00:02:23.850
They said find the measure of
angle RPS, which is 2 times

00:02:23.850 --> 00:02:28.530
x plus 22 or 2 times
9 plus 22, which

00:02:28.530 --> 00:02:32.390
is 18 plus 22, which
is equal to 40.

00:02:32.390 --> 00:02:36.210
So the measure of angle
RPS is 40 degrees.


Find measure of complementary angles

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.990
So we're told that ray OL
is perpendicular to ray ON.

00:00:04.990 --> 00:00:07.640
So OL is perpendicular to ON.

00:00:07.640 --> 00:00:10.020
So we know that this right
over here is a right angle,

00:00:10.020 --> 00:00:11.600
or it's a 90-degree angle.

00:00:11.600 --> 00:00:15.190
Then they tell us that
the measure of angle LOM

00:00:15.190 --> 00:00:18.060
is equal to 2x plus 46.

00:00:18.060 --> 00:00:22.360
So LOM-- so this angle
right over here--

00:00:22.360 --> 00:00:24.937
is equal to 2x plus 46.

00:00:28.100 --> 00:00:31.000
Then they tell us that
the measure of angle MON

00:00:31.000 --> 00:00:34.480
is equal to 3x minus 6.

00:00:34.480 --> 00:00:38.440
So this angle right
over here is 3x minus 6.

00:00:38.440 --> 00:00:40.730
And so they ask
us find angle MON.

00:00:40.730 --> 00:00:43.339
So we need to find this
angle right over here.

00:00:43.339 --> 00:00:45.380
And we'd be able to find
it if we knew what x is.

00:00:45.380 --> 00:00:47.505
If we knew what x is, then
it's going to be 3 times

00:00:47.505 --> 00:00:48.720
that minus 6.

00:00:48.720 --> 00:00:50.330
Now, the one thing
that we know is

00:00:50.330 --> 00:00:52.020
that if we add up
these two angles,

00:00:52.020 --> 00:00:53.710
these are adjacent angles.

00:00:53.710 --> 00:00:56.880
And their outside rays
form a right angle.

00:00:56.880 --> 00:01:00.130
So these two angles are going
to add up to 90 degrees.

00:01:00.130 --> 00:01:02.820
And I guess we can assume here
that we're dealing in degrees.

00:01:02.820 --> 00:01:12.990
So let's say that 2x
plus 46 plus 3x minus 6

00:01:12.990 --> 00:01:17.410
is going to add
up to 90 degrees.

00:01:17.410 --> 00:01:19.610
These two angles
are complementary.

00:01:19.610 --> 00:01:21.830
So it's going to add
up to 90 degrees,

00:01:21.830 --> 00:01:23.164
or it's going to add up to 90.

00:01:23.164 --> 00:01:24.580
And now, we just
have to simplify.

00:01:24.580 --> 00:01:25.460
We have 2 x's.

00:01:25.460 --> 00:01:26.690
We have another 3 x's.

00:01:26.690 --> 00:01:29.720
So you add those two, and
you're going to get 5 x's.

00:01:29.720 --> 00:01:32.419
And then you have 46.

00:01:32.419 --> 00:01:33.710
And you're going to subtract 6.

00:01:33.710 --> 00:01:36.700
So it's going to be
plus 40 is equal to 90.

00:01:39.220 --> 00:01:41.550
Subtract 40 from both sides.

00:01:41.550 --> 00:01:42.450
So let's do that.

00:01:42.450 --> 00:01:45.060
Subtract 40 from both sides.

00:01:45.060 --> 00:01:49.140
And on the left-hand side,
you're just left with a 5x.

00:01:49.140 --> 00:01:53.220
And on the right-hand side,
you're left with a 50.

00:01:53.220 --> 00:01:55.950
Now, just divide
both sides by 5.

00:01:55.950 --> 00:02:00.630
And we are left with
x is equal to 10.

00:02:00.630 --> 00:02:02.240
Now, our answer is not 10.

00:02:02.240 --> 00:02:04.280
We've just figured
out what x is.

00:02:04.280 --> 00:02:07.470
What they ask us for is what's
the measure of angle MON?

00:02:07.470 --> 00:02:11.770
Well, we already know that
the measure of angle MON

00:02:11.770 --> 00:02:23.300
is equal to 3x minus 6, which
is equal to 3 times 10 minus 6--

00:02:23.300 --> 00:02:27.430
I want to do that minus 6
in that green color-- which

00:02:27.430 --> 00:02:32.550
is equal to 24 degrees.


Finding the vertex of a parabola example

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.000
I have an equation right here.

00:00:02.000 --> 00:00:03.500
It's a second degree equation.

00:00:03.500 --> 00:00:04.760
It's a quadratic.

00:00:04.760 --> 00:00:07.280
And I know its graph is
going to be a parabola.

00:00:07.280 --> 00:00:10.495
Just as a review, that means it
looks something like this or it

00:00:10.495 --> 00:00:12.850
looks something like that.

00:00:12.850 --> 00:00:17.470
Because the coefficient on the
x squared term here is positive,

00:00:17.470 --> 00:00:20.130
I know it's going to be an
upward opening parabola.

00:00:20.130 --> 00:00:24.372
And I am curious about the
vertex of this parabola.

00:00:24.372 --> 00:00:26.080
And if I have an upward
opening parabola,

00:00:26.080 --> 00:00:27.880
the vertex is going to
be the minimum point.

00:00:27.880 --> 00:00:29.421
If I had a downward
opening parabola,

00:00:29.421 --> 00:00:31.490
then the vertex would
be the maximum point.

00:00:31.490 --> 00:00:37.410
So I'm really trying
to find the x value.

00:00:37.410 --> 00:00:39.910
I don't know actually where
this does intersect the x-axis

00:00:39.910 --> 00:00:41.150
or if it does it all.

00:00:41.150 --> 00:00:44.050
But I want to find
the x value where

00:00:44.050 --> 00:00:47.520
this function takes
on a minimum value.

00:00:47.520 --> 00:00:49.970
Now, there's many
ways to find a vertex.

00:00:49.970 --> 00:00:53.060
Probably the easiest,
there's a formula for it.

00:00:53.060 --> 00:00:55.680
And we talk about where that
comes from in multiple videos,

00:00:55.680 --> 00:00:58.270
where the vertex of a
parabola or the x-coordinate

00:00:58.270 --> 00:01:00.170
of the vertex of the parabola.

00:01:00.170 --> 00:01:02.710
So the x-coordinate
of the vertex

00:01:02.710 --> 00:01:07.030
is just equal to
negative b over 2a.

