Optimization with Calculus 3

WEBVTT
Kind: captions
Language: en

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This next optimization problem
that I've been sent seems fun.

00:00:03.780 --> 00:00:07.350
I've actually never seen this
problem before, and so I'm

00:00:07.350 --> 00:00:09.170
even more excited to do it.

00:00:09.170 --> 00:00:12.620
So it says, a wire of
length 100 centimeters

00:00:12.620 --> 00:00:14.470
is cut into two pieces.

00:00:14.470 --> 00:00:16.115
Let me draw this wire.

00:00:19.000 --> 00:00:23.450
So that's our wire, and it's
going to be cut into 2 pieces.

00:00:23.450 --> 00:00:25.200
Let me read the whole thing
before I draw anything.

00:00:25.200 --> 00:00:28.760
One is bent to form a square,
and the other is bent to form

00:00:28.760 --> 00:00:30.640
an equilateral triangle.

00:00:30.640 --> 00:00:34.400
Where should the cut be made if
A, the sum of the two areas is

00:00:34.400 --> 00:00:36.530
to be a minimum or a maximum?

00:00:36.530 --> 00:00:39.610
And it says, allow the
possibility of no cut.

00:00:39.610 --> 00:00:40.910
Fascinating.

00:00:40.910 --> 00:00:43.870
So this is a 100
centimeters long.

00:00:43.870 --> 00:00:45.440
Doesn't matter if it's
centimeters or meters

00:00:45.440 --> 00:00:46.730
or whatever, right?

00:00:46.730 --> 00:00:49.230
So it's 100 long.

00:00:49.230 --> 00:00:52.540
Could be 100 miles long.

00:00:52.540 --> 00:00:54.630
And we're going to cut
it at some point.

00:00:54.630 --> 00:00:56.130
We'll cut it there.

00:00:59.080 --> 00:01:00.510
Well, I shouldn't have
done that, because I was

00:01:00.510 --> 00:01:02.205
going to use x and y.

00:01:02.205 --> 00:01:04.330
But actually, let's use a
and b, since I made the

00:01:04.330 --> 00:01:05.500
cut look like an x.

00:01:05.500 --> 00:01:08.575
So let's say that that is a
centimeters from the left

00:01:08.575 --> 00:01:10.630
hand side of the wire.

00:01:10.630 --> 00:01:12.990
And then what is the distance
from the right hand

00:01:12.990 --> 00:01:13.530
side of the wire?

00:01:13.530 --> 00:01:16.420
I could call it b, but even
easier would be to say, if

00:01:16.420 --> 00:01:18.760
this is a, then this is
going to be 100 minus a.

00:01:24.200 --> 00:01:26.620
And they say, I'm going to take
one of the wires, let me just

00:01:26.620 --> 00:01:30.150
say the left hand wire, and I'm
going to take it into, and

00:01:30.150 --> 00:01:31.550
I'm going to form a square.

00:01:31.550 --> 00:01:32.870
Which is not a rectangle.

00:01:32.870 --> 00:01:33.610
A square.

00:01:33.610 --> 00:01:36.900
And then the other wire
is going to form an

00:01:36.900 --> 00:01:40.150
equilateral triangle.

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OK.

00:01:40.610 --> 00:01:41.090
So let's do it.

00:01:41.090 --> 00:01:43.620
And actually, I can already
sense that it'll be easier

00:01:43.620 --> 00:01:46.240
to manipulate this one.

00:01:46.240 --> 00:01:47.850
So I'm going to make the
equilateral triangle out of

00:01:47.850 --> 00:01:50.290
this side, and make the square
out of the side, just because I

00:01:50.290 --> 00:01:52.800
have a sense it will make
the math a little easier.

00:01:52.800 --> 00:01:55.690
So I'm going to take this wire
of length a and form an

00:01:55.690 --> 00:01:57.800
equilateral triangle
out of it, right?

00:01:57.800 --> 00:01:59.790
What is the triangle
going to look like?

00:01:59.790 --> 00:02:01.780
It's going to look like, and
I'm going to arbitrarily

00:02:01.780 --> 00:02:04.090
switch colors now.

00:02:04.090 --> 00:02:07.720
So this is my
equilateral triangle.

00:02:07.720 --> 00:02:09.500
All the sides are equal.

00:02:09.500 --> 00:02:12.840
And how long are each of
the sides going to be?

00:02:12.840 --> 00:02:15.810
Well, the whole perimeter
is a, so each of

00:02:15.810 --> 00:02:16.690
these have to be a/3.

00:02:26.840 --> 00:02:30.175
And then I also have a
square with the leftovers.

00:02:33.500 --> 00:02:37.250
I'm not using the the square
tool-- so I have it,

00:02:37.250 --> 00:02:39.780
I should use it.

00:02:39.780 --> 00:02:43.020
Oh, I don't want to
do it like that.

00:02:43.020 --> 00:02:45.670
OK, so there's my square.

00:02:45.670 --> 00:02:49.320
And if the whole perimeter
is 100 minus a, what is

00:02:49.320 --> 00:02:50.380
the length of each side?

00:02:50.380 --> 00:02:53.780
It's going to be this
divided by 4, right?

00:02:53.780 --> 00:02:56.630
Because all the
sides are equal.

00:02:56.630 --> 00:02:59.590
So what's this length
divided by 4?

00:02:59.590 --> 00:03:05.190
100 minus a divided by 4
is 25 minus a/4, right?

00:03:05.190 --> 00:03:13.430
So this side is 25 minus a/4,
this side is 25 minus a/4, and

00:03:13.430 --> 00:03:17.340
this is the same thing, 25
minus a/4, and this

00:03:17.340 --> 00:03:20.180
is 25 minus a/4.

00:03:20.180 --> 00:03:24.990
And they want us to find what
value of a, they say, the

00:03:24.990 --> 00:03:26.550
sum of the 2 areas, right?

00:03:26.550 --> 00:03:34.870
They want to know what value of
a minimizes the sum of the 2

00:03:34.870 --> 00:03:37.785
areas and maximize the
sum of the 2 areas.

00:03:37.785 --> 00:03:39.330
So what are the sum
of the 2 areas?

00:03:39.330 --> 00:03:40.380
What's the are of
this triangle?

00:03:40.380 --> 00:03:43.440
I always forget the formula for
the area of an equilateral

00:03:43.440 --> 00:03:46.460
triangle, so we'll have to
kind of derive it right now.

00:03:46.460 --> 00:03:50.300
So what I do, is I always draw
something like that, and then

00:03:50.300 --> 00:03:52.110
we know that this length
right here is a/6.

00:03:56.170 --> 00:03:57.700
I'm just doing the
halfway point.

00:03:57.700 --> 00:03:59.230
The whole thing is a/3.

00:03:59.230 --> 00:04:01.550
And this is a 30-60-90
triangle, right?

00:04:01.550 --> 00:04:04.520
That's 30, this is 60, so
this is going to be square

00:04:04.520 --> 00:04:05.700
root of 3 times this.

00:04:05.700 --> 00:04:08.120
So the height, let me do
that here, just so you

00:04:08.120 --> 00:04:08.700
know what I'm doing.

00:04:08.700 --> 00:04:12.750
This is a some geometry I'm
drawing half the triangle.

00:04:12.750 --> 00:04:14.990
This would be a/6.

00:04:14.990 --> 00:04:17.960
That's a/3.

00:04:17.960 --> 00:04:27.180
And so this would be square
root of 3 times this.

00:04:27.180 --> 00:04:34.890
So square root of
3 over 6 times a.

00:04:34.890 --> 00:04:36.370
So what's the area?

00:04:36.370 --> 00:04:41.540
Well, the area of this half
triangle, is base times

00:04:41.540 --> 00:04:43.730
height times 1/2.

00:04:43.730 --> 00:04:45.370
So the area of the whole
triangle is just going to be

00:04:45.370 --> 00:04:48.150
this base times this height.

00:04:48.150 --> 00:04:55.530
So the area of the triangle
is equal to this base

00:04:55.530 --> 00:04:57.050
times that height.

00:04:57.050 --> 00:05:03.200
So its area over 6, that's
this, times the height, times

00:05:03.200 --> 00:05:07.370
square root of 3 over 6, area.

00:05:07.370 --> 00:05:09.690
And just so you remember,
area of a triangle is

00:05:09.690 --> 00:05:11.390
1/2 base times height.

00:05:11.390 --> 00:05:14.220
So if we did 1/2 base times
height, we would get the area

00:05:14.220 --> 00:05:15.740
of this triangle, right?

00:05:15.740 --> 00:05:17.590
But the area of the
entire triangle is 2

00:05:17.590 --> 00:05:18.770
of these triangles.

00:05:18.770 --> 00:05:20.890
And so that's why I
got rid of the 1/2.

00:05:20.890 --> 00:05:24.450
Because it's 2 times
1/2 base times height.

00:05:24.450 --> 00:05:25.660
Hopefully that
doesn't confuse you.

00:05:25.660 --> 00:05:27.520
It's just a little
bit of geometry.

00:05:27.520 --> 00:05:33.920
Anyway, the area of the
triangle is equal to the square

00:05:33.920 --> 00:05:38.270
root of 3 over 36 a squared.

00:05:38.270 --> 00:05:40.190
And what's the area
of the square?

00:05:40.190 --> 00:05:41.380
That's easier.

00:05:41.380 --> 00:05:43.250
That's just base times height.

00:05:43.250 --> 00:05:49.510
So area of the square is
equal to 25 minus a over

00:05:49.510 --> 00:05:54.740
4, that's just base
times height, squared.

00:05:54.740 --> 00:06:03.370
And that is equal to 625
minus 2 times-- so it's

00:06:03.370 --> 00:06:09.020
20, let's see, 25 over
4 minus 50 over 4 a?

00:06:09.020 --> 00:06:09.360
Right.

00:06:09.360 --> 00:06:20.240
Minus 50 over 4a, plus
a squared over 16.

00:06:20.240 --> 00:06:22.070
So that's the area
of the square.

00:06:22.070 --> 00:06:27.080
But we want to know the
combined area of both of them.

00:06:27.080 --> 00:06:29.470
So what's the function
for the combined area?

00:06:29.470 --> 00:06:34.150
So the combined area, and I'll
do it in a bold color, area of

00:06:34.150 --> 00:06:36.300
both of them, I'll say area
combined, that's equal to

00:06:36.300 --> 00:06:38.750
the area of the square plus
the area of the triangle.

00:06:38.750 --> 00:06:45.060
So the area of the
square is, let's see.

00:06:45.060 --> 00:07:01.050
It's a squared over 16 minus 50
over 4 a plus 625, and we want

00:07:01.050 --> 00:07:07.450
to add the area of the
triangle, plus square root

00:07:07.450 --> 00:07:11.570
of 3 over 36 a squared.

00:07:11.570 --> 00:07:13.570
And that's a function of a.

00:07:13.570 --> 00:07:15.260
We could it put it like that.

00:07:15.260 --> 00:07:16.800
So now we have to optimize it.

00:07:16.800 --> 00:07:19.930
And we want to find the
minimum point and the maximum

00:07:19.930 --> 00:07:21.310
point, if they exist.

00:07:21.310 --> 00:07:23.315
Let me clear up some space.

00:07:29.300 --> 00:07:31.970
The area of the square is
already implicitly there.

00:07:34.690 --> 00:07:37.940
I could actually clear
up all of this here.

00:07:37.940 --> 00:07:41.910
And so what is the derivative?

00:07:41.910 --> 00:07:46.670
a prime, the combined--
actually, I hope you

00:07:46.670 --> 00:07:47.960
don't mind, I'm going
to erase that too.

00:07:47.960 --> 00:07:49.630
Because I think I'm going to
run out of space and just get

00:07:49.630 --> 00:07:52.110
all bunched up, otherwise.

00:07:52.110 --> 00:07:54.246
That's not what I wanted to do.

00:07:54.246 --> 00:07:56.520
But you know we're
talking about.

00:07:56.520 --> 00:08:00.930
We already have figured out the
area the combined entities.

00:08:00.930 --> 00:08:02.410
So now it's just pure calculus.

00:08:02.410 --> 00:08:04.740
All the geometry is done.

00:08:04.740 --> 00:08:11.680
So what is the derivative of
the combined areas as a

00:08:11.680 --> 00:08:15.530
function of a is equal to the
derivative of this expression?

00:08:15.530 --> 00:08:22.920
So it's 2 times 16,
2/16, so it's a over 8.

00:08:22.920 --> 00:08:25.560
So let me write this.

00:08:25.560 --> 00:08:29.780
a over 8, let me draw a
line here, I don't want

00:08:29.780 --> 00:08:31.480
things to get confused.

00:08:31.480 --> 00:08:42.240
a over 8 minus 50/4, this
derivative 625 is just zero,

00:08:42.240 --> 00:08:52.450
plus 2 over square root
of 3 over 18 a, right?

00:08:52.450 --> 00:08:55.030
And so we want to find the
minimum, or the maximum,

00:08:55.030 --> 00:08:56.300
values for this.

