Representing systems of any number of equations with matrices

WEBVTT
Kind: captions
Language: en

00:00:00.170 --> 00:00:01.070
- [Instructor] In a previous video,

00:00:01.070 --> 00:00:02.240
we saw that if you have a system

00:00:02.240 --> 00:00:04.900
of three equations with
three unknowns like this,

00:00:04.900 --> 00:00:08.360
you can represent it as
a matrix vector equation,

00:00:08.360 --> 00:00:10.520
where this matrix right over here

00:00:10.520 --> 00:00:13.740
is a three-by-three matrix.

00:00:13.740 --> 00:00:16.180
That is essentially a coefficient matrix.

00:00:16.180 --> 00:00:18.300
It has all of the coefficients of the Xs,

00:00:18.300 --> 00:00:21.350
the Ys, and the Zs as its various columns.

00:00:21.350 --> 00:00:24.660
and then you're going to
multiply that times this vector,

00:00:24.660 --> 00:00:27.070
which is really the vector
of the unknown variables,

00:00:27.070 --> 00:00:29.570
and this is a three-by-one vector.

00:00:29.570 --> 00:00:30.830
And then you would result

00:00:30.830 --> 00:00:35.450
in this other three-by-one
vector, which is a vector

00:00:35.450 --> 00:00:40.050
that contains these constant
terms right over here.

00:00:40.050 --> 00:00:42.160
What we're gonna do in
this video is recognize

00:00:42.160 --> 00:00:44.570
that you can generalize this phenomenon.

00:00:44.570 --> 00:00:46.340
It's not just true with a system

00:00:46.340 --> 00:00:48.580
of three equations with three unknowns.

00:00:48.580 --> 00:00:52.490
It actually generalizes to
N equations with N unknowns.

00:00:52.490 --> 00:00:55.600
But just to appreciate that
that is indeed the case,

00:00:55.600 --> 00:00:59.610
let us look at a system of two
equations with two unknowns.

00:00:59.610 --> 00:01:04.020
So let's say you had 2x
plus y is equal to nine,

00:01:04.020 --> 00:01:09.020
and we had 3x minus y is equal to five.

00:01:09.230 --> 00:01:10.630
I encourage you, pause this video

00:01:10.630 --> 00:01:12.610
and think about how that
would be represented

00:01:12.610 --> 00:01:15.253
as a matrix vector equation.

00:01:16.520 --> 00:01:19.520
All right, now let's
work on this together.

00:01:19.520 --> 00:01:23.710
So this is a system of two
equations with two unknowns.

00:01:23.710 --> 00:01:27.010
So the matrix that represents
the coefficients is going

00:01:27.010 --> 00:01:29.390
to be a two-by-two matrix

00:01:29.390 --> 00:01:31.680
and then that's going to be multiplied

00:01:31.680 --> 00:01:35.200
by a vector that represents
the unknown variables.

00:01:35.200 --> 00:01:37.090
We have two unknown variables over here.

00:01:37.090 --> 00:01:40.120
So this is going to be
a two-by-one vector,

00:01:40.120 --> 00:01:42.910
and then that's going
to be equal to a vector

00:01:42.910 --> 00:01:45.520
that represents the constants
on the right-hand side,

00:01:45.520 --> 00:01:46.810
and obviously we have two of those.

00:01:46.810 --> 00:01:49.430
So that's going to be a
two-by-one vector as well.

00:01:49.430 --> 00:01:51.870
And then we can do exactly what we did

00:01:51.870 --> 00:01:54.410
in that previous example
in a previous video.

00:01:54.410 --> 00:01:58.850
The coefficients on the
X terms, two and three,

00:01:58.850 --> 00:02:02.570
and then we have the
coefficients on the Y terms.

00:02:02.570 --> 00:02:04.530
This would be a positive one

00:02:04.530 --> 00:02:06.930
and then this would be a negative one.

00:02:06.930 --> 00:02:09.570
And then we multiply it times the vector

00:02:09.570 --> 00:02:12.490
of the variables, X, Y,

00:02:12.490 --> 00:02:14.700
and then last but not least,

00:02:14.700 --> 00:02:19.200
you have this nine and this
five over here, nine and five.

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And I encourage you and multiply this out.

00:02:22.040 --> 00:02:25.390
Multiply this matrix times this vector.

00:02:25.390 --> 00:02:28.560
And when you do that and you
still set up this equality,

00:02:28.560 --> 00:02:30.620
you're going to see that
it essentially turns

00:02:30.620 --> 00:02:32.770
into this exact same system

00:02:32.770 --> 00:02:36.010
of two equations and two unknowns.

00:02:36.010 --> 00:02:37.650
Now, what's interesting about this is

00:02:37.650 --> 00:02:40.730
that we see a generalizable form.

00:02:40.730 --> 00:02:43.860
In general, you can represent a system

00:02:43.860 --> 00:02:47.890
of N equations and N unknowns in the form.

00:02:47.890 --> 00:02:51.740
Sum N-by-N matrix A,

00:02:51.740 --> 00:02:53.400
N by N,

00:02:53.400 --> 00:02:58.340
times sum N-by-one vector X.

00:02:58.340 --> 00:02:59.523
This isn't just the variable X.

00:02:59.523 --> 00:03:03.950
This is a vector X that
has N dimensions to it.

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So times sum N-by-one vector X is going

00:03:08.610 --> 00:03:13.530
to be equal to sum N-by-one vector B.

00:03:14.480 --> 00:03:17.360
These are the letters that
people use by convention.

00:03:17.360 --> 00:03:19.030
This is going to be N by one.

00:03:19.030 --> 00:03:21.890
And so you can see in
these different scenarios.

00:03:21.890 --> 00:03:24.850
In that first one, this is
a three-by-three matrix.

