Relating trig function through angle rotations

WEBVTT
Kind: captions
Language: en

00:00:00.786 --> 00:00:03.728
Let's say that I've got some angle theta,

00:00:03.728 --> 00:00:07.430
some angle theta right over here.

00:00:07.430 --> 00:00:09.208
And I'm drawing it on our unit circle

00:00:09.208 --> 00:00:10.376
with a typical convention

00:00:10.376 --> 00:00:12.125
that we started with a ray

00:00:12.125 --> 00:00:13.913
that's along the positive X axis,

00:00:13.913 --> 00:00:15.542
and the terminal side of this angle,

00:00:15.542 --> 00:00:17.907
is the terminal side of the angle,

00:00:17.907 --> 00:00:19.875
where it intersects the unit circle,

00:00:19.875 --> 00:00:21.799
determines essentially the sine

00:00:21.799 --> 00:00:24.460
and cosine of that theta,

00:00:24.460 --> 00:00:26.419
so the cosine of theta is the X...

00:00:26.419 --> 00:00:27.411
is the...

00:00:27.411 --> 00:00:28.954
let me just set a color
I haven't used before.

00:00:28.954 --> 00:00:32.202
The cosine of theta is the X coordinate

00:00:32.202 --> 00:00:33.943
of where this terminal ray

00:00:33.943 --> 00:00:35.245
intersects the unit circle.

00:00:35.245 --> 00:00:37.141
Or another way of thinking about it is

00:00:37.141 --> 00:00:41.034
the cosine of theta is the length of

00:00:41.034 --> 00:00:44.065
what I'm drawing in
purple right over here.

00:00:44.065 --> 00:00:45.058
It's this length.

00:00:45.058 --> 00:00:46.850
That length right over there

00:00:46.850 --> 00:00:49.820
is cosine of theta,

00:00:49.820 --> 00:00:52.197
and the sine of theta is the Y coordinate.

00:00:52.197 --> 00:00:53.434
Or another way of thinking about it

00:00:53.434 --> 00:00:57.864
the sine of theta is the length of

00:00:57.864 --> 00:01:00.109
this line right over here.

00:01:00.109 --> 00:01:02.503
The how high you are above the X axis,

00:01:02.503 --> 00:01:04.613
that is essentially the Y coordinate,

00:01:04.613 --> 00:01:07.981
and so the length of that is sine theta.

00:01:07.981 --> 00:01:09.400
And this makes sense,

00:01:09.400 --> 00:01:12.176
this actually shows why
the unit circle definition

00:01:12.176 --> 00:01:15.425
is an extension of the
Soh Cah Toa definition.

00:01:15.425 --> 00:01:16.644
Remember, Soh Cah Toa.

00:01:16.644 --> 00:01:17.537
Let me write it down.

00:01:17.537 --> 00:01:18.800
Soh Cah Toa.

00:01:18.800 --> 00:01:28.128
Soh Cah Toa.

00:01:28.128 --> 00:01:29.231
So sine is opposite over hypotenuse.

00:01:29.231 --> 00:01:31.659
So if I want to do the sine of theta,

00:01:31.659 --> 00:01:33.145
what's it going to be?

00:01:33.145 --> 00:01:35.996
So if I think about the sine of theta,

00:01:35.996 --> 00:01:38.366
sine of theta by the
Soh Cah Toa definition,

00:01:38.366 --> 00:01:39.378
it's going to be equal to

00:01:39.378 --> 00:01:41.606
the length of the opposite side.

00:01:41.606 --> 00:01:43.856
Well, we're saying that
that's sine of theta,

00:01:43.856 --> 00:01:46.373
it's sine of theta,

00:01:46.373 --> 00:01:47.695
over the hypotenuse.

00:01:47.695 --> 00:01:48.582
Well the hypotenuse here,

00:01:48.582 --> 00:01:49.481
this is a unit circle,

00:01:49.481 --> 00:01:51.386
so it's going to be one.

00:01:51.386 --> 00:01:53.319
So this shows that this is consistent.

00:01:53.319 --> 00:01:55.387
Or another way of thinking about it,

00:01:55.387 --> 00:02:00.390
sine of theta is equal
to the opposite side

00:02:00.390 --> 00:02:02.479
over the hypotenuse.

00:02:02.479 --> 00:02:04.038
In this case it's going to be equal to

00:02:04.038 --> 00:02:05.160
the opposite side,

00:02:05.160 --> 00:02:06.158
and what's the hypotenuse?

00:02:06.158 --> 00:02:07.043
This is a unit circle,

00:02:07.043 --> 00:02:08.549
so it's going to be one.

00:02:08.549 --> 00:02:10.135
In this case, sine of theta is equal

00:02:10.135 --> 00:02:12.112
to the length of the opposite side.

00:02:12.112 --> 00:02:13.232
The length of the opposite side

00:02:13.232 --> 00:02:14.713
is equal to sine theta.

00:02:14.713 --> 00:02:16.694
And same exact logic.

00:02:16.694 --> 00:02:21.292
The cosine of theta is equal to

00:02:21.292 --> 00:02:23.436
adjacent over hypotenuse,

00:02:23.436 --> 00:02:27.963
is equal to adjacent over hypotenuse.

00:02:27.963 --> 00:02:29.046
And so that's...

00:02:29.046 --> 00:02:30.678
since the hypotenuse is equal to one,

00:02:30.678 --> 00:02:32.520
it's just the length of the adjacent side,

00:02:32.520 --> 00:02:35.412
so cosine of theta is the
length of the adjacent side.