00:08:56.300 --> 00:09:01.860
So let's set this
equation as equal to 0.

00:09:01.860 --> 00:09:06.600
So we get, let me
see what we can do.

00:09:06.600 --> 00:09:08.170
We could add the
a coefficients.

00:09:08.170 --> 00:09:19.216
So we have 1/8 plus square root
of 3 over 18 a, right, I just

00:09:19.216 --> 00:09:22.890
added that to that, minus 50/4.

00:09:22.890 --> 00:09:25.420
And we want to set this
derivative equal to zero.

00:09:25.420 --> 00:09:27.880
So let me switch
colors arbitrarily.

00:09:27.880 --> 00:09:32.600
And so we get, let me add these
two fractions, so 8 and 18.

00:09:32.600 --> 00:09:35.350
What's their common
denominator?

00:09:35.350 --> 00:09:36.420
8 and 9.

00:09:39.350 --> 00:09:40.520
8 times 9, is it 72?

00:09:45.400 --> 00:09:49.120
And 1/8 is the same
thing as 9/72.

00:09:49.120 --> 00:09:58.810
And so 18 goes into 72 4 times,
so that's plus 4 square roots

00:09:58.810 --> 00:10:07.370
of 3, and all that times
a is equal to 50/4.

00:10:07.370 --> 00:10:09.740
And now we can multiply both
sides times the reciprocal of

00:10:09.740 --> 00:10:15.300
this coefficient, and we get--
maybe I made a careless

00:10:15.300 --> 00:10:17.650
mistake, but the numbers
aren't coming out too clean.

00:10:17.650 --> 00:10:19.210
But maybe they're not meant to.

00:10:19.210 --> 00:10:30.770
a is equal to 50/4
times 72 over 9 plus

00:10:30.770 --> 00:10:33.650
4 square roots of 3.

00:10:33.650 --> 00:10:35.360
Let's see if we can
cancel anything out.

00:10:35.360 --> 00:10:43.380
This 4 becomes, this becomes an
18, and so we get a is equal

00:10:43.380 --> 00:10:54.950
to 50 times 18 divided by 9
plus 4 square roots of 3.

00:10:54.950 --> 00:10:58.870
Now let's see if that
makes any kind of sense.

00:10:58.870 --> 00:10:59.980
And I think I'm going
to have to get the

00:10:59.980 --> 00:11:01.080
calculator for this one.

00:11:07.320 --> 00:11:09.910
So what should I do first?

00:11:09.910 --> 00:11:12.380
Let me figure out what the
denominator's value is.

00:11:12.380 --> 00:11:26.070
so 3 square root times 4 equals
plus 9 is equal to 15.928.

00:11:26.070 --> 00:11:31.030
So that a is equal to,
what's 50 times 18,

00:11:31.030 --> 00:11:34.560
550, 50, 90, right?

00:11:34.560 --> 00:11:36.270
5 times 8 is 40, right?

00:11:36.270 --> 00:11:45.090
So that's 900 over 15.928,
so let me do that, let

00:11:45.090 --> 00:11:48.990
me just invert that.

00:11:48.990 --> 00:11:56.420
Times 900.

00:11:56.420 --> 00:12:00.660
It equals 56.5.

00:12:00.660 --> 00:12:05.690
So a is equal to 56.5.

00:12:05.690 --> 00:12:08.040
So at least we got a
number that makes sense.

00:12:08.040 --> 00:12:12.070
It would have been unfortunate
if the point that we have to

00:12:12.070 --> 00:12:17.440
cut this wire was some number
larger, or God forbid we had

00:12:17.440 --> 00:12:18.480
a negative number there.

00:12:18.480 --> 00:12:20.600
But we actually got a positive
number, and it's a number

00:12:20.600 --> 00:12:22.640
between 0 and 100.

00:12:22.640 --> 00:12:25.060
So we are probably right.

00:12:25.060 --> 00:12:28.460
But the question is, is this
a minimum value, or is

00:12:28.460 --> 00:12:30.060
this a maximum value?

00:12:30.060 --> 00:12:32.550
And to figure that out, well,
let's see if we're concave

00:12:32.550 --> 00:12:35.160
upwards or downwards
at this point.

00:12:35.160 --> 00:12:36.470
So let's think about it.

00:12:36.470 --> 00:12:38.860
Let's take the second
derivative of our

00:12:38.860 --> 00:12:41.230
combined area function.

00:12:41.230 --> 00:12:46.600
So this was our combined
area function.

00:12:46.600 --> 00:12:47.660
We had set it equal to 0.

00:12:47.660 --> 00:12:49.840
Let's take the second
derivative of that.

00:12:49.840 --> 00:12:55.830
Well that just equals, the
derivative of this equals, 1/8

00:12:55.830 --> 00:13:01.840
plus square root of 3 over 18,
well, times a, but that

00:13:01.840 --> 00:13:02.560
just [? switches ?]

00:13:02.560 --> 00:13:03.950
coefficient, and then
that goes to 0.

00:13:03.950 --> 00:13:09.890
So that equals a prime
prime as a function of a.

00:13:09.890 --> 00:13:11.540
The combined areas
are a function of a.

00:13:11.540 --> 00:13:14.620
It's just a constant number,
and it is a positive

00:13:14.620 --> 00:13:15.660
constant number.

00:13:15.660 --> 00:13:18.290
So what does that tell us,
given that that is a positive

00:13:18.290 --> 00:13:22.790
constant Number That tells us
that at any value of a, the

00:13:22.790 --> 00:13:24.870
second derivative is positive.

00:13:24.870 --> 00:13:28.680
The second derivative is
positive tells us that we are

00:13:28.680 --> 00:13:34.400
concave upwards, really over
this whole combined

00:13:34.400 --> 00:13:36.960
area function, we're
concave upwards.

00:13:36.960 --> 00:13:37.610
And how do we know that?

00:13:37.610 --> 00:13:38.170
Well, the [UNINTELLIGIBLE]

00:13:38.170 --> 00:13:41.610
is positive, the slope is
always increasing, right?

00:13:41.610 --> 00:13:45.900
And that tells us that any
value where the slope a 0,

00:13:45.900 --> 00:13:49.230
which was this one, is
actually a minimum point.

00:13:51.860 --> 00:13:53.790
So a is equal to 56.5.

00:13:53.790 --> 00:13:59.930
So if we cut it at 56.5, we are
actually at a minimum point.

00:13:59.930 --> 00:14:03.270
So how would we maximize
the combined value?

00:14:03.270 --> 00:14:05.130
Well, maybe here, we have
to just take one of the

00:14:05.130 --> 00:14:06.440
boundary conditions.

00:14:06.440 --> 00:14:10.380
So maybe we could set a as
equal to 0, or we could set a

00:14:10.380 --> 00:14:13.960
as equal to 100, and figure out
what the values are, and one

00:14:13.960 --> 00:14:16.620
of those have to be
our maximum value.

00:14:16.620 --> 00:14:19.470
So what happens when
a is equal to zero?

00:14:22.420 --> 00:14:25.110
So this was our function
to begin with.

00:14:25.110 --> 00:14:30.630
What is our combined area
function, if a is 0?

00:14:30.630 --> 00:14:39.520
Well, this is 0, this is 0,
that is 0, so it equals 625.

00:14:39.520 --> 00:14:41.280
And what does that
translate into?

00:14:41.280 --> 00:14:44.490
That's the case where the
entire wire is used to make a

00:14:44.490 --> 00:14:47.660
square, and none of the wire
is used to make a triangle.

00:14:47.660 --> 00:14:51.670
In that situation, our combined
area, and all the area coming

00:14:51.670 --> 00:14:54.140
from the square, would be 625.

00:14:54.140 --> 00:15:07.030
Well, what happens when
a is equal to 100?

00:15:07.030 --> 00:15:13.460
Well then, if this is equal
to 100, then the area

00:15:13.460 --> 00:15:14.310
of the square is 0.

00:15:14.310 --> 00:15:16.010
So we can actually even
ignore these terms.

00:15:16.010 --> 00:15:18.800
We actually know that this will
evaluate to 0, because we are

00:15:18.800 --> 00:15:20.760
committing no wire
to the square.

00:15:20.760 --> 00:15:23.440
And so what is the
area of the triangle?

00:15:23.440 --> 00:15:28.010
Well, this term right here was
the area of the triangle, so it

00:15:28.010 --> 00:15:36.620
equaled the square root of 3
over 36 times 100 squared.

00:15:36.620 --> 00:15:38.550
So what is that?

00:15:38.550 --> 00:15:40.610
Let's see if we can do that.

00:15:40.610 --> 00:15:48.440
So 3 square root divided by
36 is equal to, and then 100

00:15:48.440 --> 00:15:49.830
squared is 10,000, right?

00:15:49.830 --> 00:15:57.170
So times 110, 1, 2, 3.

00:15:57.170 --> 00:16:00.890
481.

00:16:00.890 --> 00:16:02.700
It equals 481.

00:16:02.700 --> 00:16:04.640
So what do we know?

00:16:04.640 --> 00:16:07.690
We know that if we cut it
exactly at 56.5, and actually

00:16:07.690 --> 00:16:10.830
there's a little bit more
precision that I didn't pay

00:16:10.830 --> 00:16:14.470
attention to, if we cut it at
roughly 56.5, the combined

00:16:14.470 --> 00:16:17.000
area, actually we haven't even
figured out what that combined

00:16:17.000 --> 00:16:19.110
area is, but we know it's a
minimum point, because we're

00:16:19.110 --> 00:16:20.770
concave upward at that point.

00:16:20.770 --> 00:16:23.510
And as a fun exercise for you,
you might want to plug it into

00:16:23.510 --> 00:16:25.200
your calculator, and figure out
what the actual area is for

00:16:25.200 --> 00:16:27.910
that point, and realize
it is the low point.

00:16:27.910 --> 00:16:29.740
So that is the minimum value.

00:16:29.740 --> 00:16:33.640
That's how we minimize the area
of the square or the triangle.

00:16:33.640 --> 00:16:37.400
Now, if we committed all the
wire to the square, we get an

00:16:37.400 --> 00:16:40.640
area of 625, and all of
it comes to the square.

00:16:40.640 --> 00:16:46.150
If we cut it here, so all of
the wire is used to make the

00:16:46.150 --> 00:16:48.660
triangle, we get area of 481.

00:16:48.660 --> 00:16:53.080
So the maximum value is when a
is equal to 0, and our combined

00:16:53.080 --> 00:16:56.220
area is 625, because we're
doing a square with the whole

00:16:56.220 --> 00:17:02.170
wire, and the minimum value is
when we actually cut it

00:17:02.170 --> 00:17:04.850
right there at 56.5.

00:17:04.850 --> 00:17:11.180
So where we commit to 56.5 to
the triangle, and we commit,

00:17:11.180 --> 00:17:15.170
you know, the remainder, so
what is that, 43.5

00:17:15.170 --> 00:17:16.330
to the square.

00:17:16.330 --> 00:17:17.890
That's when we minimize it.

00:17:17.890 --> 00:17:20.840
That was that frankly a
fascinating problem, and I hope

00:17:20.840 --> 00:17:23.500
I got the problem right, and
even if I made a careless

00:17:23.500 --> 00:17:26.030
mistake, I think you understand
how to do the problem.

00:17:26.030 --> 00:17:27.260
I will see you in
the next video.


Optimization with Calculus 1

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.550
Let's use what we've learned
about finding the minimums and

00:00:03.550 --> 00:00:06.950
maximums of functions to solve
some optimization problems.

00:00:06.950 --> 00:00:09.160
And you might be wondering,
Sal, what is an

00:00:09.160 --> 00:00:10.580
optimization problem.

00:00:10.580 --> 00:00:13.310
And my reply to you is well,
the problems we're about to do

00:00:13.310 --> 00:00:15.420
are optimization problems, and
I think you'll understand

00:00:15.420 --> 00:00:17.610
why they're called
optimization problems.

00:00:17.610 --> 00:00:20.810
Anyway so the first problem
is find two numbers whose

00:00:20.810 --> 00:00:22.810
product is a negative 16.

00:00:22.810 --> 00:00:29.030
So the product is
equal to minus 16.

00:00:29.030 --> 00:00:33.520
And it says, and the sum of
whose squares is a minimum.

00:00:33.520 --> 00:00:41.820
So sum of squares is minimum.

00:00:41.820 --> 00:00:44.390
So if you haven't figured out
yet, these optimization

00:00:44.390 --> 00:00:50.020
problems all impose some type
of constraints, and then say,

00:00:50.020 --> 00:00:53.220
for what numbers do we get a
maximum value or minimum value.

00:00:56.000 --> 00:00:57.990
So let's do this problem, and
then we'll do a couple of more.

00:00:57.990 --> 00:01:03.120
So two numbers whose product
is negative 16 and the sum

00:01:03.120 --> 00:01:05.240
of their squares is minimum.