00:03:24.850 --> 00:03:26.870
We could call that A,

00:03:26.870 --> 00:03:29.580
and then we could call this the vector X,

00:03:29.580 --> 00:03:31.760
and then we could call this the vector B.

00:03:31.760 --> 00:03:32.850
Now in that second scenario,

00:03:32.850 --> 00:03:34.800
we could call this the matrix A,

00:03:34.800 --> 00:03:36.850
we could call this the vector X,

00:03:36.850 --> 00:03:39.320
and then we could call this the vector B,

00:03:39.320 --> 00:03:42.820
but we can generalize
that to N dimensions.

00:03:42.820 --> 00:03:44.760
And as I talked about
in the previous video,

00:03:44.760 --> 00:03:47.880
what's interesting about this
is you could think about,

00:03:47.880 --> 00:03:49.890
for example, in this
system of two equations

00:03:49.890 --> 00:03:52.300
with two unknowns, as all
right, I have a line here,

00:03:52.300 --> 00:03:53.490
I have a line here,

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and X and Y represent the
intersection of those lines.

00:03:57.640 --> 00:03:58.850
But when you represent it this way,

00:03:58.850 --> 00:04:00.580
you could also imagine it as saying,

00:04:00.580 --> 00:04:04.470
okay, I have some unknown
vector in the coordinate plane

00:04:04.470 --> 00:04:07.460
and I'm transforming it using this matrix

00:04:07.460 --> 00:04:10.030
to get this vector nine five.

00:04:10.030 --> 00:04:12.410
And so I have to figure out what vector,

00:04:12.410 --> 00:04:15.510
when transformed in this
way, gets us to nine five,

00:04:15.510 --> 00:04:18.160
and we also thought about it
in the three-by-three case.

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What three-dimensional vector,
when transformed in this way,

00:04:20.950 --> 00:04:23.360
gets us to this vector right over here?

00:04:23.360 --> 00:04:24.910
And so that hints,

00:04:24.910 --> 00:04:27.410
that foreshadows where
we might be able to go.

00:04:27.410 --> 00:04:30.530
If we can unwind this
transformation somehow,

00:04:30.530 --> 00:04:33.850
then we can figure out what
these unknown vectors are.

00:04:33.850 --> 00:04:37.270
And if we can do it in two
dimensions or three dimensions,

00:04:37.270 --> 00:04:40.010
why not be able to do it in N dimensions?

00:04:40.010 --> 00:04:41.980
Which you'll see is actually very useful

00:04:41.980 --> 00:04:43.650
if you ever become a data scientist,

00:04:43.650 --> 00:04:45.220
or you go into computer science,

00:04:45.220 --> 00:04:47.670
or if you go into computer
graphics of some kind.


Probability with permutations & combinations example: taste testing

WEBVTT
Kind: captions
Language: en

00:00:00.210 --> 00:00:02.080
- [Instructor] We're told
that Samara is setting up

00:00:02.080 --> 00:00:05.130
an olive tasting
competition for a festival.

00:00:05.130 --> 00:00:07.630
From 15 distinct varieties,

00:00:07.630 --> 00:00:11.160
Samara will choose three
different olive oils

00:00:11.160 --> 00:00:12.870
and blend them together.

00:00:12.870 --> 00:00:14.990
A contestant will taste the blend

00:00:14.990 --> 00:00:18.870
and try to identify which
three of the 15 varieties

00:00:18.870 --> 00:00:20.860
were used to make it.

00:00:20.860 --> 00:00:24.020
Assume that a contestant
can't taste any difference

00:00:24.020 --> 00:00:26.050
and is randomly guessing.

00:00:26.050 --> 00:00:27.560
What is the probability

00:00:27.560 --> 00:00:29.730
that a contestant correctly guesses

00:00:29.730 --> 00:00:33.090
which three varieties were used?

00:00:33.090 --> 00:00:35.840
So pause this video and see
if you can think about that.

00:00:35.840 --> 00:00:37.330
And if you can just come
up with the expression,

00:00:37.330 --> 00:00:38.640
you don't have to compute it.

00:00:38.640 --> 00:00:41.493
That is probably good enough,
at least for our purposes.

00:00:42.420 --> 00:00:45.420
All right, now let's work
through this together.

00:00:45.420 --> 00:00:48.040
So we know several things here.

00:00:48.040 --> 00:00:51.410
We have 15 distinct varieties

00:00:51.410 --> 00:00:55.860
and we are choosing
three of those varieties.

00:00:55.860 --> 00:00:57.080
And anytime we're talking

00:00:57.080 --> 00:00:59.470
about probability and combinatorics,

00:00:59.470 --> 00:01:02.230
it's always interesting to
say, "Does order matter?

00:01:02.230 --> 00:01:04.440
Does it matter what order that Samara

00:01:04.440 --> 00:01:07.590
is picking those three from the 15?"

00:01:07.590 --> 00:01:09.000
It doesn't look like it matters.

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It looks like we just have to think

00:01:10.610 --> 00:01:11.740
about what three they are.

00:01:11.740 --> 00:01:15.390
It doesn't matter what order
either she picked them in,

00:01:15.390 --> 00:01:18.410
or the order in which the
contestant guesses them in.

00:01:18.410 --> 00:01:20.290
And so if you think about the total number

00:01:20.290 --> 00:01:24.540
of ways of picking three
things from a group of 15,

00:01:24.540 --> 00:01:28.913
you could write that as 15, choose three.

00:01:30.700 --> 00:01:33.020
Once again, this is
just shorthand notation

00:01:33.020 --> 00:01:35.610
for how many combinations are there,

00:01:35.610 --> 00:01:38.630
so you can pick three
things from a group of 15?

00:01:38.630 --> 00:01:40.657
So some of you might
have been tempted to say,

00:01:40.657 --> 00:01:43.730
"Hey, let me think
about permutations here.

00:01:43.730 --> 00:01:46.340
And I have 15 things.