00:01:07.030 --> 00:01:10.360
And the negative b, you're just
talking about the coefficient,

00:01:10.360 --> 00:01:14.030
or b is the coefficient
on the first degree term,

00:01:14.030 --> 00:01:15.830
is on the coefficient
on the x term.

00:01:15.830 --> 00:01:18.150
And a is the coefficient
on the x squared term.

00:01:18.150 --> 00:01:21.555
So this is going to be
equal to b is negative 20.

00:01:21.555 --> 00:01:28.295
So it's negative
20 over 2 times 5.

00:01:34.360 --> 00:01:36.930
Well, this is going to
be equal to positive 20

00:01:36.930 --> 00:01:39.920
over 10, which is equal to 2.

00:01:39.920 --> 00:01:42.470
And so to find the y
value of the vertex,

00:01:42.470 --> 00:01:45.900
we just substitute
back into the equation.

00:01:45.900 --> 00:01:51.710
The y value is going
to be 5 times 2 squared

00:01:51.710 --> 00:01:56.520
minus 20 times 2 plus 15,
which is equal to let's see.

00:01:56.520 --> 00:02:02.300
This is 5 times 4, which is 20,
minus 40, which is negative 20,

00:02:02.300 --> 00:02:03.942
plus 15 is negative 5.

00:02:03.942 --> 00:02:06.400
So just like that, we're able
to figure out the coordinate.

00:02:06.400 --> 00:02:09.740
This coordinate right over here
is the point 2, negative 5.

00:02:12.660 --> 00:02:14.730
Now it's not so
satisfying just to plug

00:02:14.730 --> 00:02:17.904
and chug a formula like this.

00:02:17.904 --> 00:02:19.570
And we'll see where
this comes from when

00:02:19.570 --> 00:02:21.060
you look at the
quadratic formula.

00:02:21.060 --> 00:02:22.530
This is the first term.

00:02:22.530 --> 00:02:26.180
It's the x value that's
halfway in between the roots.

00:02:26.180 --> 00:02:28.350
So that's one way
to think about it.

00:02:28.350 --> 00:02:30.330
But another way to do
it, and this probably

00:02:30.330 --> 00:02:32.990
will be of more lasting
help for you in your life,

00:02:32.990 --> 00:02:34.700
because you might
forget this formula.

00:02:34.700 --> 00:02:37.530
It's really just try to
re-manipulate this equation

00:02:37.530 --> 00:02:40.070
so you can spot
its minimum point.

00:02:40.070 --> 00:02:42.370
And we're going to do that
by completing the square.

00:02:42.370 --> 00:02:43.470
So let me rewrite that.

00:02:47.259 --> 00:02:49.300
And what I'll do is out
of these first two terms,

00:02:49.300 --> 00:02:51.680
I'll factor out a 5, because I
want to complete a square here

00:02:51.680 --> 00:02:53.370
and I'm going to leave
this 15 out to the right,

00:02:53.370 --> 00:02:55.578
because I'm going to have
to manipulate that as well.

00:02:55.578 --> 00:03:00.770
So it is 5 times x
squared minus 4x.

00:03:00.770 --> 00:03:05.040
And then I have
this 15 out here.

00:03:05.040 --> 00:03:07.650
And I want to write this
as a perfect square.

00:03:07.650 --> 00:03:09.470
And we just have
to remind ourselves

00:03:09.470 --> 00:03:13.550
that if I have x plus
a squared, that's

00:03:13.550 --> 00:03:18.730
going to be x squared
plus 2ax plus a squared.

00:03:18.730 --> 00:03:25.130
So if I want to turn something
that looks like this, 2ax,

00:03:25.130 --> 00:03:26.920
into a perfect
square, I just have

00:03:26.920 --> 00:03:29.830
to take half of this coefficient
and square it and add it

00:03:29.830 --> 00:03:32.530
right over here in order
to make it look like that.

00:03:32.530 --> 00:03:35.340
So I'm going to do
that right over here.

00:03:35.340 --> 00:03:38.980
So if I take half of negative
4, that's negative 2.

00:03:38.980 --> 00:03:42.810
If I square it, that is
going to be positive 4.

00:03:42.810 --> 00:03:44.220
I have to be very careful here.

00:03:44.220 --> 00:03:46.750
I can't just willy nilly
add a positive 4 here.

00:03:46.750 --> 00:03:47.816
I have equality here.

00:03:47.816 --> 00:03:49.440
If they were equal
before adding the 4,

00:03:49.440 --> 00:03:51.689
then they're not going to
be equal after adding the 4.

00:03:51.689 --> 00:03:53.490
So I have to do proper
accounting here.

00:03:53.490 --> 00:03:56.120
I either have to add 4 to both
sides or I should be careful.

00:03:56.120 --> 00:03:58.580
I have to add the same
amount to both sides

00:03:58.580 --> 00:04:01.387
or subtract the
same amount again.

00:04:01.387 --> 00:04:03.220
Now, the reason why I
was careful there is I

00:04:03.220 --> 00:04:05.850
didn't just add 4 to the right
hand side of the equation.

00:04:05.850 --> 00:04:08.900
Remember, the 4 is
getting multiplied by 5.

00:04:08.900 --> 00:04:12.440
I have added 20 to the right
hand side of the equation.

00:04:12.440 --> 00:04:14.450
So if I want to make
this balance out,

00:04:14.450 --> 00:04:16.570
if I want the equality
to still be true,

00:04:16.570 --> 00:04:20.660
I either have to
now add 20 to y or I

00:04:20.660 --> 00:04:22.930
have to subtract 20 from
the right hand side.

00:04:22.930 --> 00:04:23.750
So I'll do that.

00:04:23.750 --> 00:04:26.300
I'll subtract 20 from
the right hand side.

00:04:26.300 --> 00:04:27.970
So I added 5 times 4.

00:04:27.970 --> 00:04:29.970
If you were to distribute
this, you'll see that.

00:04:29.970 --> 00:04:34.320
I could have literally, up
here, said hey, I'm adding 20

00:04:34.320 --> 00:04:36.110
and I'm subtracting 20.

00:04:36.110 --> 00:04:38.390
This is the exact same
thing that I did over here.

00:04:38.390 --> 00:04:41.950
If you distribute the 5, it
becomes 5x squared minus 20x

00:04:41.950 --> 00:04:44.930
plus 20 plus 15 minus 20.

00:04:44.930 --> 00:04:47.100
Exactly what's up here.

00:04:47.100 --> 00:04:48.750
The whole point of
this is that now I

00:04:48.750 --> 00:04:50.890
can write this in
an interesting way.

00:04:50.890 --> 00:04:57.430
I could write this as y is equal
to 5 times x minus 2 squared,

00:04:57.430 --> 00:05:00.820
and then 15 minus 20 is minus 5.