00:01:05.240 --> 00:01:08.730
So call those two
numbers x and y.

00:01:08.730 --> 00:01:12.610
So their product is minus
16, so we know that x times

00:01:12.610 --> 00:01:15.870
y is equal to minus 16.

00:01:15.870 --> 00:01:17.670
And then what's the
sum of their squares.

00:01:17.670 --> 00:01:21.170
Let's just say the some
of squares, so I'll

00:01:21.170 --> 00:01:24.050
just call that s.

00:01:24.050 --> 00:01:27.010
The sum of their squares
is going to be equal to x

00:01:27.010 --> 00:01:31.070
squared plus y squared.

00:01:31.070 --> 00:01:33.240
And so this is the function
that we want to optimize.

00:01:33.240 --> 00:01:36.730
And in this case we want to
find the minimum value.

00:01:36.730 --> 00:01:38.830
So how do we do this problem?

00:01:38.830 --> 00:01:41.510
Well the first thing we want to
do, we want to minimize this

00:01:41.510 --> 00:01:44.430
function, but right now it's a
function of two variables,

00:01:44.430 --> 00:01:47.140
and we haven't done
multivariable calculus yet.

00:01:47.140 --> 00:01:50.970
So it would be easier if we
could express this as a

00:01:50.970 --> 00:01:53.560
function of one variable, and
then we could optimize it with

00:01:53.560 --> 00:01:55.475
respect that variable, and we
could substitute backwards

00:01:55.475 --> 00:01:56.240
et cetera, et cetera.

00:01:56.240 --> 00:01:58.620
So how can we write the
sum of the squares as a

00:01:58.620 --> 00:01:59.980
function of one variable?

00:01:59.980 --> 00:02:02.880
Well that's where this
equation comes in handy.

00:02:02.880 --> 00:02:04.700
We could substitute
either for x or y.

00:02:04.700 --> 00:02:07.890
Let's just substitute y, so we
could rewrite this equation

00:02:07.890 --> 00:02:16.580
dividing both sides by x as y
is equal to minus 16 over x.

00:02:19.640 --> 00:02:21.120
I just divided both
sides of this by x.

00:02:21.120 --> 00:02:24.270
If we take this and substitute
it here, we get the sum of the

00:02:24.270 --> 00:02:28.150
squares of the two numbers
is going to be equal to x

00:02:28.150 --> 00:02:30.120
squared plus y squared.

00:02:30.120 --> 00:02:33.380
Well y is this, so what's
this thing squared?

00:02:33.380 --> 00:02:39.540
Minus 16 squared is
256 over x squared.

00:02:39.540 --> 00:02:42.830
So that's the sum of the
squares of the two numbers.

00:02:42.830 --> 00:02:46.940
And so how do we learn to find
minimums or maximums, or in

00:02:46.940 --> 00:02:48.560
general optimize the function?

00:02:48.560 --> 00:02:52.000
Well in general, a function
hits a minimum or maximum

00:02:52.000 --> 00:02:54.480
point, especially a local
minimum or maximum point,

00:02:54.480 --> 00:02:57.510
when it's derivative is 0.

00:02:57.510 --> 00:03:00.450
So let's take the derivative of
this and set it equal to 0, and

00:03:00.450 --> 00:03:01.990
hopefully we find
a minimum point.

00:03:01.990 --> 00:03:04.670
And then we'll later show that
it really is a minimum point.

00:03:04.670 --> 00:03:07.440
So let's take the derivative
of this with respect to x.

00:03:07.440 --> 00:03:08.880
I'll just write s prime.

00:03:08.880 --> 00:03:13.030
We could have written ds
dx, but save on notation.

00:03:13.030 --> 00:03:14.965
Save on my electronic ink.

00:03:14.965 --> 00:03:19.610
So s prime is equal to-- so
there's a slope of this

00:03:19.610 --> 00:03:25.410
function at any time, at any
x-- 2x plus-- this is the

00:03:25.410 --> 00:03:30.720
same thing as 256 times
x to the negative 2.

00:03:30.720 --> 00:03:32.270
So what will be the derivative?

00:03:32.270 --> 00:03:37.070
The derivative is
going to be 256.

00:03:37.070 --> 00:03:40.310
We multiply times exponent
which is minus 2, and then we

00:03:40.310 --> 00:03:44.150
decrement the exponent by 1 ,
so we get x to the minus 3.

00:03:44.150 --> 00:03:53.710
So the derivative is equal to
2x plus-- no, not plus-- 2x

00:03:53.710 --> 00:04:00.350
minus 512 times x
to the minus 3.

00:04:00.350 --> 00:04:02.450
And we want to find a local
minimum or maximum, or

00:04:02.450 --> 00:04:03.700
maybe a global one.

00:04:03.700 --> 00:04:06.630
So we want to figure out where
the slope is equal to 0, so

00:04:06.630 --> 00:04:09.410
this is the slope, so let's
set this equal to 0.

00:04:09.410 --> 00:04:20.740
So we get 2x minus 512x to
the minus 3 is equal to 0.

00:04:20.740 --> 00:04:23.850
Let's see, we can manipulate
that around a little bit.

00:04:23.850 --> 00:04:27.700
We get 2x-- let me do it
in a different color,

00:04:27.700 --> 00:04:28.680
I'm tired of the yellow.

00:04:32.510 --> 00:04:38.740
So you get going up here, we
get-- just let's add 512 to

00:04:38.740 --> 00:04:50.310
both sides-- so we get 2x is
equal to 512x to the minus 3,

00:04:50.310 --> 00:04:52.190
right, I just added
512 to both sides.

00:04:52.190 --> 00:04:55.840
Now let's multiply both sides
of this equation times x to the

00:04:55.840 --> 00:04:58.150
third, just to get rid of this.

00:04:58.150 --> 00:05:00.270
So if we multiply both sides
of this equation by x to

00:05:00.270 --> 00:05:07.550
the third, we get 2x to the
fourth is equal to 512.

00:05:07.550 --> 00:05:09.460
Right, x to the third
times x to the minus

00:05:09.460 --> 00:05:11.070
3, that just equals 1.

00:05:11.070 --> 00:05:13.720
Divide both sides of this
by 2, you get x to the

00:05:13.720 --> 00:05:17.770
fourth is equal to 256.

00:05:17.770 --> 00:05:21.510
And if it's not immediately
clear what this is-- you know,

00:05:21.510 --> 00:05:23.950
x is going to be the fourth
root of 256-- if you don't know

00:05:23.950 --> 00:05:26.040
that immediately, you can
think, well, what's the

00:05:26.040 --> 00:05:27.780
square root of 256.

00:05:27.780 --> 00:05:28.850
Well it's 16.

00:05:28.850 --> 00:05:30.540
And then what's the
square root of that.

00:05:30.540 --> 00:05:33.870
Well the square root of
16 is plus or minus.

00:05:33.870 --> 00:05:36.335
So the square root of this is
plus or minus 16, but then

00:05:36.335 --> 00:05:40.070
a square root of at least a
plus 16 is plus or minus 4.

00:05:40.070 --> 00:05:45.420
So this x is equal
to plus or minus 4.

00:05:45.420 --> 00:05:48.130
If you don't believe me, take
either minus 4 or positive

00:05:48.130 --> 00:05:51.580
4 and take it to the fourth
power and you will get 256.

00:05:51.580 --> 00:05:55.160
So if x is plus or
minus 4, what is y?

00:05:55.160 --> 00:06:00.290
Well we just know that y
is minus 16 divided by x,

00:06:00.290 --> 00:06:02.050
so we could get x is 4.

00:06:02.050 --> 00:06:05.230
If x is 4, then what is y?

00:06:05.230 --> 00:06:09.660
It's minus 16 divided by 4,
then y is equal to minus 4.

00:06:09.660 --> 00:06:13.650
And of course if x was minus 4,
then y is minus 16 divided by

00:06:13.650 --> 00:06:16.150
minus 4, y would be equal to 4.

00:06:16.150 --> 00:06:18.200
So it actually doesn't
matter which one we take.

00:06:18.200 --> 00:06:19.960
The two numbers are
4 and negative 4.

00:06:19.960 --> 00:06:23.550
It doesn't matter what
we call them, x or y.

00:06:23.550 --> 00:06:25.920
The two numbers are
4 and minus 4.

00:06:25.920 --> 00:06:27.600
And we are actually
done the problem.

00:06:27.600 --> 00:06:30.010
These are the two numbers where
the sum of the squares-- well

00:06:30.010 --> 00:06:31.460
actually, no we, are not done.

00:06:31.460 --> 00:06:35.010
We know that we've reached some
type of minimum or maximum

00:06:35.010 --> 00:06:37.740
point here, and since they ask
for minimum, and this was the

00:06:37.740 --> 00:06:41.160
only point that we found where
the slope was 0, we probably

00:06:41.160 --> 00:06:42.250
solved the problem.

00:06:42.250 --> 00:06:45.140
But let's prove to ourselves
that we're really at a minimum

00:06:45.140 --> 00:06:50.420
point on this function
when x is equal to 4.

00:06:50.420 --> 00:06:51.750
And how do we do that?

00:06:51.750 --> 00:06:54.690
Well, we can take the second
derivative of this function,

00:06:54.690 --> 00:06:57.400
see what the second derivative
is at x equals 4, and then see

00:06:57.400 --> 00:07:01.590
if we're concave upwards
or concave downwards.

00:07:01.590 --> 00:07:02.755
This is the first derivative.

00:07:08.750 --> 00:07:10.063
What is the second derivative?

00:07:13.280 --> 00:07:21.480
s prime prime is equal to 2,
and then minus 3 times minus

00:07:21.480 --> 00:07:25.000
512, so that's plus,
and what is that?

00:07:25.000 --> 00:07:34.020
1,536 Right, plus 1,536
times x to the minus 4.

00:07:40.230 --> 00:07:44.700
I actually you know should be
writing s as a function of x.

00:07:44.700 --> 00:07:47.350
I shouldn't be lazy
with my notation.

00:07:47.350 --> 00:07:49.400
These are all the function of
x, but you get the point.

00:07:49.400 --> 00:07:56.160
So what is s prime prime
when x is equal to 4?

00:07:56.160 --> 00:08:01.350
Well it's equal
to 2 plus 1,536.

00:08:01.350 --> 00:08:03.950
And then what's 4 to
the minus 4 power?

00:08:03.950 --> 00:08:07.790
We get 1/256.

00:08:07.790 --> 00:08:09.140
And we don't even have
to calculate this.

00:08:09.140 --> 00:08:11.160
All we care about is this
value positive or negative.

00:08:11.160 --> 00:08:12.295
It And it's clearly positive.

00:08:12.295 --> 00:08:14.480
We have a positive number plus
a positive number divided

00:08:14.480 --> 00:08:15.080
by a positive number.

00:08:15.080 --> 00:08:16.940
So we have a positive
number here.

00:08:16.940 --> 00:08:18.630
So what do we know?

00:08:18.630 --> 00:08:26.270
We know at the point x is equal
to 4, this function here-- our

00:08:26.270 --> 00:08:30.410
sum of squares function-- is
concave upwards, and we know

00:08:30.410 --> 00:08:32.650
that because the second
derivative at x is equal

00:08:32.650 --> 00:08:34.050
to 4 is positive.

00:08:34.050 --> 00:08:36.310
And what does concave
upwards look like?

00:08:36.310 --> 00:08:39.290
Concave upwards
looks like this.

00:08:39.290 --> 00:08:42.960
And so x equals 4 is there, and
so we are at a minimum point.

00:08:42.960 --> 00:08:45.740
And so we know that the sum
of squares are minimum.

00:08:45.740 --> 00:08:48.520
And if you really want to make
sure that you're not missing

00:08:48.520 --> 00:08:51.840
something or we haven't done
this wrong, just play around,

00:08:51.840 --> 00:08:55.310
think of other numbers whose
product is minus 16 and figure

00:08:55.310 --> 00:08:56.540
out their sum of squares.

00:08:56.540 --> 00:08:59.910
And I can pretty much guarantee
you that it will be larger than

00:08:59.910 --> 00:09:03.540
the sum of the squares
of 4 of minus 4 and 4.

00:09:03.540 --> 00:09:05.260
So actually, what are they?

00:09:05.260 --> 00:09:06.590
What is s of 4?

00:09:06.590 --> 00:09:08.960
What is the sum of squares
of 4 and minus 4?

00:09:08.960 --> 00:09:13.340
It's 4 squared is
16 plus 16 is 32.

00:09:13.340 --> 00:09:14.045
That's the sum.

00:09:14.045 --> 00:09:15.430
So let's try some
other numbers.

00:09:15.430 --> 00:09:23.010
What is the sum of 1 squared
plus minus 16 squared.

00:09:23.010 --> 00:09:27.180
Those two have the product
minus 16, 1 and minus 16,

00:09:27.180 --> 00:09:28.020
that is equal to what?

00:09:28.020 --> 00:09:29.280
257.

00:09:29.280 --> 00:09:30.300
Much higher number.