00:02:35.412 --> 00:02:37.534
So this is all a little bit of review,

00:02:37.534 --> 00:02:39.244
just showing how the
unit circle definition

00:02:39.244 --> 00:02:41.770
is an extension of the
Soh Cah Toa definition.

00:02:41.770 --> 00:02:43.278
But now let's do something interesting.

00:02:43.278 --> 00:02:44.714
This is the angle theta.

00:02:44.714 --> 00:02:46.264
Let's think about the angle theta

00:02:46.264 --> 00:02:48.509
plus pi over two.

00:02:48.752 --> 00:02:52.075
So the angle theta plus pi over two.

00:02:52.075 --> 00:02:54.150
So if I were to essentially
add pi over two to this,

00:02:54.150 --> 00:02:56.265
I'm going to get a ray
that is perpendicular

00:02:56.265 --> 00:02:57.542
to the first ray,

00:02:57.542 --> 00:02:58.376
pi over two.

00:02:58.376 --> 00:02:59.252
If we think in degrees,

00:02:59.252 --> 00:03:01.004
pi over two radians,

00:03:01.004 --> 00:03:02.165
so when I say theta plus pi over two,

00:03:02.165 --> 00:03:03.325
I'm talking in radians.

00:03:03.325 --> 00:03:04.668
Pi over two radians is

00:03:04.668 --> 00:03:06.458
equivalent to 90 degrees.

00:03:06.458 --> 00:03:08.856
So we're essentially
adding 90 degrees to it.

00:03:08.856 --> 00:03:11.236
So this angle right over here,

00:03:11.236 --> 00:03:12.538
that angle right over here is

00:03:12.538 --> 00:03:16.882
theta plus pi over two.

00:03:16.882 --> 00:03:19.755
Now, what I want to explore in this video,

00:03:19.755 --> 00:03:21.537
and I guess this is the
interesting part of the video,

00:03:21.537 --> 00:03:26.635
is can we relate sin of theta plus

00:03:26.635 --> 00:03:31.976
pi over two to somehow sine of theta

00:03:31.976 --> 00:03:33.624
or cosine of theta?

00:03:33.624 --> 00:03:35.428
I encourage you to pause this video

00:03:35.428 --> 00:03:36.763
and try to think this through on your own

00:03:36.763 --> 00:03:38.912
before I work it out.

00:03:39.413 --> 00:03:40.712
Well let's think about what sine of theta

00:03:40.712 --> 00:03:41.865
plus pi over two is.

00:03:41.865 --> 00:03:43.712
We know from the unit circle definition,

00:03:43.712 --> 00:03:45.756
the sine of this angle,

00:03:45.756 --> 00:03:46.915
which is theta plus pi over two,

00:03:46.915 --> 00:03:48.574
is the Y coordinate.

00:03:48.574 --> 00:03:51.687
It's that, it's this
value right over here.

00:03:51.687 --> 00:03:53.244
Or another way of thinking about it,

00:03:53.244 --> 00:03:59.951
it's the length of this line in magenta.

00:03:59.951 --> 00:04:04.432
This right over here is the sine of theta,

00:04:04.432 --> 00:04:06.671
plus pi over two.

00:04:07.373 --> 00:04:08.982
So that right over there.

00:04:09.498 --> 00:04:10.767
Now how does that relate

00:04:10.767 --> 00:04:13.591
to what we have over here?

00:04:13.591 --> 00:04:14.649
Well when you look at it,

00:04:14.649 --> 00:04:16.190
it looks like we just took this triangle,

00:04:16.190 --> 00:04:17.421
and we just kind of...

00:04:17.421 --> 00:04:18.653
we rotated it.

00:04:18.653 --> 00:04:20.887
We rotated it counter
clockwise by 90 degrees,

00:04:20.887 --> 00:04:22.765
which essentially what we did do.

00:04:22.765 --> 00:04:24.347
Because we took this terminal side,

00:04:24.347 --> 00:04:26.195
and we added 90 degrees to it,

00:04:26.195 --> 00:04:27.936
or pi over two radians.

00:04:28.170 --> 00:04:28.837
And if you want to get

00:04:28.837 --> 00:04:30.263
a little bit more rigorous about it,

00:04:30.263 --> 00:04:34.238
if this whole white angle here is

00:04:34.238 --> 00:04:36.014
theta plus pi over two,

00:04:36.014 --> 00:04:37.431
and the part that's in the first quadrant

00:04:37.431 --> 00:04:38.971
is pi over two,

00:04:38.971 --> 00:04:40.734
then this part right over here,

00:04:40.734 --> 00:04:43.534
that must be equal to theta.

00:04:43.534 --> 00:04:44.859
And if we think about it,

00:04:44.859 --> 00:04:47.007
if we try to relate the side

00:04:47.007 --> 00:04:48.576
this side that I've put in magenta

00:04:48.576 --> 00:04:49.890
relative to this angle theta

00:04:49.890 --> 00:04:51.834
using the Soh Cah Toa definition,

00:04:51.834 --> 00:04:54.728
here, relative to this
angle theta in yellow,

00:04:54.728 --> 00:04:56.656
this is the adjacent side.

00:04:56.888 --> 00:04:58.426
So let's think about it a little bit.

00:04:58.580 --> 00:04:59.734
So if we were..