00:05:00.820 --> 00:05:03.390
So the whole point of this is
now to be able to inspect this.

00:05:03.390 --> 00:05:06.810
When does this equation
hit a minimum value?

00:05:06.810 --> 00:05:10.830
Well, we know that this
term right over here

00:05:10.830 --> 00:05:12.423
is always going to
be non-negative.

00:05:14.805 --> 00:05:16.180
Or we could say
it's always going

00:05:16.180 --> 00:05:18.760
to be greater than
or equal to 0.

00:05:18.760 --> 00:05:20.760
This whole thing is going
to hit a minimum value

00:05:20.760 --> 00:05:25.740
when this term is equal
to 0 or when x equals 2.

00:05:25.740 --> 00:05:29.160
When x equals 2, we're going
to hit a minimum value.

00:05:29.160 --> 00:05:30.690
And when x equals
2, what happens?

00:05:30.690 --> 00:05:35.440
Well, this whole term is 0
and y is equal to negative 5.

00:05:35.440 --> 00:05:39.060
The vertex is 2, negative 5.


Examples using pythagorean identities to simplify trigonometric

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:02.650
Let's do some
examples simplifying

00:00:02.650 --> 00:00:04.590
trigonometric expressions.

00:00:04.590 --> 00:00:10.210
So let's say that I have 1
minus sine squared theta,

00:00:10.210 --> 00:00:16.190
and this whole thing times
cosine squared theta.

00:00:16.190 --> 00:00:18.110
So how could I simplify this?

00:00:18.110 --> 00:00:19.820
Well the one thing
that we do know--

00:00:19.820 --> 00:00:21.550
and this is the most
fundamental trig identity,

00:00:21.550 --> 00:00:23.383
this comes straight out
of the unit circle--

00:00:23.383 --> 00:00:27.470
is that cosine squared
theta plus sine squared

00:00:27.470 --> 00:00:29.520
theta is equal to 1.

00:00:29.520 --> 00:00:32.790
And then, if we subtract sine
squared theta from both sides,

00:00:32.790 --> 00:00:36.300
we get cosine squared
theta is equal to 1

00:00:36.300 --> 00:00:39.530
minus sine squared theta.

00:00:39.530 --> 00:00:40.750
So we have two options.

00:00:40.750 --> 00:00:43.580
We could either replace this
1 minus sine squared theta

00:00:43.580 --> 00:00:45.670
with the cosine
squared theta, or we

00:00:45.670 --> 00:00:47.300
could replace this
cosine squared theta

00:00:47.300 --> 00:00:49.335
with the 1 minus
sine squared theta.

00:00:49.335 --> 00:00:51.210
Well I'd prefer to do
the former because this

00:00:51.210 --> 00:00:52.810
is a more complicated
expression.

00:00:52.810 --> 00:00:58.510
So if I can replace this with
the cosine squared theta,

00:00:58.510 --> 00:01:00.499
then I think I'm
simplifying this.

00:01:00.499 --> 00:01:01.040
So let's see.

00:01:01.040 --> 00:01:03.850
This will be cosine
squared theta times

00:01:03.850 --> 00:01:06.690
another cosine squared theta.

00:01:06.690 --> 00:01:09.796
And so all of this is going to
simplify to cosine theta times

00:01:09.796 --> 00:01:12.170
cosine theta times cosine
theta times cosine theta, well,

00:01:12.170 --> 00:01:16.780
that's just going to be
cosine to the fourth of theta.

00:01:16.780 --> 00:01:19.930
Let's do another example.

00:01:19.930 --> 00:01:31.350
Let's say that we have sine
squared theta, all of that

00:01:31.350 --> 00:01:37.920
over 1 minus sine squared theta.

00:01:37.920 --> 00:01:40.674
What is this going
to be equal to?

00:01:40.674 --> 00:01:42.590
Well we already know
that 1 minus sine squared

00:01:42.590 --> 00:01:44.730
theta is the same thing
as cosine squared theta.

00:01:44.730 --> 00:01:50.020
So it's going to be sine
squared theta over-- this thing

00:01:50.020 --> 00:01:52.050
is the same thing as
cosine squared theta,

00:01:52.050 --> 00:01:56.250
we just saw that-- over
cosine squared theta, which

00:01:56.250 --> 00:01:58.540
is going to be equal
to-- you could view

00:01:58.540 --> 00:02:05.010
this as sine theta over cosine
theta whole quantity squared.

00:02:05.010 --> 00:02:06.970
Well what's sine over cosine?

00:02:06.970 --> 00:02:08.160
That's tangent.

00:02:08.160 --> 00:02:12.670
So this is equal to
tangent squared theta.

00:02:12.670 --> 00:02:14.600
Let's do one more example.

00:02:14.600 --> 00:02:26.944
Let's say that we have cosine
squared theta plus 1 minus--

00:02:26.944 --> 00:02:28.360
actually, let's
make it this way--

00:02:28.360 --> 00:02:32.185
plus 1 plus sine squared theta.

00:02:35.420 --> 00:02:37.947
What is this going to be?

00:02:37.947 --> 00:02:40.030
Well you might be tempted,
especially with the way

00:02:40.030 --> 00:02:41.446
I wrote the colors,
to think, hey,

00:02:41.446 --> 00:02:43.875
is there some identity for
1 plus sine squared theta?

00:02:43.875 --> 00:02:45.630
But this is really
all about rearranging

00:02:45.630 --> 00:02:48.640
it to realize that, gee, by
the unit circle definition,

00:02:48.640 --> 00:02:52.730
I know what cosine squared theta
plus sine squared theta is.

00:02:52.730 --> 00:02:54.820
Cosine squared theta
plus sine squared

00:02:54.820 --> 00:02:57.500
theta, for any given theta,
is going to be equal to 1.

00:02:57.500 --> 00:03:00.580
So this is going to be
equal to 1 plus this 1 right

00:03:00.580 --> 00:03:06.300
over here, which is equal to 2.


Double angle formula for cosine example

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.926
We have triangle ABC here, which
looks like a right triangle.

00:00:03.926 --> 00:00:05.300
And we know it's
a right triangle

00:00:05.300 --> 00:00:08.720
because 3 squared plus 4
squared is equal to 5 squared.

00:00:08.720 --> 00:00:12.050
And they want us to figure out
what cosine of 2 times angle

00:00:12.050 --> 00:00:12.970
ABC is.

00:00:12.970 --> 00:00:16.620
So that's this angle-- ABC.

00:00:16.620 --> 00:00:18.850
Well, we can't
immediately evaluate that,

00:00:18.850 --> 00:00:22.400
but we do know what the
cosine of angle ABC is.