00:01:46.340 --> 00:01:49.180
And from that, I wanna
figure out how many ways

00:01:49.180 --> 00:01:54.160
can I pick three things that
actually has order mattering?"

00:01:54.160 --> 00:01:56.010
But this would be the
situation where we're talking

00:01:56.010 --> 00:01:58.960
about the contestant actually
having to maybe guess

00:01:58.960 --> 00:02:02.120
in the same order in which the varieties

00:02:02.120 --> 00:02:04.710
were originally blended,
or something like that,

00:02:04.710 --> 00:02:05.543
but we're not doing that,

00:02:05.543 --> 00:02:08.400
we just care about getting
the right three varieties.

00:02:08.400 --> 00:02:11.930
So this will tell us
the total number of ways

00:02:11.930 --> 00:02:14.930
that you can pick three out of 15.

00:02:14.930 --> 00:02:16.170
And so what's the probability

00:02:16.170 --> 00:02:17.870
that the contestant correctly guesses

00:02:17.870 --> 00:02:19.430
which three varieties were used?

00:02:19.430 --> 00:02:21.610
Well, the contestant is
going to be guessing one

00:02:21.610 --> 00:02:24.450
out of the possible
number of scenarios here.

00:02:24.450 --> 00:02:29.450
So the probability would be
one over 15, choose three.

00:02:30.200 --> 00:02:31.860
And if you wanted to compute this,

00:02:31.860 --> 00:02:35.310
this would be equal to one over,

00:02:35.310 --> 00:02:37.930
now, how many ways can you
pick three things from 15?

00:02:37.930 --> 00:02:39.530
And of course there is a formula here,

00:02:39.530 --> 00:02:41.470
but I always like to reason through it.

00:02:41.470 --> 00:02:44.450
Well, you could say, "All
right, if there's three slots,

00:02:44.450 --> 00:02:46.910
there's 15 different varieties

00:02:46.910 --> 00:02:49.260
that could've gone into that first slot,

00:02:49.260 --> 00:02:52.310
and then there's 14 that could
go into that second slot,

00:02:52.310 --> 00:02:55.720
and then there's 13 that can
go into that third slot."

00:02:55.720 --> 00:02:56.760
But then we have to remember

00:02:56.760 --> 00:02:59.600
that it doesn't matter
what order we pick them in.

00:02:59.600 --> 00:03:02.730
So how many ways can you
rearrange three things?

00:03:02.730 --> 00:03:04.700
Well, it would be three factorial,

00:03:04.700 --> 00:03:08.050
or three times two times one.

00:03:08.050 --> 00:03:12.580
So this would be the same thing
as three times two times one

00:03:12.580 --> 00:03:17.580
over 15 times 14 times 13.

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See, I can simplify this,

00:03:19.190 --> 00:03:22.330
divide numerator and denominator by two,

00:03:22.330 --> 00:03:25.850
divide numerator and denominator by three.

00:03:25.850 --> 00:03:30.423
This is going to be equal
to one over 35 times 13.

00:03:32.030 --> 00:03:37.030
This is going to be one over
350 plus 105, which is 455.

00:03:42.090 --> 00:03:44.103
And we are done.


Vector word problem: resultant force

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:01.730
- [Instructor] We're
told that a metal ball

00:00:01.730 --> 00:00:04.290
lies on a flat, horizontal surface.

00:00:04.290 --> 00:00:07.800
It is attracted by two
magnets placed around it.

00:00:07.800 --> 00:00:12.800
We're told that the first
magnets force on the ball is 5 N.

00:00:14.220 --> 00:00:17.700
We're then told the second
magnets force on the ball

00:00:17.700 --> 00:00:20.858
is 3 N in a direction

00:00:20.858 --> 00:00:25.650
that is 100 degree rotation
from the first magnets force.

00:00:25.650 --> 00:00:27.080
And we can see that drawn here.

00:00:27.080 --> 00:00:29.950
This is the first magnets force, it's 5 N.

00:00:29.950 --> 00:00:34.110
And then the second magnets force is 3 N

00:00:34.110 --> 00:00:37.130
at 100 degree angle, 100 degree rotation

00:00:37.130 --> 00:00:39.460
from the first magnets force.

00:00:39.460 --> 00:00:42.040
Now they're asking us a
few interesting questions.

00:00:42.040 --> 00:00:46.000
What is the combined strength
of the magnets' pulls?

00:00:46.000 --> 00:00:47.110
And then they also say,

00:00:47.110 --> 00:00:49.930
what is the direction of
the magnets' combined pulls,

00:00:49.930 --> 00:00:53.350
relative to the direction
of the first magnets pull?

00:00:53.350 --> 00:00:56.340
So I encourage you, pause this video

00:00:56.340 --> 00:00:58.250
and have a go at this on your own

00:00:58.250 --> 00:01:00.100
before we work through this together.

00:01:01.290 --> 00:01:03.440
All right, now let's work
through this together.

00:01:03.440 --> 00:01:05.120
So they're really saying,

00:01:05.120 --> 00:01:07.210
if I take the sum of these two vectors,

00:01:07.210 --> 00:01:09.450
what is gonna be the
resultant force vector?

00:01:09.450 --> 00:01:10.810
What is going to be the magnitude

00:01:10.810 --> 00:01:12.457
of that result in force vector?

00:01:12.457 --> 00:01:16.690
And what is its direction going to be?

00:01:16.690 --> 00:01:18.340
There's two ways we could approach this.

00:01:18.340 --> 00:01:20.440
We could break it down
each of these vectors

00:01:20.440 --> 00:01:22.260
into their respective components,

00:01:22.260 --> 00:01:24.330
and then add the respect of components,

00:01:24.330 --> 00:01:26.750
and then from that figure
out what the magnitude

00:01:26.750 --> 00:01:27.583
and direction is.