00:04:59.734 --> 00:05:01.942
so what deals with the
adjacent and the hypotenuse,

00:05:01.942 --> 00:05:02.857
and in this case of course

00:05:02.857 --> 00:05:04.225
our hypotenuse has length one,

00:05:04.225 --> 00:05:05.767
this is a unit circle,

00:05:05.986 --> 00:05:09.045
what cosine deals with
adjacent and hypotenuse?

00:05:09.189 --> 00:05:10.793
So we could say that the cosine

00:05:10.793 --> 00:05:12.367
of this theta

00:05:12.367 --> 00:05:15.345
so the cosine of that theta

00:05:15.345 --> 00:05:17.975
is equal to the adjacent side,

00:05:17.975 --> 00:05:19.248
the length of the adjacent side

00:05:19.248 --> 00:05:20.223
which we already know

00:05:20.223 --> 00:05:24.374
is sine of theta plus pi over two.

00:05:24.539 --> 00:05:25.164
Let me write it this way.

00:05:25.164 --> 00:05:31.625
Sine of theta plus pi over two.

00:05:31.824 --> 00:05:33.518
over the hypotenuse.

00:05:33.570 --> 00:05:34.550
Over the hypotenuse,

00:05:34.550 --> 00:05:35.588
which is just one,

00:05:35.588 --> 00:05:37.219
so that doesn't change its value.

00:05:37.482 --> 00:05:38.768
So that was pretty neat.

00:05:38.768 --> 00:05:40.025
Just like that,

00:05:40.025 --> 00:05:40.961
we were able to come up

00:05:40.961 --> 00:05:42.381
with a pretty neat relationship

00:05:42.381 --> 00:05:44.218
between cosine and sine.

00:05:44.218 --> 00:05:46.048
The cosine of theta

00:05:46.048 --> 00:05:48.645
is equal to sine of
theta plus pi over two,

00:05:48.645 --> 00:05:51.360
or you could say sine of
theta plus pi over two

00:05:51.360 --> 00:05:55.381
is equal to cosine of theta.

00:05:55.381 --> 00:05:56.945
Now what I encourage you to do is,

00:05:56.945 --> 00:05:57.912
after this video,

00:05:57.912 --> 00:06:00.504
see if you can come up with other results.

00:06:00.504 --> 00:06:01.609
Think about what happens to...

00:06:01.609 --> 00:06:04.593
what sine of theta relates to?

00:06:04.593 --> 00:06:06.503
Or what cosine of theta

00:06:06.503 --> 00:06:09.163
plus pi over two might relate to?

00:06:09.163 --> 00:06:10.455
So I encourage you to explore that

00:06:10.455 --> 00:06:12.234
on your own.


Law of cosines to determine grade

WEBVTT
Kind: captions
Language: en

00:00:00.699 --> 00:00:01.627
Voiceover:Let's say you're studying

00:00:01.627 --> 00:00:02.900
some type of a little hill

00:00:02.900 --> 00:00:05.107
or rock formation right over here.

00:00:05.107 --> 00:00:06.628
And you're able to figure
out the dimensions.

00:00:06.628 --> 00:00:08.251
You know that from this
point to this point

00:00:08.251 --> 00:00:10.355
along the base, straight
along level ground,

00:00:10.355 --> 00:00:11.711
is 60 meters.

00:00:11.711 --> 00:00:13.722
You know the steeper side,

00:00:13.722 --> 00:00:15.795
steeper I guess surface or edge

00:00:15.795 --> 00:00:17.370
of this cliff or whatever you

00:00:17.370 --> 00:00:19.201
wanna call it, is 20 meters.

00:00:19.201 --> 00:00:20.712
And then the longer side here,

00:00:20.712 --> 00:00:24.176
I guess the less steep
side, is 50 meters long.

00:00:24.176 --> 00:00:25.369
So you're able to measure that.

00:00:25.369 --> 00:00:26.984
But now what you wanna do is use your

00:00:26.984 --> 00:00:29.106
knowledge of trigonometry,

00:00:29.106 --> 00:00:30.664
given this information,

00:00:30.664 --> 00:00:32.672
to figure out how steep is this side.

00:00:32.672 --> 00:00:34.120
What is the actual inclination

00:00:34.120 --> 00:00:35.616
relative to level ground?

00:00:35.616 --> 00:00:37.257
Or another way of thinking about it,

00:00:37.257 --> 00:00:40.977
what is this angle theta right over there?

00:00:41.799 --> 00:00:42.983
And I encourage you to pause the video

00:00:42.983 --> 00:00:44.244
and think about it on your own.

00:00:45.522 --> 00:00:47.430
Well it might be ringing a bell.

00:00:47.430 --> 00:00:50.515
Well you know three sides of a triangle

00:00:50.515 --> 00:00:52.891
and then we want to figure out an angle.

00:00:52.891 --> 00:00:54.475
And so the thing that
jumps out in my head,

00:00:54.475 --> 00:00:57.002
well maybe the law of
cosines could be useful.

00:00:57.002 --> 00:01:00.483
Let me just write out the law of cosines,

00:01:00.483 --> 00:01:02.341
before we try to apply it to this

00:01:02.341 --> 00:01:04.270
triangle right over here.

00:01:04.270 --> 00:01:06.453
So the law of cosines tells us

00:01:06.453 --> 00:01:09.397
that C-squared is equal to A-squared,

00:01:09.397 --> 00:01:12.923
plus B-squared, minus two A B,

00:01:12.923 --> 00:01:16.430
times the cosine of theta.