00:00:22.400 --> 00:00:27.810
We know that the cosine of
angle ABC-- well, cosine

00:00:27.810 --> 00:00:29.680
is just adjacent
over hypotenuse.

00:00:29.680 --> 00:00:32.159
It's going to be equal to 3/5.

00:00:32.159 --> 00:00:39.700
And similarly, we know what
the sine of angle ABC is.

00:00:39.700 --> 00:00:41.530
That's opposite over hypotenuse.

00:00:41.530 --> 00:00:43.310
That is 4/5.

00:00:43.310 --> 00:00:46.200
So if we could break this
down into just cosines of ABC

00:00:46.200 --> 00:00:49.120
and sines of ABC, then we'll
be able to evaluate it.

00:00:49.120 --> 00:00:52.340
And lucky for us, we have a
trig identity at our disposal

00:00:52.340 --> 00:00:55.410
that does exactly that.

00:00:55.410 --> 00:01:00.310
We know that the cosine
of 2 times an angle

00:01:00.310 --> 00:01:03.390
is equal to cosine
of that angle squared

00:01:03.390 --> 00:01:05.560
minus sine of that
angle squared.

00:01:05.560 --> 00:01:07.710
And we've proved
this in other videos,

00:01:07.710 --> 00:01:10.040
but this becomes very
helpful for us here.

00:01:10.040 --> 00:01:13.545
Because now we know
that the cosine--

00:01:13.545 --> 00:01:15.560
Let me do this in
a different color.

00:01:15.560 --> 00:01:21.460
Now, we know that the
cosine of angle ABC

00:01:21.460 --> 00:01:23.670
is going to be equal
to-- oh, sorry.

00:01:23.670 --> 00:01:25.810
It's the cosine of 2
times the angle ABC.

00:01:25.810 --> 00:01:27.360
That's what we care about.

00:01:27.360 --> 00:01:30.310
2 times the angle
ABC is going to be

00:01:30.310 --> 00:01:40.480
equal to the cosine of angle ABC
squared minus sine of the angle

00:01:40.480 --> 00:01:42.050
ABC squared.

00:01:45.770 --> 00:01:47.740
And we know what
these things are.

00:01:47.740 --> 00:01:52.060
This thing right
over here is just

00:01:52.060 --> 00:01:55.470
going to be equal
to 3/5 squared.

00:01:55.470 --> 00:01:57.320
Cosine of angle a ABC is 3/5.

00:01:57.320 --> 00:01:58.640
So we're going to square it.

00:01:58.640 --> 00:02:01.740
And this right over here
is just 4/5 squared.

00:02:01.740 --> 00:02:05.590
So it's minus 4/5 squared.

00:02:05.590 --> 00:02:14.120
And so this simplifies
to 9/25 minus 16/25,

00:02:14.120 --> 00:02:17.320
which is equal to 7/25.

00:02:21.515 --> 00:02:22.015
Sorry.

00:02:22.015 --> 00:02:23.021
It's negative.

00:02:23.021 --> 00:02:24.020
Got to be careful there.

00:02:24.020 --> 00:02:27.010
16 is larger than 9.

00:02:27.010 --> 00:02:29.299
Negative 7/25.

00:02:29.299 --> 00:02:30.840
Now, one thing that
might jump at you

00:02:30.840 --> 00:02:33.370
is, why did I get a
negative value here

00:02:33.370 --> 00:02:36.260
when I doubled the angle here?

00:02:36.260 --> 00:02:38.852
Because the cosine was
clearly a positive number.

00:02:38.852 --> 00:02:41.310
And there you just have to
think of the unit circle-- which

00:02:41.310 --> 00:02:45.910
we already know the unit circle
definition of trig functions

00:02:45.910 --> 00:02:49.080
is an extension of the
Sohcahtoa definition.

00:02:49.080 --> 00:02:49.830
X-axis.

00:02:49.830 --> 00:02:50.740
Y-axis.

00:02:50.740 --> 00:02:52.830
Let me draw a unit circle here.

00:02:52.830 --> 00:02:54.430
My best attempt.

00:02:54.430 --> 00:02:57.340
So that's our unit circle.

00:02:57.340 --> 00:03:01.775
So this angle right over here
looks like something like this.

00:03:04.380 --> 00:03:06.610
And so you see its
x-coordinate-- which

00:03:06.610 --> 00:03:08.940
is the cosine of that
angle-- looks positive.

00:03:08.940 --> 00:03:11.640
But then, if you were
to double this angle,

00:03:11.640 --> 00:03:15.965
it would take you out
someplace like this.

00:03:15.965 --> 00:03:18.090
And then, you see-- by the
unit circle definition--

00:03:18.090 --> 00:03:21.850
the x-coordinate, we are now
sitting in the second quadrant.

00:03:21.850 --> 00:03:23.520
And the x-coordinate
can be negative.

00:03:23.520 --> 00:03:26.667
And that's essentially what
happened in this problem.


Cosine addition identity example

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:04.019
We've got triangle ABC here.

00:00:04.019 --> 00:00:05.320
It looks like a right triangle.

00:00:05.320 --> 00:00:08.320
We can verify that it is because
it satisfies the Pythagorean

00:00:08.320 --> 00:00:08.820
theorem.

00:00:08.820 --> 00:00:12.595
8 squared is 64 plus
15 squared is 225.

00:00:12.595 --> 00:00:16.090
64 plus 225 is 289,
which is 17 squared.

00:00:16.090 --> 00:00:18.000
So it satisfies the
Pythagorean theorem.

00:00:18.000 --> 00:00:22.280
So we know that this right
over here is a right triangle.

00:00:22.280 --> 00:00:24.060
But what they're
asking us is-- what

00:00:24.060 --> 00:00:27.520
is the cosine of
angle ABC, which

00:00:27.520 --> 00:00:31.436
is this angle right over
here plus 60 degrees.

00:00:31.436 --> 00:00:32.810
Now, just like
this, I don't have

00:00:32.810 --> 00:00:34.510
any clear way of evaluating it.

00:00:34.510 --> 00:00:37.600
But we do have some trig
identities in our toolkit

00:00:37.600 --> 00:00:39.660
that we could use to
express this in a way

00:00:39.660 --> 00:00:41.590
that we might be
able to evaluate it.

00:00:41.590 --> 00:00:47.290
In particular, we know
that the cosine of A plus B

00:00:47.290 --> 00:00:51.270
is equal to the cosine of
A times the cosine of B

00:00:51.270 --> 00:00:56.884
minus the sine of A
times the sine of B.

00:00:56.884 --> 00:00:58.550
So we could do the
exact same thing here

00:00:58.550 --> 00:01:10.990
with the cosine of angle
ABC plus 60 degrees.