00:09:30.300 --> 00:09:33.600
You could try negative 8 and
2, or negative 2 and 8, and

00:09:33.600 --> 00:09:36.480
everything you'll find will
have a higher sum of squares

00:09:36.480 --> 00:09:38.330
than 4 and minus 4.

00:09:38.330 --> 00:09:39.710
Anyway.

00:09:39.710 --> 00:09:41.560
I was actually planning on
doing three problems in this

00:09:41.560 --> 00:09:45.030
video, but I'm reaching the 10
minute limit so I will finish

00:09:45.030 --> 00:09:47.230
this video now, and I will
do two more of these

00:09:47.230 --> 00:09:47.960
optimization problems.

00:09:47.960 --> 00:09:48.320
See you soon.


Optimization with Calculus 2

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:03.090
This next problem is a bit of
a classic and I've actually

00:00:03.090 --> 00:00:06.740
gotten multiple requests to do
some variation of this problem.

00:00:06.740 --> 00:00:12.190
So it says find the volume of
the largest open box-- It can

00:00:12.190 --> 00:00:16.210
be made from a piece of
cardboard-- 24 inches square.

00:00:16.210 --> 00:00:19.910
So let me draw my
piece of cardboard.

00:00:19.910 --> 00:00:23.200
I don't want to
draw it filled in.

00:00:23.200 --> 00:00:27.120
So my piece of cardboard
is going to be a square.

00:00:27.120 --> 00:00:28.260
I'll draw it pretty big,
because I want you

00:00:28.260 --> 00:00:29.300
to visualize it.

00:00:29.300 --> 00:00:30.640
And it's 24, what did they say?

00:00:30.640 --> 00:00:32.595
24 inches square.

00:00:32.595 --> 00:00:39.370
So it is 24 by 24 square
by cutting equal squares

00:00:39.370 --> 00:00:41.230
from the corners and
turning up the sides.

00:00:41.230 --> 00:00:42.050
So what are they saying?

00:00:42.050 --> 00:00:43.290
Well, they're saying
we're going to cut equal

00:00:43.290 --> 00:00:44.820
squares from the corner.

00:00:44.820 --> 00:00:49.850
So let's say we cut-- let me
see if I can draw this well

00:00:49.850 --> 00:00:56.690
--so I'm going to cut a square
here, and I'm going to cut an

00:00:56.690 --> 00:01:01.810
equal size square there-- draw
this neatly --and then cut an

00:01:01.810 --> 00:01:07.100
equal size square there,
and then cut an equal

00:01:07.100 --> 00:01:08.123
size square there.

00:01:12.770 --> 00:01:15.810
So I'm going to cut out those
corners and then I'm going

00:01:15.810 --> 00:01:18.540
to fold up the sides.

00:01:18.540 --> 00:01:21.930
Just so you can visualize it,
so those dotted lines are where

00:01:21.930 --> 00:01:24.120
I would actually fold the box,
and then I would end

00:01:24.120 --> 00:01:25.100
up with an open box.

00:01:25.100 --> 00:01:27.310
And let's see what the
dimensions would be.

00:01:27.310 --> 00:01:29.470
And that's actually what the
whole crux of the problem is.

00:01:29.470 --> 00:01:33.840
So how big of a squares should
I cut out in order to maximize

00:01:33.840 --> 00:01:35.900
the volume up the box?

00:01:35.900 --> 00:01:38.220
Well, these are equal sided
squares that I'm cutting

00:01:38.220 --> 00:01:39.945
out, so let's say
that each side is x.

00:01:39.945 --> 00:01:44.405
So if this x, then this x, this
is x, this is x, this is x,

00:01:44.405 --> 00:01:48.360
this is x, this is x,
this is x, this is x.

00:01:48.360 --> 00:01:50.880
And then what is going to be
the volume of our box when

00:01:50.880 --> 00:01:52.730
we fold up everything?

00:01:52.730 --> 00:01:54.860
Let me see, so I'll
draw the base.

00:01:54.860 --> 00:01:56.610
So this dotted square
right here, that's

00:01:56.610 --> 00:01:59.630
going to be the base.

00:01:59.630 --> 00:02:02.150
I'll draw it at an angle.

00:02:02.150 --> 00:02:03.050
That's the base.

00:02:03.050 --> 00:02:05.700
And what are the
dimensions of the base?

00:02:05.700 --> 00:02:09.400
Well, this base is going to be
the length of this side of the

00:02:09.400 --> 00:02:12.750
cardboard minus this x
minus this x, right?

00:02:12.750 --> 00:02:18.510
So this base right here, that
is going to be 24 minus 2x.

00:02:18.510 --> 00:02:21.010
It's just the whole length
minus this minus that;

00:02:21.010 --> 00:02:22.360
that's how long that is.

00:02:22.360 --> 00:02:24.400
And of course, this is going to
be the same length, this is

00:02:24.400 --> 00:02:25.810
going to be the same length.

00:02:25.810 --> 00:02:28.130
All of these sides of the
base are 24 minus 2x.

00:02:32.910 --> 00:02:37.100
And then this side is
also 24 minus 2x.

00:02:37.100 --> 00:02:40.840
And then what is going to
be the height of this open

00:02:40.840 --> 00:02:42.410
box that we're creating?

00:02:42.410 --> 00:02:44.770
Let me draw the sides of it.

00:02:47.770 --> 00:02:52.100
It's open, which means it
doesn't have a top to it.

00:02:52.100 --> 00:02:54.420
So we have an opening here,
so you can kind of see

00:02:54.420 --> 00:02:56.720
the inside of the box.

00:02:56.720 --> 00:02:57.990
And what is this height?

00:02:57.990 --> 00:03:04.850
Well, when you fold these
flaps up the height is x.

00:03:04.850 --> 00:03:07.742
If you can think about it--
here this is really just a bit

00:03:07.742 --> 00:03:10.600
of visualization --this side
right here, that you can view

00:03:10.600 --> 00:03:12.660
as this thing right here.

00:03:12.660 --> 00:03:15.380
I can even do it in a
different color for you.

00:03:15.380 --> 00:03:19.040
You can view this side,
this front facing side

00:03:19.040 --> 00:03:21.450
as this right here.

00:03:21.450 --> 00:03:23.156
I'll just do a couple; you
I don't have to do all of

00:03:23.156 --> 00:03:24.580
the side, I don't think.

00:03:24.580 --> 00:03:30.120
You could view this side
as this side right here.

00:03:30.120 --> 00:03:30.390
Right?

00:03:30.390 --> 00:03:31.330
It's folded up.

00:03:31.330 --> 00:03:34.660
And then this backside is this
backside, and that side there

00:03:34.660 --> 00:03:36.320
is that and this is the base.

00:03:36.320 --> 00:03:40.710
Anyway, so that is the volume
of our open box and we

00:03:40.710 --> 00:03:42.840
want to maximize it.

00:03:42.840 --> 00:03:43.710
How do we do that?

00:03:43.710 --> 00:03:47.070
Let's write the volume as a
function of x and then take its

00:03:47.070 --> 00:03:49.100
derivative, figure out where
the derivative is 0, and hope

00:03:49.100 --> 00:03:51.070
that that's a maximum point and
we'll prove it by taking the

00:03:51.070 --> 00:03:52.500
second derivative and seeing.

00:03:52.500 --> 00:03:54.870
Now we actually want it to be
concave downwards because

00:03:54.870 --> 00:03:57.250
concave downwards looks
like that, that means you

00:03:57.250 --> 00:03:59.000
found a maximum point.

00:03:59.000 --> 00:04:03.480
So we know that the volume as a
function of x-- I like writing

00:04:03.480 --> 00:04:07.240
the big v's --the volume of the
function of x is that side

00:04:07.240 --> 00:04:09.700
times-- you know, you could
view it as depth times

00:04:09.700 --> 00:04:10.980
length times height.

00:04:10.980 --> 00:04:21.020
So it's x, the height, times
the depth, 24 minus 2x, times

00:04:21.020 --> 00:04:25.360
the width, 24 minus 2x.

00:04:25.360 --> 00:04:27.440
Now let's see if I can
multiply this out.

00:04:27.440 --> 00:04:29.600
It's probably the hardest
part about this problem.

00:04:29.600 --> 00:04:33.980
So x times-- what's
24 times 24?

00:04:33.980 --> 00:04:39.400
I want to say it's-- actually
kind of sad, but I've made so

00:04:39.400 --> 00:04:42.280
many careless mistakes
recently that I don't

00:04:42.280 --> 00:04:42.970
want to make any more.

00:04:42.970 --> 00:04:43.870
I want to get the right number.

00:04:43.870 --> 00:04:47.100
For 4 times 4 is 16.

00:04:47.100 --> 00:04:49.780
4 times 2 is 8, 96.

00:04:49.780 --> 00:04:50.670
0.

00:04:50.670 --> 00:04:54.380
2 times 24 is 48, 6.

00:04:54.380 --> 00:04:57.840
9 plus 8 is 17.

00:04:57.840 --> 00:04:59.630
576: that's what I
thought it was.

00:04:59.630 --> 00:05:02.400
I should have memorized my
times tables up to 25.

00:05:02.400 --> 00:05:09.580
But anyway, 24 times 24 is 576.

00:05:09.580 --> 00:05:15.630
And then we'll have 2 times 24,
so that's 48 x, but we have

00:05:15.630 --> 00:05:20.420
them twice so it's minus 96x.

00:05:20.420 --> 00:05:25.130
Because minus 2x times 24 and
then minus 2x times 24, so

00:05:25.130 --> 00:05:29.220
that's 48 to plus
48, so minus 96x.

00:05:29.220 --> 00:05:34.900
And then finally the last
two: plus 4x squared.

00:05:34.900 --> 00:05:38.890
And now we can multiply the x's
out and we get the volume of

00:05:38.890 --> 00:05:45.040
the open box is-- let me put
the x term first --4x to the

00:05:45.040 --> 00:05:49.390
third-- I'm just multiplying
this out --minus 96x

00:05:49.390 --> 00:05:54.320
squared plus 576x.

00:05:54.320 --> 00:05:57.180
Let me erase this.

00:05:57.180 --> 00:05:59.240
I should have been able
to do that in my head.

00:05:59.240 --> 00:06:04.920
Anyway, I'm not too proud to
show that I didn't know that.

00:06:04.920 --> 00:06:06.470
Let's see, so let
me erase that.

00:06:12.260 --> 00:06:14.200
And so what do we want to do?

00:06:14.200 --> 00:06:16.940
We want to take the derivative
of this, figure out at what x

00:06:16.940 --> 00:06:21.250
values do we have a 0 slope,
and then test to see if those

00:06:21.250 --> 00:06:22.470
were maximum or minimum points.

00:06:22.470 --> 00:06:24.820
If we have a maximum point at
that x value, and if it's a

00:06:24.820 --> 00:06:27.520
global maximum, that we have
found the x value that

00:06:27.520 --> 00:06:29.980
optimizes the volume.

00:06:29.980 --> 00:06:32.070
You might want to graph these
and experiment with them and

00:06:32.070 --> 00:06:33.750
get a more intuitive
sense of it.

00:06:33.750 --> 00:06:36.100
But that's really what we did
when we found minima and

00:06:36.100 --> 00:06:38.960
maxima, and we did concave
upwards and downwards

00:06:38.960 --> 00:06:40.930
and all of that.

00:06:40.930 --> 00:06:50.165
Anyway, the derivative v prime
of x is equal to 12x squared--

00:06:50.165 --> 00:07:05.180
this part is the fun part --12x
squared minus 192x plus 576.

00:07:05.180 --> 00:07:06.550
That says 576.

00:07:06.550 --> 00:07:08.870
I know it's hard to read;
it's hard to read for me.

00:07:08.870 --> 00:07:10.710
So let's figure out
where this equals 0.

00:07:10.710 --> 00:07:18.460
So we want to know where
12x squared minus 192x

00:07:18.460 --> 00:07:23.220
plus 576 is equal 0.

00:07:23.220 --> 00:07:26.050
And the easiest thing to do
here is just divide the whole

00:07:26.050 --> 00:07:27.410
thing by 12, both sides.

00:07:27.410 --> 00:07:29.110
Because when you divide 0
by 12 you still get 0.

00:07:29.110 --> 00:07:35.150
So you get x squared minus--
what's 192 divided by 12?

00:07:35.150 --> 00:07:37.725
Well, 192 was 24 times 4.

00:07:46.480 --> 00:07:47.100
Is that right?

00:07:47.100 --> 00:07:47.840
24?

00:07:47.840 --> 00:07:56.760
No, 24 times 16 is 192.

00:07:56.760 --> 00:07:59.300
I've been making so many
careless mistakes.

00:07:59.300 --> 00:08:01.240
12 goes into 192, 1.

00:08:01.240 --> 00:08:02.480
12, 72.

00:08:02.480 --> 00:08:02.700
All right.

00:08:02.700 --> 00:08:03.440
16.