00:01:27.583 --> 00:01:29.160
And we do that in other videos,

00:01:29.160 --> 00:01:31.430
or we could take the geometric approach.

00:01:31.430 --> 00:01:32.680
So that's what we're gonna do here.

00:01:32.680 --> 00:01:34.030
And to help us with that,

00:01:34.030 --> 00:01:36.770
we're gonna use what we've
called the Parallelogram rule

00:01:36.770 --> 00:01:38.730
which is really the same idea

00:01:38.730 --> 00:01:42.030
as the head to tail addition of vectors.

00:01:42.030 --> 00:01:44.030
I can take the 3 N vector,

00:01:44.030 --> 00:01:45.510
I can shift it over,

00:01:45.510 --> 00:01:48.750
so its tails at the
head of the 5 N vector.

00:01:48.750 --> 00:01:51.930
It would look something like this.

00:01:51.930 --> 00:01:54.590
So this is 3 N right over there.

00:01:54.590 --> 00:01:57.100
And then I could also
go the other way around.

00:01:57.100 --> 00:01:59.000
I could take the 3 N vector first,

00:01:59.000 --> 00:02:01.330
and take the tail of the 5 N vector

00:02:01.330 --> 00:02:06.270
at the head of the 3 N vector
and shift it like this.

00:02:06.270 --> 00:02:08.230
You can add an either direction

00:02:09.100 --> 00:02:10.940
and either way you look at it,

00:02:10.940 --> 00:02:13.590
when you start at the tails
and you get to the head

00:02:13.590 --> 00:02:15.360
of the second vector,

00:02:15.360 --> 00:02:17.920
you're going to have a resultant force

00:02:17.920 --> 00:02:19.130
that looks like this,

00:02:19.130 --> 00:02:22.390
which is the diagonal
of this parallelogram.

00:02:22.390 --> 00:02:24.600
So there we go.

00:02:24.600 --> 00:02:28.900
And let me just call that our
force vector right over there.

00:02:28.900 --> 00:02:32.190
So if we can figure out
the length of this line

00:02:32.190 --> 00:02:33.680
of this diagonal right over here,

00:02:33.680 --> 00:02:37.200
that would be the magnitude
of this force vector.

00:02:37.200 --> 00:02:38.950
Now, how can we do that?

00:02:38.950 --> 00:02:40.290
Well, let's just think geometrically

00:02:40.290 --> 00:02:44.100
what else we can figure out
about what's going on over here?

00:02:44.100 --> 00:02:45.810
This is a parallelogram.

00:02:45.810 --> 00:02:49.000
So if this is 100 degree
angle right over here,

00:02:49.000 --> 00:02:53.850
this angle right over here is
also going to be 100 degree.

00:02:53.850 --> 00:02:57.420
And we also know that
these two opposite angles

00:02:57.420 --> 00:02:59.930
are also gonna have the same
measure right over here.

00:02:59.930 --> 00:03:02.850
And we also know that the
sum of all of the angles

00:03:02.850 --> 00:03:06.850
in a quadrilateral are
going to be 360 degrees.

00:03:06.850 --> 00:03:08.930
So these two make up 200 degrees,

00:03:08.930 --> 00:03:11.610
we have 160 degrees left
that have to be split

00:03:11.610 --> 00:03:13.290
between that one and that one.

00:03:13.290 --> 00:03:15.310
So we know that this is 80 degrees,

00:03:15.310 --> 00:03:17.700
and we know that this is 80 degrees.

00:03:17.700 --> 00:03:19.320
Well, how does that help us?

00:03:19.320 --> 00:03:21.730
So we know the length of this brown side,

00:03:21.730 --> 00:03:24.950
we know the length of
this side right over here.

00:03:24.950 --> 00:03:26.860
We know the angle between them,

00:03:26.860 --> 00:03:28.030
and what we're trying to do,

00:03:28.030 --> 00:03:31.310
is figure out the length of
the side opposite this angle,

00:03:31.310 --> 00:03:33.200
opposite this 80 degree angle.

00:03:33.200 --> 00:03:37.370
And some of you might remember
the Law of Cosines here.

00:03:37.370 --> 00:03:39.880
And the Law of Cosines I always imagine it

00:03:39.880 --> 00:03:42.950
as an adaptation of the
Pythagorean theorem,

00:03:42.950 --> 00:03:46.340
so that we can deal with
non-right triangles.

00:03:46.340 --> 00:03:49.080
And the Law of Cosines will tell us

00:03:49.080 --> 00:03:52.380
that the magnitude, I'll
just write it over here,

00:03:52.380 --> 00:03:54.770
the magnitude of this vector,

00:03:54.770 --> 00:03:57.130
which is the length of this diagonal,

00:03:57.130 --> 00:04:01.650
is going to be equal
to the square root of,

00:04:01.650 --> 00:04:04.830
we're going to have this side squared,

00:04:04.830 --> 00:04:07.910
so let me write 3 squared,

00:04:07.910 --> 00:04:09.730
plus this side squared,

00:04:09.730 --> 00:04:13.580
plus 5 squared minus

00:04:13.580 --> 00:04:15.863
2 times this side.

00:04:16.910 --> 00:04:19.800
So times 3 times that side.

00:04:19.800 --> 00:04:24.800
So times 5 times the cosine of 80 degrees.

00:04:27.620 --> 00:04:30.450
And so let's get our calculator
out to calculate that.

00:04:30.450 --> 00:04:34.510
I'll start with taking
the cosine of 80 degrees,

00:04:34.510 --> 00:04:38.370
then I'm just gonna multiply
that times looks like 30.

00:04:38.370 --> 00:04:41.890
So times 30 is equal to that.

00:04:41.890 --> 00:04:44.610
Let's put it a little negative there.

00:04:44.610 --> 00:04:49.610
And then to that, I'm going
to add 25 and 9, which is 34.