00:01:10.990 --> 00:01:13.810
This is going to be equal
to-- and let me just write out

00:01:13.810 --> 00:01:16.010
the whole thing--
the cosine of angle

00:01:16.010 --> 00:01:21.830
ABC times the cosine of 60
degrees minus the sine of angle

00:01:21.830 --> 00:01:25.810
ABC times the sine
of 60 degrees.

00:01:25.810 --> 00:01:30.490
So this right over
here is angle ABC.

00:01:30.490 --> 00:01:33.680
This is angle ABC.

00:01:33.680 --> 00:01:39.250
And this is 60 degrees,
and this is 60 degrees.

00:01:39.250 --> 00:01:42.170
So let's evaluate
each of these things.

00:01:42.170 --> 00:01:46.660
What is the cosine of angle
ABC going to be equal to?

00:01:46.660 --> 00:01:48.000
I'll do that over here.

00:01:48.000 --> 00:01:50.960
The cosine of angle ABC.

00:01:50.960 --> 00:01:53.440
Well, we could go
back to Sohcahtoa.

00:01:53.440 --> 00:01:54.490
Let me write it down.

00:01:57.970 --> 00:02:05.720
Cosine is defined as the
adjacent-- for this angle--

00:02:05.720 --> 00:02:07.950
the adjacent over
the hypotenuse.

00:02:07.950 --> 00:02:13.100
So cosine of angle ABC
is equal to 15 over 17.

00:02:13.100 --> 00:02:15.480
So this right over
here is 15 over 17.

00:02:15.480 --> 00:02:18.940
What is the cosine
of 60 degrees?

00:02:18.940 --> 00:02:22.840
Well, for there, we have to turn
back to our knowledge of 30,

00:02:22.840 --> 00:02:24.570
60, 90 triangles.

00:02:24.570 --> 00:02:27.540
So if I have a triangle
like this-- so let

00:02:27.540 --> 00:02:30.230
me do my best
here-- to construct

00:02:30.230 --> 00:02:32.540
a 30, 60, 90 right triangle.

00:02:32.540 --> 00:02:34.430
So that's a 60-degree side.

00:02:34.430 --> 00:02:36.254
This is a 30-degree side.

00:02:36.254 --> 00:02:38.170
We know-- and we've seen
this multiple times--

00:02:38.170 --> 00:02:40.610
if the hypotenuse has
length 1, the side

00:02:40.610 --> 00:02:43.014
opposite the 30-degree
side is one half.

00:02:43.014 --> 00:02:44.930
And then, the side
opposite the 60-degree side

00:02:44.930 --> 00:02:46.840
is square root of 3 times this.

00:02:46.840 --> 00:02:49.920
So it's square root of 3 over 2.

00:02:49.920 --> 00:02:51.770
So the cosine of 60
degrees-- if you're

00:02:51.770 --> 00:02:53.690
looking at this side
right over here.

00:02:53.690 --> 00:02:57.110
And let me write it in a
color I haven't used yet.

00:02:57.110 --> 00:03:00.210
So I care about the
cosine of 60 degrees.

00:03:00.210 --> 00:03:03.590
The cosine of 60
degrees is going

00:03:03.590 --> 00:03:06.640
to be equal to, once again,
adjacent over hypotenuse.

00:03:06.640 --> 00:03:08.740
One half over 1.

00:03:08.740 --> 00:03:10.860
it's going to be equal
to one half over 1, which

00:03:10.860 --> 00:03:13.240
is equal to one half.

00:03:13.240 --> 00:03:16.260
Now, let's think about the sine.

00:03:16.260 --> 00:03:22.240
Now, let's think about
the sine of angle ABC.

00:03:22.240 --> 00:03:25.270
So that's this one
right over here.

00:03:25.270 --> 00:03:27.250
So we have our
triangle right here.

00:03:27.250 --> 00:03:29.390
Sine is opposite
over hypotenuse.

00:03:29.390 --> 00:03:31.690
So opposite has length 8.

00:03:31.690 --> 00:03:33.580
Over hypotenuse, is 17.

00:03:33.580 --> 00:03:36.330
So it's equal to 8 over 17.

00:03:36.330 --> 00:03:40.860
And then, finally,
we need to figure out

00:03:40.860 --> 00:03:45.870
what the sign of 60 degrees is.

00:03:45.870 --> 00:03:48.540
Well, for the 60-degree
side of this right triangle,

00:03:48.540 --> 00:03:50.840
opposite over hypotenuse.

00:03:50.840 --> 00:03:52.440
So square root of
3 over 2 over 1,

00:03:52.440 --> 00:03:54.946
which is just the
square root of 3 over 2.

00:03:54.946 --> 00:03:56.320
So we have all of
the information

00:03:56.320 --> 00:03:58.030
we need to evaluate it.

00:03:58.030 --> 00:04:00.290
So this was the
sine of 60 degrees.

00:04:00.290 --> 00:04:01.960
This whole thing is
going to evaluate

00:04:01.960 --> 00:04:08.840
to cosine of angle ABC is
15 over 17 times cosine

00:04:08.840 --> 00:04:11.210
of 60 degrees is one half.

00:04:11.210 --> 00:04:14.230
So times one half.

00:04:14.230 --> 00:04:15.650
And then, we're
going to subtract

00:04:15.650 --> 00:04:19.940
sine of ABC, which is 8 over 17.

00:04:19.940 --> 00:04:21.680
And then, times
sine of 60, which

00:04:21.680 --> 00:04:23.590
is square root of 3 over 2.

00:04:23.590 --> 00:04:26.640
So times the square
root of 3 over 2.

00:04:26.640 --> 00:04:28.852
And now, we just have to
simplify it a little bit.

00:04:28.852 --> 00:04:30.810
So this is going to be
equal to-- if I multiply

00:04:30.810 --> 00:04:32.820
one half times this, let's see.

00:04:32.820 --> 00:04:39.180
It's going to be 15 over
34 minus-- and let's see.

00:04:39.180 --> 00:04:40.300
We're dividing by 2.

00:04:40.300 --> 00:04:42.600
So it's 4/17ths.

00:04:42.600 --> 00:04:49.337
I'll write this 4 square
roots of 3 over 17.

00:04:49.337 --> 00:04:51.170
And that's about as
simple as I could do it.

00:04:51.170 --> 00:04:55.340
If I wanted, I could have a
common denominator here of 34.

00:04:55.340 --> 00:04:57.590
And then, I could add the
2, so it could be 8 squared.

00:04:57.590 --> 00:05:02.130
So 3 over 34, but that still
won't simplify it that much.

00:05:02.130 --> 00:05:05.830
So this is a
reasonable good answer

00:05:05.830 --> 00:05:07.100
for what they are asking for.