00:08:03.440 --> 00:08:04.100
I should have been
able to do that.

00:08:04.100 --> 00:08:12.070
Anyway, minus 16x plus-- well,
this was 24 times 24, so

00:08:12.070 --> 00:08:14.230
this should be 12 times 48.

00:08:14.230 --> 00:08:18.210
So this should be plus
48 is equal to 0.

00:08:18.210 --> 00:08:22.190
And we could write 24
times 24 is 5 plus 76.

00:08:22.190 --> 00:08:24.330
So 12 times 48.

00:08:24.330 --> 00:08:24.920
Right.

00:08:24.920 --> 00:08:26.940
So now we just have
to factor this.

00:08:26.940 --> 00:08:32.660
So what two numbers that when I
add them I get minus 16 and

00:08:32.660 --> 00:08:38.970
then when I multiply them
I get 48, positive 48?

00:08:38.970 --> 00:08:43.170
Let's see 6 times-- I want
to make sure I did this; so

00:08:43.170 --> 00:08:47.160
12 goes into 72 six times.

00:08:47.160 --> 00:08:48.580
I just want to make sure I
have the numbers right.

00:08:48.580 --> 00:08:50.500
As you can tell, I don't do
these problems ahead of time,

00:08:50.500 --> 00:08:53.180
because I want you see that I
go through the pain as you do.

00:08:53.180 --> 00:08:53.823
So let me make sure.

00:08:53.823 --> 00:09:01.640
I have 12 goes into
576 four times.

00:09:01.640 --> 00:09:07.480
48, and you get 96.

00:09:07.480 --> 00:09:09.030
12 goes into-- right.

00:09:09.030 --> 00:09:10.680
48 times, 12 goes into that.

00:09:10.680 --> 00:09:13.060
I shouldn't have had to do
that, but I've been making

00:09:13.060 --> 00:09:14.256
so many careless mistakes.

00:09:14.256 --> 00:09:15.900
I haven't eaten dinner yet.

00:09:15.900 --> 00:09:19.000
So anyway, maybe I'm just
missing the easy factor, right?

00:09:19.000 --> 00:09:22.300
Because I have, what numbers
you get 12 times-- oh, actually

00:09:22.300 --> 00:09:23.610
I just figured it out.

00:09:23.610 --> 00:09:26.230
12 times 4.

00:09:26.230 --> 00:09:28.140
Sometimes when your brain
is in calculus mode, the

00:09:28.140 --> 00:09:29.440
algebra II gets difficult.

00:09:29.440 --> 00:09:34.620
So this is the same thing
as x minus 4 times x

00:09:34.620 --> 00:09:39.440
minus 12 is equal to 0.

00:09:39.440 --> 00:09:44.620
So at x is equal to 4 we're at
some type of minima or maxima

00:09:44.620 --> 00:09:45.850
where it's some critical point.

00:09:45.850 --> 00:09:48.060
And x is equal to 12.

00:09:48.060 --> 00:09:50.850
And let me tell you something:
we don't even have to look at

00:09:50.850 --> 00:09:51.680
the secondary derivative here.

00:09:51.680 --> 00:09:53.560
What happens when
x is equal to 12?

00:09:53.560 --> 00:09:55.520
What will happen to the
volume of the box?

00:09:55.520 --> 00:10:00.080
If x is equal to 12, what is
the length of this base?

00:10:00.080 --> 00:10:01.800
It's 0.

00:10:01.800 --> 00:10:04.460
Because the length of the
base is 24 minus 2x.

00:10:04.460 --> 00:10:05.710
So this would be 0,
this would be 0.

00:10:05.710 --> 00:10:09.140
So this is the point where we
have a volume of 0, so you

00:10:09.140 --> 00:10:11.450
know that this is going
to be a minimum point.

00:10:11.450 --> 00:10:14.090
And if you want to verify it,
take the second derivative of

00:10:14.090 --> 00:10:16.690
the volume and evaluate the
second derivative of the volume

00:10:16.690 --> 00:10:21.720
at 12 and you will realize that
we are concave upwards and that

00:10:21.720 --> 00:10:23.360
x equals twelve is
the minimum point.

00:10:23.360 --> 00:10:26.472
So you already know, in
all probability, the

00:10:26.472 --> 00:10:29.120
answer is x equals 4.

00:10:29.120 --> 00:10:30.900
And if you really wanted to
verify it, you can take the

00:10:30.900 --> 00:10:35.250
second derivative, add x equals
4, and make sure that we are

00:10:35.250 --> 00:10:37.470
concave downwards
at that point.

00:10:37.470 --> 00:10:39.400
Because if we want a maximum
point, we want to be some

00:10:39.400 --> 00:10:41.180
place where the graph
looks something like that.

00:10:41.180 --> 00:10:43.060
So we're at a maximum point.

00:10:43.060 --> 00:10:45.070
So what's the
second derivative?

00:10:45.070 --> 00:10:58.730
v prime prime of x is
equal to 24x minus 192.

00:10:58.730 --> 00:11:00.050
And what's 24 times 4?

00:11:00.050 --> 00:11:02.070
It's 96, right?

00:11:02.070 --> 00:11:08.246
So v prime prime of
4 is equal to 96.

00:11:08.246 --> 00:11:08.800
Right?

00:11:08.800 --> 00:11:14.110
4 times 24 minus 192,
which equals minus 96.

00:11:14.110 --> 00:11:18.470
So the second derivative at
this point is negative, which

00:11:18.470 --> 00:11:21.490
means that we are concave
downwards, which means that

00:11:21.490 --> 00:11:23.630
this is a maximum point.

00:11:23.630 --> 00:11:27.390
And once again, I challenge you
to find another value of x

00:11:27.390 --> 00:11:29.270
where you get a larger value.

00:11:29.270 --> 00:11:32.400
And just to get an intuition,
how large will this box be?

00:11:32.400 --> 00:11:37.270
Well, if x is 4, then
each side is 24 minus 8.

00:11:37.270 --> 00:11:43.160
This would be 16 by 16 by 4.

00:11:43.160 --> 00:11:47.110
So that would be the optimal
volume for that box.

00:11:47.110 --> 00:11:49.900
Hope you found that useful, and
I will see you in the next

00:11:49.900 --> 00:11:52.605
video where I will do another
optimization problem, and

00:11:52.605 --> 00:11:53.540
this one is especially fun.

00:11:53.540 --> 00:11:53.930
See you soon.


Singular matrices

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.800
Perhaps even more interesting
than finding the inverse of a

00:00:03.800 --> 00:00:07.210
matrix is trying to determine
when an inverse of a matrix

00:00:07.210 --> 00:00:09.950
doesn't exist. Or when
it's undefined.

00:00:09.950 --> 00:00:14.865
And a square matrix for which
there is no inverse, of which

00:00:14.865 --> 00:00:17.340
an inverse is undefined is
called a singular matrix.

00:00:17.340 --> 00:00:19.970
So let's think about what a
singular matrix will look

00:00:19.970 --> 00:00:23.440
like, and how that applies to
the different problems that

00:00:23.440 --> 00:00:25.140
we've address using matrices.

00:00:25.140 --> 00:00:27.410
So if I had the other 2
by 2, because that's

00:00:27.410 --> 00:00:28.420
just a simpler example.

00:00:28.420 --> 00:00:31.470
But it carries over into really
any size square matrix.

00:00:31.470 --> 00:00:33.530
So let's take our
2 by 2 matrix.

00:00:33.530 --> 00:00:38.320
And the elements are
a, b, c and d.

00:00:38.320 --> 00:00:41.220
What's the inverse
of that matrix?

00:00:41.220 --> 00:00:43.510
This hopefully is a bit of
second nature to you now.

00:00:43.510 --> 00:00:51.765
It's 1 over the determinant of
a, times the adjoint of a.

00:00:51.765 --> 00:00:53.780
And in this case, you just
switch these two terms. So you

00:00:53.780 --> 00:00:55.200
have a d and an a.

00:00:55.200 --> 00:00:56.480
And you make these two
terms negative.

00:00:56.480 --> 00:01:00.710
So you have minus
c and minus b.

00:01:00.710 --> 00:01:04.420
So my question to you is, what
will make this entire

00:01:04.420 --> 00:01:06.410
expression undefined?

00:01:06.410 --> 00:01:09.260
Well it doesn't matter what
numbers I have. If I have

00:01:09.260 --> 00:01:11.740
numbers here that make a
defined, then I can obviously

00:01:11.740 --> 00:01:13.470
swap them or make them negative,
and it won't change

00:01:13.470 --> 00:01:15.030
this part of the expression.

00:01:15.030 --> 00:01:20.130
But what would create a problem
is if we attempted to

00:01:20.130 --> 00:01:21.410
divide by 0 here.

00:01:21.410 --> 00:01:26.100
If the determinant of the
matrix A were undefined.

00:01:26.100 --> 00:01:40.560
So A inverse is undefined, if
and only if-- and in math they

00:01:40.560 --> 00:01:46.770
sometimes write it if with two
f's-- if and only if the

00:01:46.770 --> 00:01:49.460
determinant of A
is equal to 0.

00:01:49.460 --> 00:01:51.910
So the other way to view that
is, if a determinant of any

00:01:51.910 --> 00:01:54.720
matrix is equal to 0, then
that matrix is a singular

00:01:54.720 --> 00:01:59.480
matrix, and it has no inverse,
or the inverse is undefined.

00:01:59.480 --> 00:02:02.950
So let's think about in
conceptual terms, at least the

00:02:02.950 --> 00:02:06.200
two problems that we've looked
at, what a 0 determinant

00:02:06.200 --> 00:02:08.500
means, and see if we can get a
little bit of intuition for

00:02:08.500 --> 00:02:11.580
why there is no inverse.

00:02:11.580 --> 00:02:12.830
So what is a 0 determinant?

00:02:12.830 --> 00:02:14.850
In this case, what's a
determinant of this 2 by 2?

00:02:14.850 --> 00:02:18.170
Well the determinant of
A is equal to what?

00:02:18.170 --> 00:02:20.900
It's equal to ad minus bc.

00:02:25.560 --> 00:02:30.360
So this matrix is singular, or
it has no inverse, if this

00:02:30.360 --> 00:02:31.680
expression is equal to 0.

00:02:31.680 --> 00:02:32.990
So let me write that
over here.

00:02:32.990 --> 00:02:40.380
So if ad is equal to bc-- or we
can just manipulate things,

00:02:40.380 --> 00:02:47.350
and we could say if a/b is equal
to c/d-- I just divided

00:02:47.350 --> 00:02:50.230
both sides by b, and divided
both sides by d-- so if the

00:02:50.230 --> 00:02:55.200
ratio of a:b is the same as the
ratio of c:d, then this

00:02:55.200 --> 00:02:57.010
will have no inverse.

00:02:57.010 --> 00:03:01.230
Or another way we could write
this expression, if a/c-- if I

00:03:01.230 --> 00:03:07.310
divide both sides by c, and
divide both sides by d-- is

00:03:07.310 --> 00:03:10.600
equal to b/d.

00:03:10.600 --> 00:03:14.360
So another way that this would
be singular is if-- and it's

00:03:14.360 --> 00:03:15.290
actually the same way.

00:03:15.290 --> 00:03:16.570
If this is true, then
this is true.

00:03:16.570 --> 00:03:17.630
These are the same.

00:03:17.630 --> 00:03:20.150
Just a little bit of algebraic
manipulation.

00:03:20.150 --> 00:03:23.690
But if the ratio of a:c is equal
to the ratio of b:d, and

00:03:23.690 --> 00:03:25.270
you can think about why
that's the same thing.

00:03:25.270 --> 00:03:26.900
The ratio of a:b being
the same thing as

00:03:26.900 --> 00:03:27.610
the ratio of c:d.

00:03:27.610 --> 00:03:29.200
But anyway, I don't want
to confuse you.

00:03:29.200 --> 00:03:33.180
But let's think about how that
translates into some of the

00:03:33.180 --> 00:03:35.350
problems that we looked at.

00:03:35.350 --> 00:03:40.900
So let's say that we wanted to
look at the problem-- Let's

00:03:40.900 --> 00:03:45.110
say that we had this matrix
representing the linear

00:03:45.110 --> 00:03:45.920
equation problem.

00:03:45.920 --> 00:03:47.000
Well, actually, this would
be either one.

00:03:47.000 --> 00:04:01.290
So I have a, b, c, d times x,
y Is equal to two other

00:04:01.290 --> 00:04:06.530
numbers that we haven't
used yet, e and f.

00:04:06.530 --> 00:04:09.850
So if we have this matrix
equation representing the

00:04:09.850 --> 00:04:11.830
linear equation problem, then
the linear equation problem

00:04:11.830 --> 00:04:23.360
would be translated a times x
plus b times y is equal to e.

00:04:23.360 --> 00:04:31.290
And c times x plus d times
y is equal to f.

00:04:31.290 --> 00:04:33.690
And we would want to see where
these two intersect.