00:01:16.430 --> 00:01:17.910
And just to remind ourselves

00:01:17.910 --> 00:01:20.262
what the A, B's, and C's are,

00:01:20.262 --> 00:01:23.869
C is the side that's
opposite the angle theta.

00:01:23.869 --> 00:01:25.405
So if I were to draw an arbitrary

00:01:25.405 --> 00:01:27.437
triangle right over here.

00:01:27.437 --> 00:01:29.818
And if this is our angle theta,

00:01:29.818 --> 00:01:33.314
then this determines that C is that side,

00:01:33.314 --> 00:01:34.640
and then A and B could be either

00:01:34.640 --> 00:01:35.728
of these two sides.

00:01:35.728 --> 00:01:37.721
So A could be that one
and B could be that one.

00:01:37.721 --> 00:01:39.368
Or the other way around.

00:01:39.368 --> 00:01:40.256
As you can see, A and B

00:01:40.256 --> 00:01:42.137
essentially have the same role in this

00:01:42.137 --> 00:01:43.257
formula right over here.

00:01:43.257 --> 00:01:46.561
This could be B or this could be A.

00:01:46.561 --> 00:01:48.139
So what we wanna do is

00:01:48.139 --> 00:01:49.633
somehow relate this angle...

00:01:49.633 --> 00:01:51.841
We wanna figure out what theta

00:01:51.841 --> 00:01:54.536
is in our little hill
example right over here.

00:01:54.536 --> 00:01:55.824
So if this is going to be theta,

00:01:55.824 --> 00:01:57.513
what is C going to be?

00:01:57.513 --> 00:02:01.320
Well C is going to be this 20 meter side.

00:02:01.320 --> 00:02:02.105
And then we could set either

00:02:02.105 --> 00:02:03.537
one of these to be A or B.

00:02:03.537 --> 00:02:05.887
We could say that this A is 50 meters

00:02:05.887 --> 00:02:08.752
and B is 60 meters.

00:02:08.752 --> 00:02:11.224
And now we could just
apply the law of cosines.

00:02:11.224 --> 00:02:13.409
So the law of cosines tells us

00:02:13.409 --> 00:02:18.584
that 20-squared is equal to A-squared,

00:02:18.584 --> 00:02:21.200
so that's 50 squared,

00:02:21.200 --> 00:02:26.664
plus B-squared, plus 60 squared,

00:02:26.664 --> 00:02:31.720
minus two times A B.

00:02:31.720 --> 00:02:39.440
So minus two times 50, times 60, times 60,

00:02:39.440 --> 00:02:44.445
times the cosine of theta.

00:02:44.445 --> 00:02:45.539
This works out well for us

00:02:45.539 --> 00:02:46.652
because they've given us everything.

00:02:46.652 --> 00:02:48.315
There's really only one unknown.

00:02:48.315 --> 00:02:49.963
There's theta here.

00:02:49.963 --> 00:02:52.547
So let's see if we can solve for theta.

00:02:52.547 --> 00:02:56.691
So 20 squared, that is 400.

00:02:56.691 --> 00:03:03.859
50 squared is 2,500.

00:03:03.859 --> 00:03:08.668
60 squared is 3,600.

00:03:08.668 --> 00:03:11.200
And then 50 times...

00:03:11.200 --> 00:03:13.443
Let's see, two times 50 is 100,

00:03:13.443 --> 00:03:17.803
times 60, this is all equal to 6,000.

00:03:17.803 --> 00:03:19.390
So let's see, if we simplify

00:03:19.390 --> 00:03:21.524
this a little bit we're going to get

00:03:21.524 --> 00:03:27.173
400 is equal to 2,500 plus 3,600.

00:03:27.173 --> 00:03:30.355
Let's see, that'd be 6,100.

00:03:30.355 --> 00:03:31.661
That's equal to 6,000...

00:03:31.661 --> 00:03:32.565
Let me do this in a new color.

00:03:32.565 --> 00:03:38.429
So when I add these two, I get 6,100.

00:03:38.429 --> 00:03:39.148
Did I do that right?

00:03:39.148 --> 00:03:39.582
Yeah.

00:03:39.582 --> 00:03:41.390
So it's 2,000 plus 3,000, plus 5,000.

00:03:41.390 --> 00:03:43.285
500 plus 600 is 1,100.

00:03:43.285 --> 00:03:50.502
So I get 6,100 minus 6,000,

00:03:50.502 --> 00:03:56.121
times the cosine of theta.

00:03:56.121 --> 00:03:57.971
And let's see, now we can subtract

00:03:57.971 --> 00:04:00.619
6,100 from both sides.

00:04:00.619 --> 00:04:04.490
So I'm just gonna subtract
6,100 from both sides

00:04:04.490 --> 00:04:07.411
so that I get closer
to isolating the theta.

00:04:07.411 --> 00:04:08.739
So let's do that.

00:04:08.739 --> 00:04:14.995
So this is going to be negative 5,700.

00:04:14.995 --> 00:04:15.898
Is that right?

00:04:15.898 --> 00:04:18.299
5,700 plus...

00:04:18.299 --> 00:04:20.450
Yes, that is right.

00:04:20.450 --> 00:04:21.899
Right, because if this
was the other way around,

00:04:21.899 --> 00:04:24.659
if this was 6,100 minus 400

00:04:24.659 --> 00:04:27.859
it would be positive 5,700.