00:05:07.100 --> 00:05:12.991
15 over 34 minus 4 square
roots of 3 over 17.


Defined and undefined matrix operations

WEBVTT
Kind: captions
Language: en

00:00:01.030 --> 00:00:05.340
So we have matrix D and matrix
B and they ask us is DB defined?

00:00:05.340 --> 00:00:08.150
Is the product D
times B defined?

00:00:08.150 --> 00:00:10.740
So D times B is
going to be defined

00:00:10.740 --> 00:00:13.317
is if-- let me make
this very clear.

00:00:13.317 --> 00:00:14.525
This is how I think about it.

00:00:14.525 --> 00:00:15.941
So let me copy and
paste this so I

00:00:15.941 --> 00:00:18.840
can do this on my scratch pad.

00:00:18.840 --> 00:00:20.700
So to answer that
question, get out

00:00:20.700 --> 00:00:23.230
the scratch pad right over here.

00:00:23.230 --> 00:00:26.800
Let me paste the
question right here.

00:00:26.800 --> 00:00:30.580
So let's think about
these two matrices.

00:00:30.580 --> 00:00:36.640
You first have matrix D. I'll
do this a nice bold D here.

00:00:36.640 --> 00:00:38.700
And it has three rows
and three columns.

00:00:38.700 --> 00:00:41.330
So it is a 3 by 3 matrix.

00:00:41.330 --> 00:00:46.250
And then you'll want to multiply
that times matrix B. Matrix

00:00:46.250 --> 00:00:52.200
B is a 2 by 2 matrix.

00:00:52.200 --> 00:00:55.500
The only way that we know to
define matrix multiplication

00:00:55.500 --> 00:00:58.330
is if these middle two
numbers are the same.

00:00:58.330 --> 00:01:03.090
If the number of columns D has
is equal to the number of rows

00:01:03.090 --> 00:01:03.860
B has.

00:01:03.860 --> 00:01:06.480
Now in this case, they clearly
do not equal each other,

00:01:06.480 --> 00:01:10.250
so matrix multiplication
is not defined here.

00:01:10.250 --> 00:01:13.322
So let's go back
there and say no.

00:01:13.322 --> 00:01:15.580
No, DB is not defined.

00:01:15.580 --> 00:01:19.200
Let's do a few more
of these examples.

00:01:19.200 --> 00:01:22.440
So then we have a 2 by 1,
you could view this is as a 2

00:01:22.440 --> 00:01:25.210
by 1 matrix or you could
view this as a column vector.

00:01:25.210 --> 00:01:27.370
This is another 2 by 1
matrix, or a column vector.

00:01:27.370 --> 00:01:29.180
Is C plus B defined.

00:01:29.180 --> 00:01:32.500
Well, matrix addition is
defined if both matrices have

00:01:32.500 --> 00:01:35.300
the exact same dimensions,
and these two matrices

00:01:35.300 --> 00:01:36.780
do have the exact
same dimensions.

00:01:36.780 --> 00:01:38.905
And the reason why is
because with matrix addition,

00:01:38.905 --> 00:01:40.590
you just add every
corresponding term.

00:01:40.590 --> 00:01:42.540
So in the sum the
top, it'll actually

00:01:42.540 --> 00:01:45.680
be 4 plus 0 over negative 2 plus
0, which is still just going

00:01:45.680 --> 00:01:48.124
to be the same thing
as this matrix up here.

00:01:48.124 --> 00:01:49.790
But what they're
asking is this defined?

00:01:49.790 --> 00:01:54.080
Absolutely, these both
are 2 by 1 matrices,

00:01:54.080 --> 00:01:56.260
so yes, it is defined.

00:01:56.260 --> 00:01:58.342
Let's do one more.

00:01:58.342 --> 00:01:59.800
So once again,
they're asking us is

00:01:59.800 --> 00:02:02.470
the product A times E defined?

00:02:02.470 --> 00:02:06.360
So here you have
a 2 by 2 matrix.

00:02:06.360 --> 00:02:09.110
Let me copy and paste this
just so we can make sure

00:02:09.110 --> 00:02:12.086
that we know what
we're talking about.

00:02:12.086 --> 00:02:14.040
So get my scratch pad out.

00:02:16.930 --> 00:02:24.100
So this top matrix right
over here, so matrix A is a 2

00:02:24.100 --> 00:02:25.740
by 2 matrix.

00:02:25.740 --> 00:02:29.160
And matrix E, so we're
going to multiply it

00:02:29.160 --> 00:02:34.910
times matrix E, which has
one row and two columns.

00:02:34.910 --> 00:02:36.900
So in this scenario
once again, the number

00:02:36.900 --> 00:02:41.470
of rows-- sorry-- the number
of columns matrix A has is two

00:02:41.470 --> 00:02:44.620
and the number of rows
matrix E has is one,

00:02:44.620 --> 00:02:46.920
so this will not be defined.

00:02:46.920 --> 00:02:49.617
These two things have to be the
same for them to be defined.

00:02:49.617 --> 00:02:51.950
Now, what is interesting is,
if you did it the other way

00:02:51.950 --> 00:02:58.770
around, if you took E
times A, let's check

00:02:58.770 --> 00:03:00.860
if this would have been defined.

00:03:00.860 --> 00:03:05.250
Matrix E is 1 by 2, one
row times two columns.

00:03:05.250 --> 00:03:11.700
Matrix A is a 2 by 2,
two rows and two columns,

00:03:11.700 --> 00:03:13.620
and so this would
have been defined.

00:03:13.620 --> 00:03:16.890
Matrix E has two
columns, which is exactly

00:03:16.890 --> 00:03:19.342
the same number of
rows that matrix A has.

00:03:19.342 --> 00:03:20.800
And this really
hits the point home

00:03:20.800 --> 00:03:23.320
that the order matters
when you multiply matrices.

00:03:23.320 --> 00:03:26.070
But for the sake of this
question, is AE defined?

00:03:26.070 --> 00:03:27.670
No, it isn't.

00:03:27.670 --> 00:03:30.800
And so we can check our
answer, no, it isn't.


Multiplying a matrix by a column vector

WEBVTT
Kind: captions
Language: en

00:00:00.930 --> 00:00:04.540
So we're multiplying
matrix A here by vector w.

00:00:04.540 --> 00:00:07.630
And they ask us
what is A times w?

00:00:07.630 --> 00:00:09.720
So let me get my
scratch pad out.

00:00:09.720 --> 00:00:11.620
And let's think about this.

00:00:11.620 --> 00:00:14.640
So we have matrix
A times vector w.

00:00:14.640 --> 00:00:18.290
So just as a reminder,
Aw-- and I'll

00:00:18.290 --> 00:00:20.313
write it in bold
right over here-- Aw.