00:04:33.690 --> 00:04:34.950
That would be the solution,
the vector

00:04:34.950 --> 00:04:37.150
solution to this equation.

00:04:37.150 --> 00:04:40.850
And so, just to get a visual
understanding of what these

00:04:40.850 --> 00:04:43.850
two lines look like, let's
put it into the

00:04:43.850 --> 00:04:45.200
slope y-intercept form.

00:04:45.200 --> 00:04:47.930
So this would become what?

00:04:47.930 --> 00:04:52.190
In this case, y is
equal to what?

00:04:52.190 --> 00:05:04.610
y is equal to minus
a/b, x plus e/b.

00:05:04.610 --> 00:05:05.620
I'm just skipping some steps.

00:05:05.620 --> 00:05:08.630
But you subtract ax
from both sides.

00:05:08.630 --> 00:05:11.580
And then divide both sides
by b, and you get that.

00:05:11.580 --> 00:05:14.365
And then this equation, if you
put it in the same form, just

00:05:14.365 --> 00:05:15.690
solve for y.

00:05:15.690 --> 00:05:35.930
You get y is equal to minus
c/d x plus f/y.

00:05:35.930 --> 00:05:40.120
So let's think about this.

00:05:40.120 --> 00:05:42.720
I should probably change colors
because it looks too--

00:05:42.720 --> 00:05:46.910
Let's think about what these two
equations would look like

00:05:46.910 --> 00:05:48.160
if this holds.

00:05:50.730 --> 00:05:53.580
And we said if this holds, then
we have no determinant,

00:05:53.580 --> 00:05:57.260
and this becomes a singular
matrix, and it has no inverse.

00:05:57.260 --> 00:05:59.990
And since it has no inverse, you
can't solve this equation

00:05:59.990 --> 00:06:02.110
by multiplying both sides by
the inverse, because the

00:06:02.110 --> 00:06:03.430
inverse doesn't exist.

00:06:03.430 --> 00:06:04.560
So let's think about this.

00:06:04.560 --> 00:06:07.410
If this is true, we have no
determinant, but what does

00:06:07.410 --> 00:06:11.200
that mean intuitively in terms
of these equations?

00:06:11.200 --> 00:06:18.540
Well if a/b is equal to c/d,
these two lines will have the

00:06:18.540 --> 00:06:20.380
same slope.

00:06:20.380 --> 00:06:21.670
They'll have the same slope.

00:06:21.670 --> 00:06:24.150
So if these two expressions are
different, then what do we

00:06:24.150 --> 00:06:25.400
know about them?

00:06:25.400 --> 00:06:27.380
If two lines that have the
same slope and different

00:06:27.380 --> 00:06:30.450
y-intercepts, they're parallel
to each other, and they will

00:06:30.450 --> 00:06:32.170
never, ever intersect.

00:06:32.170 --> 00:06:46.000
So let me draw that, just so you
get the-- this top line--

00:06:46.000 --> 00:06:47.690
They don't have to be positive
numbers, but since this has a

00:06:47.690 --> 00:06:51.130
negative, I'll draw it
as a negative slope.

00:06:51.130 --> 00:06:54.740
So that's the first line.

00:06:54.740 --> 00:07:00.310
And its y-intercept
will be e/b.

00:07:02.830 --> 00:07:05.960
That's this line right here.

00:07:05.960 --> 00:07:10.970
And then the second line-- let
me do it in another color-- I

00:07:10.970 --> 00:07:12.800
don't know if it's going to be
above or below that line, but

00:07:12.800 --> 00:07:14.610
it's going to be parallel.

00:07:14.610 --> 00:07:15.910
It'll look something
like this.

00:07:19.750 --> 00:07:23.520
And that line's y-intercept--
so that's this line-- that

00:07:23.520 --> 00:07:28.570
line's y intercept is
going to be f/y.

00:07:28.570 --> 00:07:32.370
So if e/b and f/y are different
terms, but both

00:07:32.370 --> 00:07:34.290
lines have the same equation,
they're going to be parallel

00:07:34.290 --> 00:07:36.280
and they'll never intersect.

00:07:36.280 --> 00:07:37.780
So there actually would
be no solution.

00:07:37.780 --> 00:07:40.800
If someone told you-- just the
traditional way that you've

00:07:40.800 --> 00:07:43.730
done it, either through
substitution, or through

00:07:43.730 --> 00:07:45.920
adding or subtracting the
linear equations-- you

00:07:45.920 --> 00:07:47.450
wouldn't be able to find a
solution where these two

00:07:47.450 --> 00:07:50.470
intersect, if a/b
is equal to c/d.

00:07:50.470 --> 00:07:53.040
So one way to view the singular
matrix is that you

00:07:53.040 --> 00:07:54.140
have parallel lines.

00:07:54.140 --> 00:07:56.610
Well then you might say, hey
Sal, but these two lines would

00:07:56.610 --> 00:07:59.490
intersect if e/b equaled f/y.

00:07:59.490 --> 00:08:01.800
If this and this were the
same, then these would

00:08:01.800 --> 00:08:03.510
actually be the identical
lines.

00:08:03.510 --> 00:08:05.730
And not only would they
intersect, they would

00:08:05.730 --> 00:08:08.060
intersect in an infinite
number of places.

00:08:08.060 --> 00:08:10.550
But still you would have
no unique solution.

00:08:10.550 --> 00:08:14.260
You'd have no one solution
to this equation.

00:08:14.260 --> 00:08:17.320
It would be true at all
values of x and y.

00:08:17.320 --> 00:08:20.290
So you can kind of view it when
you apply the matrices to

00:08:20.290 --> 00:08:21.970
this problem.

00:08:21.970 --> 00:08:24.780
The matrix is singular, if the
two lines that are being

00:08:24.780 --> 00:08:29.960
represented are either parallel,
or they are the

00:08:29.960 --> 00:08:31.230
exact same line.

00:08:31.230 --> 00:08:33.559
They're parallel and not
intersecting at all.

00:08:33.559 --> 00:08:36.450
Or they are the exact same line,
and they intersect at an

00:08:36.450 --> 00:08:40.880
infinite number of points.

00:08:40.880 --> 00:08:42.159
And so it kind of makes
sense that the A

00:08:42.159 --> 00:08:44.039
inverse wasn't defined.

00:08:44.039 --> 00:08:47.610
So let's think about this in
the context of the linear

00:08:47.610 --> 00:08:50.070
combinations of vectors.

00:08:50.070 --> 00:08:51.970
That's not what I wanted
to use to erase it.

00:08:59.250 --> 00:09:01.970
So when we think of this problem
in terms of linear

00:09:01.970 --> 00:09:06.220
combination of factors, we can
think of it like this.

00:09:06.220 --> 00:09:15.010
That this is the same thing as
the vector ac times x plus the

00:09:15.010 --> 00:09:25.500
vector bd times y, is equal
to the vector ef.

00:09:25.500 --> 00:09:27.180
So let's think about
it a little bit.

00:09:27.180 --> 00:09:30.110
We're saying, is there some
combination of the vector ac

00:09:30.110 --> 00:09:34.510
and the vector bd that
equals the vector ef.

00:09:34.510 --> 00:09:39.440
But we just said that if we have
no inverse here, we know

00:09:39.440 --> 00:09:41.890
that because the determinant
is 0.

00:09:41.890 --> 00:09:45.400
And if the determinant is 0,
then we know in this situation

00:09:45.400 --> 00:09:50.580
that a/c must equal b/d.

00:09:50.580 --> 00:09:53.120
So a/c is equal to b/d.

00:09:53.120 --> 00:09:56.050
So what does that tell us?

00:09:56.050 --> 00:09:58.730
Well let me draw it.

00:09:58.730 --> 00:10:00.710
And maybe numbers would
be more helpful here.

00:10:00.710 --> 00:10:02.950
But I think you'll get
the intuition.

00:10:02.950 --> 00:10:05.390
I'll just draw the
first quadrant.

00:10:05.390 --> 00:10:09.282
I'll just assume both vectors
are in the first quadrant.

00:10:09.282 --> 00:10:10.835
Let me draw.

00:10:17.990 --> 00:10:19.590
The vector ac.

00:10:19.590 --> 00:10:20.755
Let's say that this is a.

00:10:20.755 --> 00:10:23.072
Let me do it in a
different color.

00:10:23.072 --> 00:10:24.860
So I'm gonna draw
the vector ac.

00:10:24.860 --> 00:10:31.540
So if this is a, and this
is c, then the vector

00:10:31.540 --> 00:10:33.870
ac looks like that.

00:10:33.870 --> 00:10:34.260
Let me draw it.

00:10:34.260 --> 00:10:36.210
I want to make this neat.

00:10:36.210 --> 00:10:40.440
The vector ac is like that.

00:10:40.440 --> 00:10:43.120
And then we have the arrow.

00:10:43.120 --> 00:10:44.950
And what would the vector
bd look like?

00:10:49.700 --> 00:10:54.310
Well the vector bd--
And I could draw

00:10:54.310 --> 00:10:55.210
it arbitrarily someplace.

00:10:55.210 --> 00:10:58.580
But we're assuming that there's
no derivative-- sorry,

00:10:58.580 --> 00:10:59.660
no determinant.

00:10:59.660 --> 00:11:01.330
Have I been saying derivative
the whole time?

00:11:01.330 --> 00:11:02.090
I hope not.

00:11:02.090 --> 00:11:03.380
Well, we're assuming
that there's no

00:11:03.380 --> 00:11:06.070
determinant to this matrix.

00:11:06.070 --> 00:11:08.010
So if there's no determinant,
we know that

00:11:08.010 --> 00:11:11.510
a/c is equal to b/d.

00:11:11.510 --> 00:11:16.120
Or another way to view it is
that c/d is equal to d/b.

00:11:16.120 --> 00:11:17.930
But what that tells you is that
both of these vectors

00:11:17.930 --> 00:11:19.350
kind of have the same slope.

00:11:19.350 --> 00:11:22.580
So if they both start at point
0, they're going to go in the

00:11:22.580 --> 00:11:23.140
same direction.

00:11:23.140 --> 00:11:25.500
They might have a different
magnitude, but they're going

00:11:25.500 --> 00:11:27.080
to go in the same direction.

00:11:27.080 --> 00:11:37.010
So if this is point b, and this
is point d, vector bd is

00:11:37.010 --> 00:11:39.540
going to be here.

00:11:39.540 --> 00:11:42.050
And if that's not obvious to
you, think a little bit about

00:11:42.050 --> 00:11:46.080
why these two vectors, if this
is true, are going to point in

00:11:46.080 --> 00:11:48.200
the same direction.

00:11:48.200 --> 00:11:52.020
So that vector is going to
essentially overlap.

00:11:52.020 --> 00:11:56.040
It's going to have the same
direction as this vector, but

00:11:56.040 --> 00:11:59.260
it's just going to have
a different magnitude.

00:11:59.260 --> 00:12:00.730
It might have the
same magnitude.

00:12:00.730 --> 00:12:04.480
So my question to you is, vector
ef, we don't know where

00:12:04.480 --> 00:12:05.600
vector ef is.

00:12:05.600 --> 00:12:08.190
Well let's pick some
arbitrary point.

00:12:08.190 --> 00:12:12.080
Let's say that this is
e, and this is f.

00:12:12.080 --> 00:12:13.700
So this is vector ef up there.

00:12:13.700 --> 00:12:17.130
Let me do it in a
different color.

00:12:17.130 --> 00:12:19.250
Vector ef, let's
say it's there.

00:12:23.420 --> 00:12:26.510
So my question to you is, if
these two vectors are in the

00:12:26.510 --> 00:12:27.310
same direction.

00:12:27.310 --> 00:12:28.850
Maybe of different magnitude.

00:12:28.850 --> 00:12:32.620
Is there any way that you can
add or subtract combinations

00:12:32.620 --> 00:12:35.140
of these two vectors to
get to this vector?

00:12:35.140 --> 00:12:37.420
Well no, you can scale these
vectors and add them.

00:12:37.420 --> 00:12:39.650
And all you're going to do is
kind of move along this line.

00:12:39.650 --> 00:12:41.610
You can get to any
other vector.

00:12:41.610 --> 00:12:43.800
There's a multiple of one
of these vectors.

00:12:43.800 --> 00:12:46.990
But because these are the exact
same direction, you

00:12:46.990 --> 00:12:49.560
can't get to any vector that's
in a different direction.

00:12:49.560 --> 00:12:53.050
So if this vector is in a
different direction, there's

00:12:53.050 --> 00:12:54.260
no solution here.

00:12:54.260 --> 00:13:00.530
If this vector just happened to
be in the same direction as

00:13:00.530 --> 00:13:03.980
this, then there would be a
solution, where you could just

00:13:03.980 --> 00:13:04.790
scale those.

00:13:04.790 --> 00:13:08.100
Actually, there would be an
infinite number of solutions

00:13:08.100 --> 00:13:09.500
in terms of x and y.

00:13:09.500 --> 00:13:13.580
But if the vector is slightly
different, in terms of its

00:13:13.580 --> 00:13:15.320
direction, then there
is no solution.