00:04:27.859 --> 00:04:29.520
Alright.

00:04:29.520 --> 00:04:31.611
And then these two of course cancel out.

00:04:31.611 --> 00:04:33.524
And this is going to be equal to

00:04:33.524 --> 00:04:39.880
negative 6,000 times the cosine of theta.

00:04:39.880 --> 00:04:48.807
Now we can divide both
sides by negative 6,000.

00:04:48.807 --> 00:04:50.316
And we get...

00:04:50.316 --> 00:04:51.859
I'm just gonna swap the sides.

00:04:51.859 --> 00:04:55.907
We get cosine of theta is equal to...

00:04:55.907 --> 00:04:57.962
Let's see we could divide the numerator

00:04:57.962 --> 00:05:01.506
and the denominator by
essentially negative 100.

00:05:01.506 --> 00:05:03.474
So these are both going
to become positive.

00:05:03.474 --> 00:05:11.300
So cosine of theta is equal to 57 over 60.

00:05:11.300 --> 00:05:13.879
And actually that can
be simplified even more.

00:05:13.879 --> 00:05:18.422
Three goes into 57, is that 19 times?

00:05:18.422 --> 00:05:19.342
Yep, so this is actually...

00:05:19.342 --> 00:05:20.374
This could be simplified.

00:05:20.374 --> 00:05:25.606
This is equal to 19 over 20.

00:05:25.606 --> 00:05:27.626
We actually didn't have to
do that simplification step

00:05:27.626 --> 00:05:29.552
because we're about to
use our calculators,

00:05:29.552 --> 00:05:32.993
but that makes the math
a little more tractable.

00:05:32.993 --> 00:05:36.112
Right, 3 goes into 57, yeah, 19 times.

00:05:36.112 --> 00:05:37.673
And so now we can take the

00:05:37.673 --> 00:05:39.873
inverse cosine of both sides.

00:05:39.873 --> 00:05:42.880
So we could get theta is equal to the

00:05:42.880 --> 00:05:55.169
inverse cosine, or the
arc cosine, of 19 over 20.

00:05:55.169 --> 00:05:56.994
So let's get our calculator out

00:05:56.994 --> 00:05:59.994
and see if we get
something that makes sense.

00:05:59.994 --> 00:06:11.593
So we wanna do the inverse
cosine of 19 over 20.

00:06:11.593 --> 00:06:13.802
And we deserve a drum roll.

00:06:13.802 --> 00:06:18.114
We get 18.19 degrees.

00:06:18.114 --> 00:06:19.355
And I already verified that my

00:06:19.355 --> 00:06:22.282
calculator is in degree mode.

00:06:22.282 --> 00:06:25.546
So it gets 18.19 degrees.

00:06:25.546 --> 00:06:26.619
So if we wanted to round,

00:06:26.619 --> 00:06:32.515
this is approximately
equal to 18.2 degrees,

00:06:32.515 --> 00:06:35.164
if we wanna round to the nearest tenth.

00:06:35.164 --> 00:06:37.018
So that essentially gives us a sense

00:06:37.018 --> 00:06:40.892
of how steep this slope actually is.


Law of sines for missing angle

WEBVTT
Kind: captions
Language: en

00:00:00.708 --> 00:00:02.931
Voiceover:Say you're out
flying kites with a friend

00:00:02.931 --> 00:00:04.073
and right at this moment

00:00:04.073 --> 00:00:06.419
you're 40 meters away from your friend

00:00:06.419 --> 00:00:09.607
and you know that the length of the kite's

00:00:09.607 --> 00:00:11.445
string is 30 meters,

00:00:11.445 --> 00:00:12.793
and you measure the angle

00:00:12.793 --> 00:00:14.708
between the kite and the ground

00:00:14.708 --> 00:00:15.700
where you're standing

00:00:15.700 --> 00:00:18.579
and you see that it's a 40 degree angle.

00:00:18.579 --> 00:00:19.678
What you're curious about is

00:00:19.678 --> 00:00:22.802
whether you can use your
powers of trigonometry

00:00:22.802 --> 00:00:25.840
to figure out the angle between the string

00:00:25.840 --> 00:00:27.919
and the ground.

00:00:27.919 --> 00:00:29.752
I encourage you to pause the video now

00:00:29.752 --> 00:00:31.613
and figure out if you can do that

00:00:31.613 --> 00:00:34.495
using just the information that you have.

00:00:34.871 --> 00:00:38.206
Whenever I see, I guess,
a non right triangle

00:00:38.206 --> 00:00:39.684
where I'm trying to
figure out some lengths

00:00:39.684 --> 00:00:41.705
of sides or some lengths of angles,

00:00:41.705 --> 00:00:45.025
I immediately think
maybe the Law of Cosine

00:00:45.025 --> 00:00:48.250
might be useful or the Law
of Sines might be useful.

00:00:48.250 --> 00:00:49.708
So, let's think about which one

00:00:49.708 --> 00:00:51.361
could be useful in this case.

00:00:51.361 --> 00:00:53.701
Law of Cosines, and I'll
just rewrite them here.

00:00:53.701 --> 00:00:55.738
The Law of Cosine is c squared is equal

00:00:55.738 --> 00:00:57.650
to a squared plus b squared

00:00:57.650 --> 00:01:02.283
minus 2ab cosine of theta.

00:01:02.283 --> 00:01:04.738
So what it's doing is it's relating

00:01:04.738 --> 00:01:07.107
3 sides of a triangle.