00:00:24.227 --> 00:00:26.310
And I could write it either
as a vector like that,

00:00:26.310 --> 00:00:28.700
or I could bold that as well.

00:00:28.700 --> 00:00:29.350
Aw.

00:00:29.350 --> 00:00:32.369
It's lowercase for a vector,
uppercase for a matrix.

00:00:32.369 --> 00:00:39.610
This is the same thing
as matrix A times vector

00:00:39.610 --> 00:00:46.590
w-- let me do that same color--
which is the same thing.

00:00:46.590 --> 00:00:55.760
This is matrix A. So it's,
A is 0, 3, 5, 5, 5, 2 times

00:00:55.760 --> 00:00:56.555
vector w.

00:00:56.555 --> 00:01:03.330
W is the vector 3, 4, 3.

00:01:03.330 --> 00:01:08.700
So we have a matrix with
two rows and three columns,

00:01:08.700 --> 00:01:13.570
so it is a 2 by 3
matrix, times a vector.

00:01:13.570 --> 00:01:15.210
And this is a column vector.

00:01:15.210 --> 00:01:17.240
It is three rows in one column.

00:01:17.240 --> 00:01:20.707
So you could view this
as a 3 by 1 matrix.

00:01:20.707 --> 00:01:23.040
And so, deciding whether this
is even a valid operation,

00:01:23.040 --> 00:01:24.706
it's just like
multiplying two matrices.

00:01:24.706 --> 00:01:28.290
You can view this column vector
is really just a 3 by 1 matrix.

00:01:28.290 --> 00:01:31.510
And the only way that matrix
multiplication is defined

00:01:31.510 --> 00:01:33.850
is if the columns
in this matrix is

00:01:33.850 --> 00:01:37.130
equal to the rows
in this matrix.

00:01:37.130 --> 00:01:40.580
And we see that is
the case, this vector.

00:01:40.580 --> 00:01:45.540
This matrix, matrix A, has
three columns: 1, 2, 3.

00:01:45.540 --> 00:01:50.140
And this and vector w
has exactly three rows.

00:01:50.140 --> 00:01:53.020
So matrix multiplication,
or matrix times

00:01:53.020 --> 00:01:55.817
vector multiplication,
is defined here.

00:01:55.817 --> 00:01:57.400
And the way that
we're going to do it,

00:01:57.400 --> 00:02:02.760
we're going to end
up with a 2 by 1.

00:02:02.760 --> 00:02:05.210
You could call it
a 2 by 1 matrix.

00:02:05.210 --> 00:02:07.600
It's going to have two
rows in one column.

00:02:07.600 --> 00:02:10.955
Or you could view it as
a column vector, when

00:02:10.955 --> 00:02:13.561
you have two rows
and one column.

00:02:13.561 --> 00:02:15.060
But either way,
let's actually think

00:02:15.060 --> 00:02:16.680
about how we would compute it.

00:02:16.680 --> 00:02:19.314
So it's going to have
two rows and one column.

00:02:19.314 --> 00:02:20.730
Then I'm going
give a lot of space

00:02:20.730 --> 00:02:23.630
so that we can actually
do the calculation.

00:02:23.630 --> 00:02:26.380
So the top entry
right over here,

00:02:26.380 --> 00:02:29.930
we're going to get
the row information

00:02:29.930 --> 00:02:34.302
from our first matrix, and
then the column information.

00:02:34.302 --> 00:02:35.510
There's only one column here.

00:02:35.510 --> 00:02:38.010
But we're going to get that
from this matrix or this vector,

00:02:38.010 --> 00:02:39.580
whatever you want to call it.

00:02:39.580 --> 00:02:48.190
So we're going to have 0 times
3, plus 3 times 4, plus 5 times

00:02:48.190 --> 00:02:50.880
3.

00:02:50.880 --> 00:02:52.940
Now the second entry
right over here

00:02:52.940 --> 00:02:57.330
is going to be the
second row here.

00:02:57.330 --> 00:03:00.000
Essentially, the dot product
of that with this vector.

00:03:00.000 --> 00:03:01.220
And if you don't know
what the dot product is,

00:03:01.220 --> 00:03:02.595
I'm essentially
about to do that.

00:03:02.595 --> 00:03:05.240
It's going to be-- you take
each corresponding term,

00:03:05.240 --> 00:03:07.800
take their product, and then
add up everything together.

00:03:07.800 --> 00:03:17.420
So you have 5 times 3, plus
5 times 4, plus 2 times 3.

00:03:17.420 --> 00:03:24.050
And this is going to
simplify to-- This top term,

00:03:24.050 --> 00:03:25.200
this is a 0.

00:03:25.200 --> 00:03:31.080
This is 12 plus 15, which is 27.

00:03:31.080 --> 00:03:37.310
And then the second term,
this is 15 plus 20 plus 6.

00:03:37.310 --> 00:03:40.260
So this is 35 plus 6 is 41.

00:03:40.260 --> 00:03:44.500
So it's a column vector,
two rows, 27 and 41.

00:03:44.500 --> 00:03:48.220
Now let's input that, 27 and 41.

00:03:48.220 --> 00:03:51.020
So we get 27.

00:03:51.020 --> 00:03:54.250
And here we can put 41,
and check our answer,

00:03:54.250 --> 00:03:56.220
and we got it right.


Multiplying a matrix by a matrix

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.210
We're given two matrices over
here, matrix E and matrix D.

00:00:04.210 --> 00:00:05.960
And they ask us,
what is ED, which

00:00:05.960 --> 00:00:08.310
is another way of saying
what is the product of matrix

00:00:08.310 --> 00:00:10.050
E and matrix D?

00:00:10.050 --> 00:00:11.590
Just so I remember
what I'm doing,

00:00:11.590 --> 00:00:12.970
let me copy and paste this.

00:00:12.970 --> 00:00:18.620
And then I'm going to get
out my little scratch pad.

00:00:18.620 --> 00:00:21.070
So let me paste that over here.

00:00:21.070 --> 00:00:23.540
So we have all the
information we needed.

00:00:23.540 --> 00:00:25.180
And so let's try
to work this out.

00:00:25.180 --> 00:00:33.840
So matrix E times matrix D,
which is equal to-- matrix E

00:00:33.840 --> 00:00:35.510
is all of this business.

00:00:35.510 --> 00:00:44.970
So it is 0, 3, 5, 5, 5, 2 times
matrix D, which is all of this.

00:00:44.970 --> 00:00:51.250
So we're going to multiply
it times 3, 3, 4, 4,

00:00:51.250 --> 00:00:54.062
negative 2, negative 2.