00:04:51.200 --> 00:04:55.280
So plus 34 is equal to that.

00:04:55.280 --> 00:04:57.380
And now I just take the
square root of all of that.

00:04:57.380 --> 00:04:59.770
And they tell us to round our
answer to the nearest tenths.

00:04:59.770 --> 00:05:03.960
So I can round this to approximately 5.4.

00:05:03.960 --> 00:05:08.423
So this is approximately 5.4 N.

00:05:10.343 --> 00:05:11.820
Now they say, what is the direction

00:05:11.820 --> 00:05:13.140
of the magnets combined poles

00:05:13.140 --> 00:05:16.470
relative to the direction
of the first magnets pole?

00:05:16.470 --> 00:05:18.070
So really what we want to do,

00:05:18.070 --> 00:05:21.340
is figure out this angle right over here,

00:05:21.340 --> 00:05:23.040
let's call that data.

00:05:23.040 --> 00:05:25.780
Well, we know what the length
of the side opposite is.

00:05:25.780 --> 00:05:29.060
So maybe we could use the Law of Sines.

00:05:29.060 --> 00:05:30.650
The Law of Sines would tell us

00:05:30.650 --> 00:05:33.630
that the sine of theta

00:05:33.630 --> 00:05:36.210
over the length of the side opposite to it

00:05:36.210 --> 00:05:39.630
is going to be equal to, let's
pick another angle we know,

00:05:39.630 --> 00:05:41.180
sine of this angle.

00:05:41.180 --> 00:05:44.310
Sine of 80 degrees

00:05:44.310 --> 00:05:47.100
over the length of the
side opposite to it.

00:05:47.100 --> 00:05:50.230
And so this is approximately 5.4.

00:05:50.230 --> 00:05:52.390
And so if we wanna solve a for theta,

00:05:52.390 --> 00:05:54.970
we can multiply both sides by 3.

00:05:54.970 --> 00:05:57.950
So we're going to get sine of theta,

00:05:57.950 --> 00:06:00.620
I'll just stay in this
purple color for simplicity,

00:06:00.620 --> 00:06:05.620
is equal to 3 times sine of 80 degrees

00:06:07.170 --> 00:06:09.510
divided by 5.4.

00:06:09.510 --> 00:06:11.580
And then we could say that theta

00:06:11.580 --> 00:06:16.450
is equal to the inverse sine
of all of this business.

00:06:16.450 --> 00:06:21.107
Three sine of 80 degrees over 5.4.

00:06:22.310 --> 00:06:26.380
So we're going to take 80
degrees, take the sine of it,

00:06:26.380 --> 00:06:29.860
we're going to multiply that by 3,

00:06:29.860 --> 00:06:33.740
divide that by 5.4, that equals that,

00:06:33.740 --> 00:06:37.780
and then I'm going to take the
inverse sine of all of that.

00:06:37.780 --> 00:06:41.210
And they want us to round
to the nearest integer.

00:06:41.210 --> 00:06:44.860
So that's approximately 33 degrees.

00:06:44.860 --> 00:06:46.220
When you do the Law of Sines,

00:06:46.220 --> 00:06:49.120
it's possible that you're also
dealing with an obtuse angle.

00:06:49.120 --> 00:06:51.070
And when you do all of
this, you get the acute one,

00:06:51.070 --> 00:06:53.010
and then you would have
to make an adjustment.

00:06:53.010 --> 00:06:54.640
But that's not what we're dealing here,

00:06:54.640 --> 00:06:58.020
so we know that this data is approximately

00:06:58.020 --> 00:06:59.840
equal to 33 degrees.

00:06:59.840 --> 00:07:02.340
So we know the magnitude of the force

00:07:02.340 --> 00:07:04.240
and we know that it forms an angle

00:07:04.240 --> 00:07:06.330
of approximately 33 degrees

00:07:06.330 --> 00:07:10.003
with the direction of the
force of that first magnet.


Vector word problem: resultant velocity

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:02.260
- [Instructor] We're
told a boat is traveling

00:00:02.260 --> 00:00:05.900
at a speed of 26 kilometers
per hour in a direction

00:00:05.900 --> 00:00:09.940
that is a 300 degree rotation from East.

00:00:09.940 --> 00:00:12.630
At a certain point it encounters a current

00:00:12.630 --> 00:00:16.070
at a speed of 15 kilometers
per hour in a direction

00:00:16.070 --> 00:00:19.720
that is a 25 degree rotation from East.

00:00:19.720 --> 00:00:22.380
Answer two questions
about the boat's velocity

00:00:22.380 --> 00:00:24.470
after it meets the current.

00:00:24.470 --> 00:00:25.850
Alright, the first question is,

00:00:25.850 --> 00:00:29.110
what is the boat's speed
after it meets the current?

00:00:29.110 --> 00:00:31.390
And it says, round your
answer to the nearest 10th.

00:00:31.390 --> 00:00:34.140
You can round intermediate
values to the nearest 100th.

00:00:34.140 --> 00:00:37.070
And what is the direction
of the boat's velocity

00:00:37.070 --> 00:00:38.670
after it meets the current?

00:00:38.670 --> 00:00:40.730
And they say the same, well,
they actually here it say,

00:00:40.730 --> 00:00:42.500
round your answer to the nearest integer

00:00:42.500 --> 00:00:45.570
and you can round intermediate
values to the nearest 100th.

00:00:45.570 --> 00:00:47.170
So like always pause this video

00:00:47.170 --> 00:00:49.020
and see if you can work through this.

00:00:50.070 --> 00:00:52.120
All right, now let's
work on this together.

00:00:52.120 --> 00:00:54.230
So first let's visualize
each of these vectors.

00:00:54.230 --> 00:00:57.610
We have this vector 26 kilometers
per hour in a direction

00:00:57.610 --> 00:01:00.270
that is a 300 degree rotation from East.