00:13:15.320 --> 00:13:18.310
There is no combination of this
vector and this vector

00:13:18.310 --> 00:13:19.980
that can add you
up to this one.

00:13:19.980 --> 00:13:22.385
And it's something for you
think about a little bit.

00:13:22.385 --> 00:13:23.360
It might be obvious to you.

00:13:23.360 --> 00:13:24.730
But another way to think about
it is, when you're trying to

00:13:24.730 --> 00:13:29.000
take sums of vectors, any other
vector, in order to move

00:13:29.000 --> 00:13:31.150
it in that direction, you have
to have a little bit of one

00:13:31.150 --> 00:13:33.470
direction and a little bit of
another direction, to get to

00:13:33.470 --> 00:13:34.230
any other vector.

00:13:34.230 --> 00:13:36.520
And if both of your ingredient
vectors are the same

00:13:36.520 --> 00:13:39.240
direction, there's no way to
get to a different one.

00:13:39.240 --> 00:13:42.450
Anyway, I'm probably just being
circular in what I'm

00:13:42.450 --> 00:13:43.240
explaining.

00:13:43.240 --> 00:13:48.150
But that hopefully gives you a
little bit of an intuition of

00:13:48.150 --> 00:13:50.800
well, one, you now know what
a singular matrix is.

00:13:50.800 --> 00:13:57.850
You know when you can not
find its inverse.

00:13:57.850 --> 00:14:01.320
You know that when the
determinant is 0, you won't

00:14:01.320 --> 00:14:01.800
find an inverse.

00:14:01.800 --> 00:14:03.730
And hopefully-- and this was
the whole point of this

00:14:03.730 --> 00:14:07.870
video-- you have an intuition
of why that is.

00:14:07.870 --> 00:14:10.050
Because if you're looking at the
vector problem, there's no

00:14:10.050 --> 00:14:12.830
way that you can find-- that
either there's no solution to

00:14:12.830 --> 00:14:14.650
finding the combination of the
vectors that get you to that

00:14:14.650 --> 00:14:16.200
vector, or there are
an infinite number.

00:14:16.200 --> 00:14:18.030
And the same thing is
true of finding the

00:14:18.030 --> 00:14:19.090
intersection of two lines.

00:14:19.090 --> 00:14:21.480
They're either parallel, or
they're the same line, if the

00:14:21.480 --> 00:14:22.640
determinant is 0.

00:14:22.640 --> 00:14:25.850
Anyway, I will see you
in the next video.


Matrices to solve a vector combination problem

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:04.280
In the last video we saw how a
matrix and figuring out its

00:00:04.280 --> 00:00:06.730
inverse can be used to solve
a system of equations.

00:00:06.730 --> 00:00:07.790
And we did a 2 by 2.

00:00:07.790 --> 00:00:09.200
And in the future, we'll
do 3 by 3's.

00:00:09.200 --> 00:00:11.170
We won't do 4 by 4's because
those take too long.

00:00:11.170 --> 00:00:13.740
But you'll see it applies to
kind of an n by n matrix.

00:00:13.740 --> 00:00:17.310
And that is probably the
application of matrices that

00:00:17.310 --> 00:00:20.420
you learn if you learn this in
your Algebra 2 class, or your

00:00:20.420 --> 00:00:22.190
Algebra 1 class.

00:00:22.190 --> 00:00:23.720
And you often wonder,
well why even do the

00:00:23.720 --> 00:00:25.120
whole matrix thing?

00:00:25.120 --> 00:00:28.570
Now I will show you another
application of matrices that

00:00:28.570 --> 00:00:31.920
actually you're more likely to
see in your linear algebra

00:00:31.920 --> 00:00:34.200
class when you take
it in college.

00:00:34.200 --> 00:00:37.030
But the really neat thing here
is, and I think this will

00:00:37.030 --> 00:00:40.220
really hit the point home, that
the matrix representation

00:00:40.220 --> 00:00:43.360
is just one way of representing
multiple types of

00:00:43.360 --> 00:00:45.550
problems. And what's really
cool is that if different

00:00:45.550 --> 00:00:47.590
problems can be represented the
same way, it kind of tells

00:00:47.590 --> 00:00:48.920
you that they're the
same problem.

00:00:48.920 --> 00:00:50.550
And that's called an isomorphism
in math.

00:00:50.550 --> 00:00:53.240
That if you can reduce one
problem into another problem,

00:00:53.240 --> 00:00:54.450
then all the work you
did with one of them

00:00:54.450 --> 00:00:55.570
applies to the other.

00:00:55.570 --> 00:00:58.330
But anyway, let's figure
out a new way that

00:00:58.330 --> 00:01:00.580
matrices can be used.

00:01:00.580 --> 00:01:03.120
So I'm going to draw
some vectors.

00:01:03.120 --> 00:01:06.880
Let's say I have the vector--
Let's call this vector a.

00:01:12.920 --> 00:01:14.920
And I'm going to just write this
is as a column vector.

00:01:14.920 --> 00:01:16.780
And all of this is
just convention.

00:01:16.780 --> 00:01:18.080
These are just human
invented things.

00:01:18.080 --> 00:01:19.320
I could have written
this diagonally.

00:01:19.320 --> 00:01:20.640
I could have written
this however.

00:01:20.640 --> 00:01:31.120
But if I say vector a
is 3, negative 6.

00:01:31.120 --> 00:01:33.265
And I view this as the x
component of the vector, and

00:01:33.265 --> 00:01:35.330
this is equal to the y component
of the vector.

00:01:35.330 --> 00:01:36.690
And then I have vector b.

00:01:41.920 --> 00:01:57.810
Vector b is equal to 2, 6.

00:01:57.810 --> 00:02:04.330
And I want to know are there
some combinations of vectors a

00:02:04.330 --> 00:02:08.949
and b-- where you can say, 5
times vector a, plus 3 times

00:02:08.949 --> 00:02:13.350
vector b, or 10 times victor a
minus 6 times vector b-- some

00:02:13.350 --> 00:02:19.420
combination of vector a and b,
where I can get vector c.

00:02:19.420 --> 00:02:31.250
And vector c is the
vector 7, 6.

00:02:31.250 --> 00:02:33.205
So let me see if I can visually
draw this problem.

00:02:33.205 --> 00:02:35.560
So let me draw the
coordinate axes.

00:02:35.560 --> 00:02:36.300
Let's see this one.

00:02:36.300 --> 00:02:37.490
3, negative 6.

00:02:37.490 --> 00:02:39.440
That'll be in quadrant--
these are both

00:02:39.440 --> 00:02:40.180
in the first quadrant.

00:02:40.180 --> 00:02:42.840
So I just want to figure
out how much of the

00:02:42.840 --> 00:02:43.900
axes I need to draw.

00:02:43.900 --> 00:02:47.580
So let's see-- Let me do
a different color.

00:02:52.640 --> 00:02:53.670
That's my y-axis.

00:02:53.670 --> 00:02:56.160
I'm not drawing the second or
third quadrants, because I

00:02:56.160 --> 00:02:58.140
don't think our vectors
show up there.

00:02:58.140 --> 00:03:01.385
And then this is the x-axis.

00:03:03.910 --> 00:03:05.365
Let me draw each of
these vectors.

00:03:08.270 --> 00:03:11.130
So first I'll do vector a.

00:03:11.130 --> 00:03:13.310
That's 3, negative 6.

00:03:13.310 --> 00:03:17.770
1, 2, 3, and then negative 6.

00:03:17.770 --> 00:03:24.170
1, 2, 3, 4, 5, 6.

00:03:24.170 --> 00:03:26.970
So it's there.

00:03:26.970 --> 00:03:31.010
So if I wanted to draw it
as a vector, usually

00:03:31.010 --> 00:03:32.260
start at the origin.

00:03:34.894 --> 00:03:36.980
And it doesn't have to start
at the origin like that.

00:03:36.980 --> 00:03:37.730
I'm just choosing to.

00:03:37.730 --> 00:03:39.990
You can move around a vector.

00:03:39.990 --> 00:03:42.190
It just has to have the
same orientation

00:03:42.190 --> 00:03:45.680
and the same magnitude.

00:03:45.680 --> 00:03:52.620
So that is vector
a for the green.

00:03:52.620 --> 00:03:56.670
Now let me do in magenta,
I'll do vector b.

00:03:56.670 --> 00:03:59.050
That is 2, 6.

00:03:59.050 --> 00:04:04.280
1, 2, 3, 4, 5, 6.

00:04:04.280 --> 00:04:12.630
So 2, 6 is right over there.

00:04:12.630 --> 00:04:14.740
And that's vector b.

00:04:22.160 --> 00:04:25.300
So it'll look like this.

00:04:25.300 --> 00:04:27.370
That's vector b.

00:04:27.370 --> 00:04:30.630
And then let me write down
vector a down there.

00:04:30.630 --> 00:04:32.790
That's vector a.

00:04:32.790 --> 00:04:35.210
And I want to take some
combination of

00:04:35.210 --> 00:04:36.720
vectors a and b.

00:04:36.720 --> 00:04:38.400
And add them up and
get vector c.

00:04:38.400 --> 00:04:39.690
So what does vector
c look like?

00:04:39.690 --> 00:04:41.620
It's 7, 6.

00:04:41.620 --> 00:04:43.290
Let me do that in purple.

00:04:43.290 --> 00:04:50.390
So 1, 2, 3, 4, 5, 6, 7.

00:04:50.390 --> 00:04:52.240
Comma 6.

00:04:52.240 --> 00:04:55.940
So 7, 6 is right over there.

00:04:55.940 --> 00:04:57.190
That's vector c.

00:05:00.400 --> 00:05:02.145
Vector c looks like that.

00:05:09.000 --> 00:05:10.730
I'm going to draw
it like that.

00:05:10.730 --> 00:05:14.640
And that's vector c.

00:05:14.640 --> 00:05:16.010
So what was the original
problem I said?

00:05:16.010 --> 00:05:19.460
I said I want to add some
multiple of vector a to some

00:05:19.460 --> 00:05:21.890
multiple of vector b,
and get vector c.

00:05:21.890 --> 00:05:24.000
And I want to see what
those multiples are.

00:05:24.000 --> 00:05:25.650
So let's say the multiple
that I multiply

00:05:25.650 --> 00:05:27.650
times vector a is x.

00:05:27.650 --> 00:05:29.210
And the multiple of
vector b is y.

00:05:29.210 --> 00:05:33.380
So I essentially want to say
that-- let me do it in another

00:05:33.380 --> 00:05:42.330
neutral color-- that vector ax--
that's how much of vector

00:05:42.330 --> 00:05:49.740
a I'm contributing-- plus vector
by-- that's how much of

00:05:49.740 --> 00:05:54.920
vector b I'm contributing--
is equal to vector c.

00:05:54.920 --> 00:05:56.200
And you know, maybe I can't.

00:05:56.200 --> 00:05:59.470
Maybe there's no combinations of
vector a and b when you add

00:05:59.470 --> 00:06:00.830
them together equal vector c.

00:06:00.830 --> 00:06:03.120
But let's see if we
can solve this.

00:06:03.120 --> 00:06:03.830
So how do we solve?

00:06:03.830 --> 00:06:05.900
So let's expand out
vectors a and b.

00:06:05.900 --> 00:06:06.710
Vector a is what?

00:06:06.710 --> 00:06:07.860
3, negative 6.

00:06:07.860 --> 00:06:17.210
So vector a, we could write
as 3, minus 6 times x.

00:06:17.210 --> 00:06:20.120
That just tells us how much
vector a we're contributing.

00:06:20.120 --> 00:06:22.120
Plus vector b, which is 2, 6.

00:06:26.790 --> 00:06:29.240
And then y is how much vector
b we're contributing.

00:06:29.240 --> 00:06:33.960
And that is equal to 7, 6.

00:06:33.960 --> 00:06:36.920
Vector c.

00:06:36.920 --> 00:06:41.440
Now this right here, this
problem can be rewritten just

00:06:41.440 --> 00:06:43.990
based on how we've defined
matrix multiplication, et

00:06:43.990 --> 00:06:47.150
cetera, et cetera, as this.

00:06:47.150 --> 00:07:07.780
As 3, minus 6, 2, 6, times
x, y, is equal to 7, 6.

00:07:07.780 --> 00:07:09.150
Now how does that work out?

00:07:09.150 --> 00:07:11.850
Well think about how matrix
multiplication works out.

00:07:11.850 --> 00:07:16.530
The way we learned matrix
multiplication, we said, 3

00:07:16.530 --> 00:07:25.250
times x, plus 2 times
y is equal to 7.

00:07:25.250 --> 00:07:28.740
3 times x plus 2 times
y is equal to 7.

00:07:28.740 --> 00:07:30.800
That's how we learned matrix
multiplication.

00:07:30.800 --> 00:07:31.700
That's the same thing here.

00:07:31.700 --> 00:07:37.170
3 times x, plus 2 times y, is
going to be equal to 7.