00:01:07.107 --> 00:01:11.798
So a, b, c to an angle.

00:01:11.798 --> 00:01:13.749
So, for example, if I do 2 sides

00:01:13.749 --> 00:01:15.300
and the angle in between them,

00:01:15.300 --> 00:01:16.666
I can figure out the third side.

00:01:16.666 --> 00:01:18.288
Or if I know all 3 sides,

00:01:18.288 --> 00:01:21.266
then I can figure out this angle.

00:01:21.266 --> 00:01:22.435
But that's not the situation

00:01:22.435 --> 00:01:23.281
that we have over here.

00:01:23.281 --> 00:01:27.730
We're trying to figure
out this question mark

00:01:27.730 --> 00:01:30.202
and we don't know 3 of the sides.

00:01:30.202 --> 00:01:31.395
We're trying to figure out an angle

00:01:31.395 --> 00:01:32.722
but we don't know 3 of the sides.

00:01:32.722 --> 00:01:34.185
The Law of Cosine just doesn't seem,

00:01:34.185 --> 00:01:35.755
at least in an obvious way,

00:01:35.755 --> 00:01:36.795
that it's going to help me.

00:01:36.795 --> 00:01:38.017
I could also try to find this angle.

00:01:38.017 --> 00:01:40.024
Once again, we don't know all 3 sides

00:01:40.024 --> 00:01:41.897
to be able to solve for the angle.

00:01:41.897 --> 00:01:44.924
So maybe Law of Sines could be useful.

00:01:44.924 --> 00:01:50.340
So the Law of Sines, the Law of Sines.

00:01:50.340 --> 00:01:52.546
Let's say that this is,

00:01:52.546 --> 00:01:54.775
the measure of this angle is a,

00:01:54.775 --> 00:01:56.440
the measure of this angle is lower case b,

00:01:56.440 --> 00:01:58.197
the measure of this angle is lower case c,

00:01:58.197 --> 00:02:00.207
length of this side is capital C,

00:02:00.207 --> 00:02:01.955
length of this side is capital A,

00:02:01.955 --> 00:02:04.283
length of this side is capital B.

00:02:04.283 --> 00:02:05.655
The Law of Sine tells us

00:02:05.655 --> 00:02:09.846
the ratio between the sine
of each of these angles

00:02:09.846 --> 00:02:11.940
and the length of the opposite side

00:02:11.940 --> 00:02:13.133
is constant.

00:02:13.133 --> 00:02:15.372
So sine of lower case a over capital A

00:02:15.372 --> 00:02:19.910
is the same as lower
case b over capital B,

00:02:19.910 --> 00:02:22.760
which is going to be the same as

00:02:22.760 --> 00:02:25.485
lower case c over capital C.

00:02:25.485 --> 00:02:27.526
Let's see if we can leverage that somehow

00:02:27.526 --> 00:02:28.892
right over here.

00:02:28.892 --> 00:02:33.157
We know this angle and the opposite side

00:02:33.157 --> 00:02:35.904
so we can write that ratio.

00:02:35.904 --> 00:02:41.297
Sine of 40 degrees over 30.

00:02:41.297 --> 00:02:43.227
Let's see. Can we say
that that's going to be

00:02:43.227 --> 00:02:45.302
equal to the sine of this angle over that?

00:02:45.302 --> 00:02:47.399
Well it would be, but we
don't know either of these

00:02:47.399 --> 00:02:49.955
so that doesn't seem like
it's going to help us.

00:02:49.955 --> 00:02:51.710
But, we do know this side.

00:02:51.710 --> 00:02:53.427
Maybe we could use the Law of Cosines

00:02:53.427 --> 00:02:54.661
to figure out this angle,

00:02:54.661 --> 00:02:56.319
because if we know 2 angles of a triangle,

00:02:56.319 --> 00:02:58.184
then we can figure out the third angle.

00:02:58.184 --> 00:02:59.659
So let's do that.

00:02:59.659 --> 00:03:02.514
Let's say that this angle
right over here is theta.

00:03:02.514 --> 00:03:04.820
We know this distance right
over here is 40 meters,

00:03:04.820 --> 00:03:11.860
so we can say that the
sine of theta over 40,

00:03:11.860 --> 00:03:13.467
this ratio is going to be the same

00:03:13.467 --> 00:03:16.232
as the sine of 40 over 30.

00:03:16.232 --> 00:03:19.141
Now we can just solve for theta.

00:03:19.141 --> 00:03:22.494
Multiplying both sides times 40,

00:03:22.494 --> 00:03:26.013
you're going to get, let's see.

00:03:26.013 --> 00:03:28.476
40 divided by 30 is 4/3.

00:03:28.476 --> 00:03:38.173
4/3 sine of 40 degrees is
equal to sine of theta,

00:03:38.173 --> 00:03:40.621
is equal to sine of theta.

00:03:40.621 --> 00:03:42.279
Now to solve for theta,
we just need to take

00:03:42.279 --> 00:03:44.175
the inverse sine of both sides.

00:03:44.175 --> 00:03:54.297
So inverse sine of 4 over
3 sine of 40 degrees.

00:03:54.297 --> 00:03:55.537
Put some parentheses here,

00:03:55.537 --> 00:03:57.670
is equal to theta.

00:03:57.670 --> 00:03:59.975
That will give us that angle here

00:03:59.975 --> 00:04:01.321
and we can use that information
and this information

00:04:01.321 --> 00:04:04.321
to figure out the angle
that we really care about.