00:01:00.270 --> 00:01:04.110
And we have this vector
15 kilometers per hour

00:01:04.110 --> 00:01:08.210
in a direction that is a 25
degree rotation from East.

00:01:08.210 --> 00:01:12.120
And so let me draw some axes here.

00:01:12.120 --> 00:01:16.990
So let's say that is my Y-axis.

00:01:16.990 --> 00:01:21.990
And then let's say that
this over here is my X-axis.

00:01:22.000 --> 00:01:26.140
And then that first vector
300 degree rotation from East,

00:01:26.140 --> 00:01:28.660
East is in the positive X direction.

00:01:28.660 --> 00:01:33.170
This would be 90 degrees,
180 degrees, 270 degrees.

00:01:33.170 --> 00:01:35.390
I'm going counter-clockwise
cause that's the convention

00:01:35.390 --> 00:01:37.280
for a positive angle.

00:01:37.280 --> 00:01:40.490
And then we'd go a little bit past 270,

00:01:40.490 --> 00:01:44.810
we would go right, right over there.

00:01:44.810 --> 00:01:48.240
And the magnitude of this vector
is 26 kilometers per hour.

00:01:48.240 --> 00:01:50.970
I'll just write a 26 right over there.

00:01:50.970 --> 00:01:54.230
And then this other vector
which is the current

00:01:54.230 --> 00:01:56.060
15 kilometers per hour in a direction

00:01:56.060 --> 00:01:58.230
that is a 25 degree rotation from East.

00:01:58.230 --> 00:02:02.880
So 25 degree rotation might
be something like this

00:02:02.880 --> 00:02:04.340
and it's going to be shorter.

00:02:04.340 --> 00:02:06.720
It's 15 kilometers per hour.

00:02:06.720 --> 00:02:10.540
So, it's going to be
roughly about that long.

00:02:10.540 --> 00:02:12.780
I'm obviously just approximating it

00:02:12.780 --> 00:02:15.860
and I'll just write 15
there for its magnitude.

00:02:15.860 --> 00:02:19.410
So we can visualize what the boat's speed

00:02:19.410 --> 00:02:21.770
and direction it is after
it meets the current.

00:02:21.770 --> 00:02:24.350
It's going to be the sum
of these two vectors.

00:02:24.350 --> 00:02:27.480
And so if we wanted to
sum these two vectors

00:02:27.480 --> 00:02:30.640
we could put the tail of one
at the head of the other.

00:02:30.640 --> 00:02:33.440
And so let's shift this
blue vector down here.

00:02:33.440 --> 00:02:36.260
So it's at the head of the red vector.

00:02:36.260 --> 00:02:38.310
So it would be something like this.

00:02:38.310 --> 00:02:43.010
And so our resulting speed
after it meets the current

00:02:43.010 --> 00:02:44.790
would look something like this.

00:02:44.790 --> 00:02:48.360
We've seen this in many
other videos so far

00:02:48.360 --> 00:02:50.410
but we don't want to just
figure it out visually.

00:02:50.410 --> 00:02:52.640
We want to actually figure
out its actual speed

00:02:52.640 --> 00:02:54.430
which would be the
magnitude of this vector

00:02:54.430 --> 00:02:55.900
and its actual direction.

00:02:55.900 --> 00:02:57.030
So what is the angle?

00:02:57.030 --> 00:02:58.620
And we could say it as a positive angle.

00:02:58.620 --> 00:03:01.410
So what the rotation,
the positive rotation

00:03:01.410 --> 00:03:05.230
from the positive X-axis or from due East.

00:03:05.230 --> 00:03:07.530
So to do that, what I'm
going to do is represent each

00:03:07.530 --> 00:03:11.430
of our original vectors in
terms of their components.

00:03:11.430 --> 00:03:14.060
And so this red vector up here

00:03:14.060 --> 00:03:17.370
and we've done this multiple
times explaining the intuition.

00:03:17.370 --> 00:03:20.680
It's X component is
going to be its magnitude

00:03:20.680 --> 00:03:23.920
26 times the cosine of this angle,

00:03:23.920 --> 00:03:26.980
cosine of 300 degrees.

00:03:26.980 --> 00:03:31.290
And it's Y component is
going to be 26 times the sine

00:03:31.290 --> 00:03:33.280
of 300 degrees.

00:03:33.280 --> 00:03:35.760
If that's unfamiliar to you,
I encourage you to review it

00:03:35.760 --> 00:03:37.600
in other videos where we first introduced

00:03:37.600 --> 00:03:40.030
the notion of components,
it comes straight out

00:03:40.030 --> 00:03:43.930
of our unit circle
definition of trig functions.

00:03:43.930 --> 00:03:47.040
And similarly, this vector right over here

00:03:47.040 --> 00:03:49.710
it's X component is going
to be its magnitude times

00:03:49.710 --> 00:03:53.150
the cosine of 25 degrees.

00:03:53.150 --> 00:03:58.150
And it's Y component is
going to be 15 times the sine

00:03:58.250 --> 00:04:00.740
of 25 degrees.

00:04:00.740 --> 00:04:03.070
And now when we have
it expressed this way,

00:04:03.070 --> 00:04:05.870
if we want to have the resulting vector,

00:04:05.870 --> 00:04:08.490
let's call the resulting vector S

00:04:08.490 --> 00:04:10.970
for maybe the resulting speed.

00:04:10.970 --> 00:04:13.970
Its components are going to
be the sum of each of these.

00:04:13.970 --> 00:04:16.120
So we can write it over here.

00:04:16.120 --> 00:04:19.860
Vector S is going to be equal to

00:04:19.860 --> 00:04:22.550
it's going to be the X
component of this red vector

00:04:22.550 --> 00:04:23.970
of our original speed vector.