00:07:37.170 --> 00:07:39.270
These x and y here are
just scalar numbers.

00:07:39.270 --> 00:07:41.380
So 3 times x plus 2 times
y is equal to 7.

00:07:41.380 --> 00:07:45.630
And then matrix multiplication
here, minus 6 times x plus 6

00:07:45.630 --> 00:07:47.030
times y is equal to 6.

00:07:47.030 --> 00:07:49.080
That's just traditional matrix
multiplication that we learned

00:07:49.080 --> 00:07:50.330
several videos ago.

00:07:50.330 --> 00:07:51.120
That's the same thing here.

00:07:51.120 --> 00:07:54.390
Minus 6x plus 6y
is equal to 6.

00:07:54.390 --> 00:07:56.560
These x's and y's are
just numbers.

00:07:56.560 --> 00:07:57.410
They're just scalar numbers.

00:07:57.410 --> 00:07:58.380
They're not vectors
or anything.

00:07:58.380 --> 00:08:01.640
We would just multiply them
times both of these numbers.

00:08:01.640 --> 00:08:03.710
So hopefully you see that this
problem is the exact same

00:08:03.710 --> 00:08:04.580
thing as this problem.

00:08:04.580 --> 00:08:08.030
And you've maybe had an a-ha
moment now, if you watched the

00:08:08.030 --> 00:08:08.700
previous video.

00:08:08.700 --> 00:08:16.070
Because this matrix also
represented the problem, where

00:08:16.070 --> 00:08:18.210
do we find the intersection
of two lines?

00:08:18.210 --> 00:08:20.220
Where the two lines-- I'm just
going to do it on the side

00:08:20.220 --> 00:08:24.290
here-- the intersection of
the two lines, 3x plus

00:08:24.290 --> 00:08:26.210
2y is equal to 7.

00:08:26.210 --> 00:08:31.800
And minus 6x plus 6y
is equal to 6.

00:08:31.800 --> 00:08:33.409
And so, I had drawn two lines.

00:08:33.409 --> 00:08:34.900
And we said, what's the
point of intersection,

00:08:34.900 --> 00:08:35.600
et cetera, et cetera.

00:08:35.600 --> 00:08:37.539
And it was represented
by this problem.

00:08:37.539 --> 00:08:39.860
But here, we have-- well I
won't say a completely

00:08:39.860 --> 00:08:41.140
different problem, because
we're learning they're

00:08:41.140 --> 00:08:43.580
actually very similar-- but here
I'm doing a problem of,

00:08:43.580 --> 00:08:46.870
I'm trying to find what
combination of the matrices a

00:08:46.870 --> 00:08:49.570
and b add up to the matrix c.

00:08:49.570 --> 00:08:53.340
But it got reduced to the same
matrix representation.

00:08:53.340 --> 00:08:56.430
And so we can solve this the
same exactly way we solved

00:08:56.430 --> 00:08:57.750
this problem.

00:08:57.750 --> 00:09:02.570
If we call this the matrix a,
let's figure out a inverse.

00:09:02.570 --> 00:09:06.190
So we get a inverse
is equal to what?

00:09:06.190 --> 00:09:08.222
It equals 1 over the
determinant of a.

00:09:08.222 --> 00:09:10.740
The determinant of
a is 3 times 6.

00:09:10.740 --> 00:09:13.850
18 minus minus 12.

00:09:13.850 --> 00:09:18.560
So that's 18 plus 12,
which is 1/30.

00:09:18.560 --> 00:09:19.850
And we did this in the
previous video.

00:09:19.850 --> 00:09:21.170
You swap these two numbers.

00:09:21.170 --> 00:09:23.150
So you get 6 and 3.

00:09:23.150 --> 00:09:25.120
And then you make these
two negatives.

00:09:25.120 --> 00:09:28.600
So you get 6 and minus 2.

00:09:28.600 --> 00:09:29.730
That's a inverse.

00:09:29.730 --> 00:09:32.590
And now to solve for x and y, we
can multiply both sides of

00:09:32.590 --> 00:09:34.210
this equation by a inverse.

00:09:34.210 --> 00:09:37.340
If you multiply a inverse times
a, this cancels out.

00:09:37.340 --> 00:09:45.790
So you get x, y is equal to
a inverse times this.

00:09:45.790 --> 00:09:53.740
It's equal to 1/30 times
6, minus 2, 6, 3.

00:09:53.740 --> 00:09:55.330
Times 7, 6.

00:09:55.330 --> 00:09:58.250
And remember, with matrices,
the order that

00:09:58.250 --> 00:09:59.830
you multiply matters.

00:09:59.830 --> 00:10:01.830
So on this side, we multiplied
a inverse on

00:10:01.830 --> 00:10:03.250
this side of the equation.

00:10:03.250 --> 00:10:06.580
So we have to do a inverse on
the left side on this side of

00:10:06.580 --> 00:10:07.530
this equation.

00:10:07.530 --> 00:10:08.360
So that's why did it here.

00:10:08.360 --> 00:10:11.570
If we did it the other way,
all bets are off.

00:10:11.570 --> 00:10:13.310
So what does this equal?

00:10:13.310 --> 00:10:17.860
This is equal to 1/30 times--
and we did this the previous

00:10:17.860 --> 00:10:21.060
problem-- 6 times 7
is 42, minus 12.

00:10:21.060 --> 00:10:22.250
30.

00:10:22.250 --> 00:10:23.410
6 times 7, 42.

00:10:23.410 --> 00:10:24.220
Plus 18.

00:10:24.220 --> 00:10:26.170
60.

00:10:26.170 --> 00:10:30.800
So that equals 1, 2.

00:10:30.800 --> 00:10:32.170
So what does this tell us?

00:10:32.170 --> 00:10:37.130
This tells us that if we have
1 times vector a, plus 2

00:10:37.130 --> 00:10:38.610
times vector b.

00:10:38.610 --> 00:10:43.360
1 times-- this is 1-- and
2 times vector b.

00:10:43.360 --> 00:10:46.420
So 1 times vector a plus
2 times vector b is

00:10:46.420 --> 00:10:47.685
equal to vector c.

00:10:47.685 --> 00:10:49.090
And let's confirm
that visually.

00:10:49.090 --> 00:10:50.820
So 1 times vector a.

00:10:50.820 --> 00:10:52.140
Well that's vector
a right there.

00:10:52.140 --> 00:10:55.410
So if we add 2 vector b's to
it, we should get vector c.

00:10:55.410 --> 00:10:57.300
So let's see if we
can do that.

00:10:57.300 --> 00:11:01.150
So if we just shift vector b
over this way, well vector

00:11:01.150 --> 00:11:03.330
let's see, vector b is
over 2 and up 6.

00:11:03.330 --> 00:11:08.020
So over 2 and up 6 would
get us there.

00:11:08.020 --> 00:11:12.230
So 1, vector b-- just doing
heads to tail visual method of

00:11:12.230 --> 00:11:16.240
adding vectors-- would
get us there.

00:11:16.240 --> 00:11:17.271
1, 2, 3.

00:11:17.271 --> 00:11:18.521
Good.

00:11:20.820 --> 00:11:23.250
No, let me see.

00:11:23.250 --> 00:11:26.780
1, 2, 3.

00:11:26.780 --> 00:11:30.340
And then vector b goes
over two more.

00:11:30.340 --> 00:11:33.140
two more.

00:11:33.140 --> 00:11:35.240
So it'll get us up 6.

00:11:35.240 --> 00:11:36.860
It's like that.

00:11:36.860 --> 00:11:40.410
So that's 1, vector b.

00:11:40.410 --> 00:11:43.550
And then if we add another-- but
we want 2 times vector b.

00:11:43.550 --> 00:11:45.180
We essentially need
two vector b's.

00:11:45.180 --> 00:11:47.820
So we had one, and then
we add another one.

00:11:47.820 --> 00:11:51.395
I think visually you see that
it does actually-- I didn't

00:11:51.395 --> 00:11:52.160
want to do it like that.

00:11:52.160 --> 00:11:55.240
I wanted to use the line
tool so it looks neat.

00:11:55.240 --> 00:11:56.830
So you add another vector b.

00:11:56.830 --> 00:11:59.840
And there you have it.

00:11:59.840 --> 00:12:01.320
That's a vector b.

00:12:01.320 --> 00:12:02.680
So it's 2 times vector b.

00:12:02.680 --> 00:12:05.530
So it's the same direction as
vector b, but it's two times

00:12:05.530 --> 00:12:06.270
the length.

00:12:06.270 --> 00:12:07.610
So we visually showed it.

00:12:07.610 --> 00:12:10.530
We solved it algabraically.

00:12:10.530 --> 00:12:13.990
But the real learning, and the
big real discovery of this

00:12:13.990 --> 00:12:18.730
whole video, is to show you that
the matrix representation

00:12:18.730 --> 00:12:23.720
can represent multiple different
problems. This was a

00:12:23.720 --> 00:12:26.210
finding the combinations
of a vector problem.

00:12:26.210 --> 00:12:28.360
And the previous one it
was figure out if

00:12:28.360 --> 00:12:30.090
two lines can intersect.

00:12:30.090 --> 00:12:32.420
But what it tells you is that
these two problems are

00:12:32.420 --> 00:12:33.980
connected in some deep way.

00:12:33.980 --> 00:12:37.830
That if we take the veneer of
reality, that underlying it,

00:12:37.830 --> 00:12:39.260
they are the same thing.

00:12:39.260 --> 00:12:41.890
And frankly, that's why math
is so interesting.

00:12:41.890 --> 00:12:45.480
Because when you realize that
two problems are really the

00:12:45.480 --> 00:12:48.540
same thing, it takes all of the
superficial human veneer

00:12:48.540 --> 00:12:49.780
away from things.

00:12:49.780 --> 00:12:52.280
Because our brains are kind of
wired to perceive the world in

00:12:52.280 --> 00:12:52.790
a certain way.

00:12:52.790 --> 00:12:56.330
But it tells us that there's
some fundamental truth,

00:12:56.330 --> 00:13:00.050
independent of our perception,
that is tying all of these

00:13:00.050 --> 00:13:01.350
different concepts together.

00:13:01.350 --> 00:13:03.800
But anyway, I don't want to
get all mystical on you.

00:13:03.800 --> 00:13:05.780
But if you do see
the mysticism in

00:13:05.780 --> 00:13:07.310
math, all the better.

00:13:07.310 --> 00:13:10.890
But hopefully you found that
pretty interesting.

00:13:10.890 --> 00:13:12.660
And actually, I know I'm going
over time, but I think this

00:13:12.660 --> 00:13:15.820
is-- A lot of people take linear
algebra, they learn how

00:13:15.820 --> 00:13:17.500
to do all of the things, and
they say, well what is the

00:13:17.500 --> 00:13:18.180
whole point of this?

00:13:18.180 --> 00:13:20.020
But this is kind of
an interesting

00:13:20.020 --> 00:13:22.250
thing to think about.

00:13:22.250 --> 00:13:27.070
We had this had vector a and
we had this vector b.

00:13:27.070 --> 00:13:29.310
And we were able to say, well
there's some combinations of

00:13:29.310 --> 00:13:31.600
the vectors a and b, that
when we added it up,

00:13:31.600 --> 00:13:34.400
we got vector c.

00:13:34.400 --> 00:13:38.250
So an interesting question is,
what are all the vectors that

00:13:38.250 --> 00:13:42.390
I could get to by adding
combinations of

00:13:42.390 --> 00:13:43.220
vectors a and b.

00:13:43.220 --> 00:13:44.600
Or adding or subtracting.

00:13:44.600 --> 00:13:46.280
Or you could say,
I could multiply

00:13:46.280 --> 00:13:47.090
them by negative numbers.

00:13:47.090 --> 00:13:47.730
But either way.

00:13:47.730 --> 00:13:51.260
What are all of the vectors
that I can get by taking

00:13:51.260 --> 00:13:53.490
linear combinations of
vectors a and b?

00:13:53.490 --> 00:13:57.600
And that's actually called the
vector space spanned by the

00:13:57.600 --> 00:13:58.360
vectors a and b.

00:13:58.360 --> 00:14:01.060
And we'll do more of that
in linear algebra.

00:14:01.060 --> 00:14:02.460
And here we're dealing
with a two

00:14:02.460 --> 00:14:04.450
dimensional Euclidean space.

00:14:04.450 --> 00:14:06.160
We could have had three
dimensional vectors.

00:14:06.160 --> 00:14:07.740
We could've had n dimensional
vectors.

00:14:07.740 --> 00:14:09.990
So it gets really, really,
really abstract.

00:14:09.990 --> 00:14:14.390
But this is, I think, a really
good toe dipping for linear

00:14:14.390 --> 00:14:14.980
algebra as well.

00:14:14.980 --> 00:14:17.110
So hopefully I haven't confused
or overwhelmed you.

00:14:17.110 --> 00:14:19.430
And I'll see you in
the next video.