00:00:54.062 --> 00:00:55.770
Now the first thing
that we have to check

00:00:55.770 --> 00:00:58.270
is whether this is
even a valid operation.

00:00:58.270 --> 00:01:01.740
Now the matrix multiplication
is a human-defined operation

00:01:01.740 --> 00:01:03.870
that just happens-- in
fact all operations are--

00:01:03.870 --> 00:01:05.950
that happen to have
neat properties.

00:01:05.950 --> 00:01:09.400
Now the way that us humans have
defined matrix multiplication,

00:01:09.400 --> 00:01:13.330
it only works when we're
multiplying our two matrices.

00:01:13.330 --> 00:01:16.000
So this right over here has
two rows and three columns.

00:01:16.000 --> 00:01:18.020
So it's a 2 by 3 matrix.

00:01:18.020 --> 00:01:22.224
And this has three rows and
two columns, it's 3 by 2.

00:01:22.224 --> 00:01:23.890
This only works-- we
could only multiply

00:01:23.890 --> 00:01:27.730
this matrix times this matrix,
if the number of columns

00:01:27.730 --> 00:01:32.600
on this matrix is equal to the
number of rows on this matrix.

00:01:32.600 --> 00:01:36.250
And in this situation it is, so
I can actually multiply them.

00:01:36.250 --> 00:01:39.240
If these two numbers weren't
equal, if the number of columns

00:01:39.240 --> 00:01:41.950
here were not equal to the
number of rows here, then

00:01:41.950 --> 00:01:45.030
this would not be a valid
operation, at least the way

00:01:45.030 --> 00:01:48.029
that we have defined
matrix multiplication.

00:01:48.029 --> 00:01:49.820
The other thing you
always have to remember

00:01:49.820 --> 00:01:53.750
is that E times D is not always
the same thing as D times E.

00:01:53.750 --> 00:01:56.220
Order matters when you're
multiplying matrices.

00:01:56.220 --> 00:01:58.620
It doesn't matter if you're
multiplying regular numbers,

00:01:58.620 --> 00:02:00.470
but it matters for matrices.

00:02:00.470 --> 00:02:02.340
But let's actually
work this out.

00:02:02.340 --> 00:02:06.740
So what we're going to get
is actually going to be a 2

00:02:06.740 --> 00:02:07.505
by 2 matrix.

00:02:07.505 --> 00:02:09.130
But I'm going to
create some space here

00:02:09.130 --> 00:02:11.255
because we're going to have
to do some computation.

00:02:11.255 --> 00:02:15.160
So this is going to
be equal to-- I'm

00:02:15.160 --> 00:02:18.300
going to make a huge
2 by 2 matrix here.

00:02:18.300 --> 00:02:20.590
So the way we get
the top left entry,

00:02:20.590 --> 00:02:21.990
the top left entry
is essentially

00:02:21.990 --> 00:02:29.770
going to be this row
times this product.

00:02:29.770 --> 00:02:31.270
If you view them
each as vectors,

00:02:31.270 --> 00:02:32.920
and you have some familiarity
with the dot product,

00:02:32.920 --> 00:02:33.590
we're essentially
going to take the dot

00:02:33.590 --> 00:02:34.800
product of that and that.

00:02:34.800 --> 00:02:37.660
And if you have no idea what
that is, I'm about to show you.

00:02:37.660 --> 00:02:43.210
This entry is going to be
0 times 3, plus 3 times

00:02:43.210 --> 00:02:47.500
3, plus 5 times 4.

00:02:50.310 --> 00:02:52.260
So that is the top left entry.

00:02:52.260 --> 00:02:54.801
And I already see that I'm going
to run out of space here, so

00:02:54.801 --> 00:02:58.160
let me move this
over to the right

00:02:58.160 --> 00:03:01.250
some space so I have
some breathing room.

00:03:01.250 --> 00:03:04.490
Now we can do the
top right entry.

00:03:04.490 --> 00:03:07.840
This was the top left, now
we're going to do the top right.

00:03:07.840 --> 00:03:12.100
So the top right
entry is going to be

00:03:12.100 --> 00:03:17.970
this row times this column.

00:03:17.970 --> 00:03:20.170
Notice the entry
is getting the row

00:03:20.170 --> 00:03:23.270
from the first matrix and the
column from the second one.

00:03:23.270 --> 00:03:25.310
That's kind of
determining its position.

00:03:25.310 --> 00:03:29.430
So, once again, is
going to be 0 times 4,

00:03:29.430 --> 00:03:37.880
plus 3 times negative 2,
plus 5 times negative 2.

00:03:37.880 --> 00:03:39.620
And we keep going.

00:03:39.620 --> 00:03:43.780
The bottom left entry is going
to be this row, the second row

00:03:43.780 --> 00:03:47.400
here times the
first column here.

00:03:47.400 --> 00:03:58.800
So it's 5 times 3, plus 5
times 3, plus 2 times 4.

00:03:58.800 --> 00:04:00.210
And we're almost done.

00:04:00.210 --> 00:04:03.710
We just need to multiply or
take the dot product of this row

00:04:03.710 --> 00:04:06.770
with this column
right over here.

00:04:06.770 --> 00:04:14.070
So it's going to be 5 times
4, plus 5 times negative 2,

00:04:14.070 --> 00:04:18.279
plus 2 times negative 2.

00:04:18.279 --> 00:04:19.949
And so this is going
to be equal to,

00:04:19.949 --> 00:04:22.550
and we could just
evaluate this now.

00:04:22.550 --> 00:04:25.370
Let's see, 0 times 3 is 0.

00:04:25.370 --> 00:04:28.370
This is 9 plus 20.

00:04:28.370 --> 00:04:30.270
This is 29.

00:04:30.270 --> 00:04:33.420
This all simplified to 29.

00:04:33.420 --> 00:04:34.940
All of this is 0.

00:04:34.940 --> 00:04:36.670
This is negative 6.

00:04:36.670 --> 00:04:38.070
And then this is minus 10.

00:04:38.070 --> 00:04:39.940
So this all simplifies
to negative 16.

00:04:43.140 --> 00:04:48.110
This right over here, it's 15
plus 15, which is 30 plus 8.

00:04:48.110 --> 00:04:51.470
So this is 38.

00:04:51.470 --> 00:04:59.180
And then finally, this
is 20 minus 10 minus 4.

00:04:59.180 --> 00:05:02.440
So this is going to be 6.

00:05:02.440 --> 00:05:05.170
So this is all going
to simplify to 6.

00:05:05.170 --> 00:05:09.178
So this all became 29,
negative 16, 38, and 6.

00:05:25.130 --> 00:05:26.800
Let's check our answer.

00:05:26.800 --> 00:05:28.457
And we got it right.