00:04:23.970 --> 00:04:28.880
So, 26 cosine of 300 degrees

00:04:28.880 --> 00:04:31.710
plus the X component of the current.

00:04:31.710 --> 00:04:36.710
So, 15 times cosine of 25 degrees
and then the Y components.

00:04:37.340 --> 00:04:40.230
Once again, I add the
corresponding Y components

00:04:40.230 --> 00:04:45.080
26 sine of 300 degrees

00:04:45.080 --> 00:04:50.080
plus 15 sine of 25 degrees.

00:04:50.290 --> 00:04:52.170
And now we could use a
calculator to figure out

00:04:52.170 --> 00:04:55.170
what these are, to say what
these approximately are.

00:04:55.170 --> 00:04:57.780
So first the X component,
we're going to take

00:04:57.780 --> 00:05:02.513
the cosine of 300 degrees, times 26,

00:05:03.380 --> 00:05:06.400
plus I'll open a parenthesis here.

00:05:06.400 --> 00:05:11.343
We're going to take the cosine
of 25 degrees, times 15,

00:05:12.280 --> 00:05:14.210
close our parentheses.

00:05:14.210 --> 00:05:19.210
And that is equal to 26.59 if
I round to the nearest 100th.

00:05:20.689 --> 00:05:23.960
26.59.

00:05:23.960 --> 00:05:25.960
And now let's do the Y component.

00:05:25.960 --> 00:05:30.960
We have the sine of 300 degrees, times 26,

00:05:32.190 --> 00:05:34.960
plus I'll open parentheses,

00:05:34.960 --> 00:05:39.960
the sine of 25 degrees
times 15, close parentheses,

00:05:40.480 --> 00:05:45.480
is equal to negative 16.18 to
round to the nearest 100th.

00:05:45.720 --> 00:05:48.690
Negative 16.18.

00:05:48.690 --> 00:05:52.200
And let's just make sure that
this makes intuitive sense.

00:05:52.200 --> 00:05:54.460
So, 26.59.

00:05:54.460 --> 00:05:56.610
So we're going to go
forward in this direction

00:05:56.610 --> 00:05:59.340
26.59 on the X direction.

00:05:59.340 --> 00:06:03.340
And then we go negative
16.18 in the Y direction.

00:06:03.340 --> 00:06:06.890
So this does seem to match our intuition

00:06:06.890 --> 00:06:09.260
when we tried to look at this visually.

00:06:09.260 --> 00:06:11.650
So we now have the X and Y components

00:06:11.650 --> 00:06:13.410
of the resulting vector

00:06:13.410 --> 00:06:14.720
but that's not what they're asking for.

00:06:14.720 --> 00:06:17.500
They're asking for the speed
which would be the magnitude

00:06:17.500 --> 00:06:19.670
of this vector right over here.

00:06:19.670 --> 00:06:24.430
And so I could write the
magnitude of that vector

00:06:24.430 --> 00:06:26.360
which is going to be its speed.

00:06:26.360 --> 00:06:28.510
We'll just use the
Pythagorean theorem here.

00:06:28.510 --> 00:06:31.750
It is going to be the
square root of this squared

00:06:31.750 --> 00:06:33.620
plus this squared, because once again

00:06:33.620 --> 00:06:35.100
this forms a right triangle here.

00:06:35.100 --> 00:06:37.340
And we review this in other videos,

00:06:37.340 --> 00:06:41.457
it's going to be the square
root of 26.59 squared

00:06:42.780 --> 00:06:47.780
plus negative 16.18 squared
which is approximately equal to,

00:06:48.810 --> 00:06:50.706
they want us to round to the nearest 10th,

00:06:50.706 --> 00:06:55.706
26.59 squared plus.

00:06:57.070 --> 00:06:58.600
And it doesn't matter that
there's a negative here

00:06:58.600 --> 00:06:59.433
cause I'm squaring it.

00:06:59.433 --> 00:07:04.370
So I'll just write 16.18
squared is equal to that.

00:07:06.770 --> 00:07:09.500
And then we want to take
the square root of that.

00:07:09.500 --> 00:07:13.693
We get 31 point, if we round
to the nearest 10th, 31.1.

00:07:15.670 --> 00:07:20.360
So it's approximately 31.1 and
we'll write the units here,

00:07:20.360 --> 00:07:25.240
kilometers per hour is
the speed the boat's speed

00:07:25.240 --> 00:07:27.340
after it meets the current.

00:07:27.340 --> 00:07:29.720
And now the second question
is what is the direction

00:07:29.720 --> 00:07:33.500
of the boat's velocity
after it meets the current?

00:07:33.500 --> 00:07:35.870
Well, one way to think about it is

00:07:35.870 --> 00:07:38.250
if we look at this angle right over here

00:07:38.250 --> 00:07:40.360
which would tell us the direction

00:07:40.360 --> 00:07:44.220
the tangent of that angle,
theta, let me write this down.

00:07:44.220 --> 00:07:47.070
Tangent of that angle theta.

00:07:47.070 --> 00:07:48.950
We know your tangent is your change in Y

00:07:48.950 --> 00:07:50.050
over your change in X.

00:07:50.050 --> 00:07:51.660
You can even view it as the slope

00:07:51.660 --> 00:07:53.570
of this vector right over here.

00:07:53.570 --> 00:07:56.130
We know what our changes in X or Y are.

00:07:56.130 --> 00:07:58.100
Those are X and Y components.

00:07:58.100 --> 00:07:59.460
So it's going to be our change in Y

00:07:59.460 --> 00:08:04.063
which is negative 16.18, over 26.59,

00:08:06.870 --> 00:08:08.320
our change in X.

00:08:08.320 --> 00:08:10.520
And so to solve for theta, we could say

00:08:10.520 --> 00:08:15.320
that theta will be equal
to the inverse tangent.