00:04:04.321 --> 00:04:06.255
So, let's get a calculator out and see if

00:04:06.255 --> 00:04:07.764
we can calculate it.

00:04:07.764 --> 00:04:10.193
Let me just verify, I am in degree mode.

00:04:10.193 --> 00:04:11.602
Very important.

00:04:11.602 --> 00:04:13.841
All right, now I'm going
to take the inverse

00:04:13.841 --> 00:04:25.562
sine of 4/3 times sine of 40 degrees,

00:04:25.562 --> 00:04:26.797
and that gets me,

00:04:26.797 --> 00:04:29.570
and I deserve a little bit of a drum roll,

00:04:29.570 --> 00:04:34.524
58, well if we round to the nearest,

00:04:34.524 --> 00:04:37.099
let's just maintain our precision here.

00:04:37.099 --> 00:04:40.308
So 58.99 degrees roughly.

00:04:40.308 --> 00:04:47.271
This is approximately
equal to 58.99 degrees.

00:04:47.271 --> 00:04:49.529
So, if that is 58.99 degrees,

00:04:49.529 --> 00:04:50.750
what is this one?

00:04:50.750 --> 00:04:53.676
It's going to 180 minus
this angle's measure

00:04:53.676 --> 00:04:55.013
minus that angle's measure.

00:04:55.013 --> 00:04:56.249
Let's calculate that.

00:04:56.249 --> 00:05:00.160
It's going to 180
degrees minus this angle,

00:05:00.160 --> 00:05:03.016
so minus 40,

00:05:03.016 --> 00:05:03.073
minus the angle that we just figured out.

00:05:03.073 --> 00:05:04.676
Actually I can get all of our precision

00:05:04.676 --> 00:05:07.383
by just typing in second answer.

00:05:07.383 --> 00:05:09.075
So that just says our previous answer

00:05:09.075 --> 00:05:10.899
so I get all that precision there

00:05:10.899 --> 00:05:15.008
and so I get 81.01 degrees.

00:05:15.008 --> 00:05:16.525
So, if I want to round to the nearest,

00:05:16.525 --> 00:05:18.333
let's say I want to round
to nearest hundredth

00:05:18.333 --> 00:05:21.685
of a degree, then I'd say 81.01 degrees.

00:05:21.685 --> 00:05:24.602
So this, this right over here,

00:05:24.602 --> 00:05:28.602
is approximately 81.01
degrees and we're done.


Law of cosines for star distance

WEBVTT
Kind: captions
Language: en

00:00:01.143 --> 00:00:02.735
Voiceover:Artemis seeks knowledge of

00:00:02.735 --> 00:00:04.602
the width of Orion's belt,

00:00:04.602 --> 00:00:08.101
which is a pattern of stars
in the Orion constellation.

00:00:08.101 --> 00:00:10.741
She has previously
discovered the distances

00:00:10.741 --> 00:00:16.804
from her house to
Alnitak, 736 lights years,

00:00:16.804 --> 00:00:20.756
and to Mintaka, 915 light years,

00:00:20.756 --> 00:00:23.758
which are the endpoints of Orion's belt.

00:00:23.758 --> 00:00:26.002
She knows the angle between these

00:00:26.002 --> 00:00:29.142
stars in the sky is three degrees.

00:00:29.142 --> 00:00:31.806
What is the width of Orion's belt?

00:00:31.806 --> 00:00:32.872
That is, what is the distance

00:00:32.872 --> 00:00:36.268
between Alnitak and Mintaka?

00:00:36.268 --> 00:00:39.209
And they want us to the
answer in light years.

00:00:39.209 --> 00:00:40.872
So let's draw a little diagram

00:00:40.872 --> 00:00:42.607
to make sure we understand
what's going on.

00:00:42.607 --> 00:00:43.670
Actually, even before we do that,

00:00:43.670 --> 00:00:44.563
I encourage you to pause

00:00:44.563 --> 00:00:46.735
this and try this on your own.

00:00:46.735 --> 00:00:48.675
Now let's make a diagram.

00:00:48.675 --> 00:00:50.798
Alright, so let's say
that this is Artemis'

00:00:50.798 --> 00:00:52.141
house right over here.

00:00:52.141 --> 00:00:53.756
This is Artemis' house.

00:00:53.756 --> 00:00:57.174
I'll say that's A for Artemis' house.

00:00:57.174 --> 00:00:58.666
And then...

00:00:58.666 --> 00:01:00.004
Alright, let me say H...

00:01:00.004 --> 00:01:01.605
Let me say this is home.

00:01:01.605 --> 00:01:03.273
This is home right over here.

00:01:03.273 --> 00:01:04.748
And we have these 2 stars.

00:01:04.748 --> 00:01:07.467
So she's looking out into the night sky

00:01:07.467 --> 00:01:09.169
and she sees these stars,

00:01:09.169 --> 00:01:14.605
Alnitak, which is 736 light years away,

00:01:14.605 --> 00:01:17.337
and obviously I'm not going
to draw this to scale.

00:01:17.337 --> 00:01:21.750
So this is Alnitak.

00:01:21.750 --> 00:01:25.521
And Mintaka.

00:01:25.521 --> 00:01:28.606
So let's say this is
Mintaka right over here.

00:01:28.606 --> 00:01:31.001
Mintaka.

00:01:31.001 --> 00:01:32.606
And we know a few things.