00:08:15.320 --> 00:08:16.930
And we'll have to think
about this for a second

00:08:16.930 --> 00:08:19.650
because this might not get us
the exact theta that we want

00:08:19.650 --> 00:08:21.160
because the inverse tangent function

00:08:21.160 --> 00:08:24.030
is going to give us something
between positive 90 degrees

00:08:24.030 --> 00:08:25.830
and negative 90 degrees.

00:08:25.830 --> 00:08:27.740
But the number we want, actually it looks

00:08:27.740 --> 00:08:31.811
like it's going to be
between 270 and 360 degrees

00:08:31.811 --> 00:08:33.650
because we're doing a,

00:08:33.650 --> 00:08:35.630
we want to think about a positive rotation

00:08:35.630 --> 00:08:38.380
instead of a negative one but
let's just try to evaluate it.

00:08:38.380 --> 00:08:40.530
The inverse tan of this,

00:08:40.530 --> 00:08:45.073
of negative 16.18, over 26.59.

00:08:47.284 --> 00:08:50.010
16.18 negative

00:08:50.910 --> 00:08:55.910
divided by 26.59 is equal to this.

00:08:57.100 --> 00:09:00.840
And now I am going to take
the inverse tangent of that.

00:09:00.840 --> 00:09:05.690
And that gets us negative 31
degrees, which makes sense.

00:09:05.690 --> 00:09:06.740
This looks intuitive sense

00:09:06.740 --> 00:09:09.230
that if you were to do
a clockwise rotation

00:09:09.230 --> 00:09:11.660
which would be a negative
angle from the positive X-axis

00:09:11.660 --> 00:09:14.290
it looks like what we
drew, but let's just go

00:09:14.290 --> 00:09:15.880
with the convention of
everything else here.

00:09:15.880 --> 00:09:17.680
And let's try to have a positive angle.

00:09:17.680 --> 00:09:21.070
So what we can do is
add 360 degrees to that

00:09:21.070 --> 00:09:22.570
to make a full rotation around.

00:09:22.570 --> 00:09:24.710
And we essentially have
the equivalent angle.

00:09:24.710 --> 00:09:29.480
So let's add 360 to that to
get that right over there.

00:09:29.480 --> 00:09:31.170
So if we round to the nearest integer

00:09:31.170 --> 00:09:35.840
we're looking at
approximately 329 degrees.

00:09:35.840 --> 00:09:40.840
So theta is approximately 329 degrees.

00:09:41.110 --> 00:09:43.680
So here, when I said theta is
equal to this I could write

00:09:43.680 --> 00:09:48.680
theta is going to be equal
to this plus 360 degrees.

00:09:49.140 --> 00:09:52.110
Now what's interesting is, I
was able to add 360 degrees

00:09:52.110 --> 00:09:54.120
to get to the exact same place.

00:09:54.120 --> 00:09:57.350
If we had a situation where
our angle was actually

00:09:57.350 --> 00:09:59.310
this angle right over
here not the situation

00:09:59.310 --> 00:10:00.270
that we actually dealt with,

00:10:00.270 --> 00:10:02.170
but if it was in the second quadrant,

00:10:02.170 --> 00:10:04.070
we would have gotten this theta.

00:10:04.070 --> 00:10:06.510
And we would have had to
be able to realize that,

00:10:06.510 --> 00:10:08.030
hey we're dealing with the second quadrant

00:10:08.030 --> 00:10:09.420
that has the same slope.

00:10:09.420 --> 00:10:11.080
So instead of adding 360 degrees

00:10:11.080 --> 00:10:12.950
we would have added 180 degrees.

00:10:12.950 --> 00:10:16.153
And we've also covered that
in other videos as well.


Help Khan Academy supercharge learning

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:01.153
- Hi everyone.

00:00:01.153 --> 00:00:02.870
Sal Khan here from Khan Academy

00:00:02.870 --> 00:00:05.400
which you probably know
is a not-for-profit

00:00:05.400 --> 00:00:08.130
with a mission of providing
a free world-class education

00:00:08.130 --> 00:00:09.810
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And not for profit means
no one owns Khan Academy,

00:00:12.450 --> 00:00:13.560
we are a public charity.

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And the way that we're able

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00:00:24.450 --> 00:00:27.180
to tens of millions of
learners around the world is

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because of philanthropic donations

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from generous folks like yourself.

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So if you're in a position to do so,

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please think about making
a donation to Khan Academy.

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As you probably know, in 2020
the world leaned more heavily

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on Khan Academy than ever before

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to keep the learning
going during the pandemic.

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We had 12 billion learning minutes

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on the platform last year.

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6 billion students coming on their own.

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6 billion is hundreds of thousands

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of teachers getting
their students to do it

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as part of the distance
learning classroom experience

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during the pandemic.

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And as now we get, hopefully, to the light

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at the end of the tunnel on the pandemic

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and hopefully over the next couple

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of months things are able to normalize,

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it's even more important
that we don't let up

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because when all of the
stuff has been thrown up

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into the air, we, one, need to make sure

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that as it falls things don't break.

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Students have accrued
gaps in their learning.

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Teachers need more support,
families need more support.

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And also there's a unique
window of time right now,

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over the next six to 12 months,
where as things come back

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to earth, not only should they not break

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but there's an opportunity to make sure

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that the post-pandemic
world could even be better

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than the pre-pandemic world.

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A world where every student
is able to fill in their gaps,

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learn at their own time and space

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and really reach whatever
that potential is,

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that every teacher has
the tools to personalize

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for their students and
feels optimally informed

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and optimally empowered
and every parent feels

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like there are resources
regardless of your zip code,

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regardless of your income

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where your children can
reach their potential.

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So I hope you can consider
making a donation.

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In particular, we're hoping

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to get at least 12,000
new donors of any level.

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Any donation makes a huge difference

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to become part of this mission

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to provide a free world-class
education for anyone anywhere.