00:01:32.606 --> 00:01:35.273
We know that this
distance between her home

00:01:35.273 --> 00:01:40.173
and Alnitak is 736 light years.

00:01:40.173 --> 00:01:42.935
So this distance right over here.

00:01:42.935 --> 00:01:44.338
So that right over there.

00:01:44.338 --> 00:01:45.837
Everything we'll do is in light years.

00:01:45.837 --> 00:01:47.707
That's 736.

00:01:47.707 --> 00:01:48.605
And the distance between

00:01:48.605 --> 00:01:54.674
her house and Mintaka is 915 light years.

00:01:54.674 --> 00:01:57.163
So it would take light 915 years

00:01:57.163 --> 00:01:58.879
to get from her house to Mintaka,

00:01:58.879 --> 00:02:01.248
or from Mintaka to her house.

00:02:01.248 --> 00:02:04.271
So this is 915 light years.

00:02:04.271 --> 00:02:05.379
And what we wanna do is figure out

00:02:05.379 --> 00:02:07.402
the width of Orion's belt,

00:02:07.402 --> 00:02:11.136
which is the distance
between Alnitak and Mintaka.

00:02:11.136 --> 00:02:15.880
So we need to figure out this distance

00:02:15.880 --> 00:02:17.335
right over here.

00:02:17.335 --> 00:02:21.506
And the one thing that they did give us

00:02:21.506 --> 00:02:23.269
is this angle.

00:02:23.269 --> 00:02:26.270
They did give us that
angle right over there.

00:02:26.270 --> 00:02:28.136
They said that the angle between

00:02:28.136 --> 00:02:30.216
these stars in the sky is three degrees.

00:02:30.216 --> 00:02:33.552
So this is three degrees right over there.

00:02:33.552 --> 00:02:36.003
So how can we figure out the distance

00:02:36.003 --> 00:02:38.406
between Alnitak and Mintaka?

00:02:38.406 --> 00:02:40.868
Let's just say that this is equal to X.

00:02:40.868 --> 00:02:42.074
This is equal to X.

00:02:42.074 --> 00:02:43.404
How do we do that?

00:02:43.404 --> 00:02:45.697
Well if we have two sides

00:02:45.697 --> 00:02:47.990
and an angle between them,

00:02:47.990 --> 00:02:50.285
we could use the law of cosines

00:02:50.285 --> 00:02:55.368
to figure out the third side.

00:02:55.368 --> 00:02:56.736
So the law of cosines,

00:02:56.736 --> 00:02:58.534
so let's just apply it.

00:02:58.534 --> 00:03:02.871
So the law of cosines tells us

00:03:02.871 --> 00:03:05.928
that X squared is going to be equal

00:03:05.928 --> 00:03:09.176
to the sum of the squares
of the other two sides.

00:03:09.176 --> 00:03:14.433
So it's going to be equal to 736 squared,

00:03:14.433 --> 00:03:28.534
plus 915 squared, minus two times 736,

00:03:28.534 --> 00:03:37.135
times 915, times the cosine of this angle.

00:03:37.135 --> 00:03:41.631
Times the cosine of three degrees.

00:03:41.631 --> 00:03:43.473
So once again,

00:03:43.473 --> 00:03:44.541
we're trying to find the length of

00:03:44.541 --> 00:03:46.501
the side opposite the three degrees.

00:03:46.501 --> 00:03:48.008
We know the other two sides,

00:03:48.008 --> 00:03:50.084
so the law of cosines, it essentially...

00:03:51.807 --> 00:03:54.061
Sorry, I just had to cough off camera

00:03:54.061 --> 00:03:56.211
because I had some peanuts
and my throat was dry.

00:03:56.211 --> 00:03:56.997
Where was I?

00:03:56.997 --> 00:03:58.324
Oh, I was saying,

00:03:58.324 --> 00:04:00.543
if we know the angle and
we know the two sides

00:04:00.543 --> 00:04:01.799
on either side of the angle,

00:04:01.799 --> 00:04:03.294
we can figure out the
length of the side opposite

00:04:03.294 --> 00:04:04.853
by the law of cosines.

00:04:04.853 --> 00:04:06.624
Where it essentially starts off not too

00:04:06.624 --> 00:04:08.215
different than the Pythagorean theorem,

00:04:08.215 --> 00:04:09.330
but then we give an adjustment

00:04:09.330 --> 00:04:12.210
because this is not an
actual right triangle.

00:04:12.210 --> 00:04:13.264
And the adjustment...

00:04:13.264 --> 00:04:16.262
So we have the 736
squared, plus 915 squared,

00:04:16.262 --> 00:04:19.426
minus two times the
product of these sides,

00:04:19.426 --> 00:04:21.674
times the cosine of this angle.

00:04:21.674 --> 00:04:23.669
Or another way we could
say, think about it is,

00:04:23.669 --> 00:04:28.722
X, let me write that,

00:04:28.722 --> 00:04:31.591
X is to equal to the square root of all

00:04:31.591 --> 00:04:33.000
of this stuff.

00:04:33.000 --> 00:04:36.174
So, I can just copy and paste that.

00:04:37.482 --> 00:04:39.115
Copy and paste.

00:04:40.126 --> 00:04:44.487
X is going to be equal to
the square root of that.

00:04:44.886 --> 00:04:48.328
And so let's get our
calculator to calculate it.

00:04:48.328 --> 00:04:51.054
And let me verify that I'm in degree mode.

00:04:51.054 --> 00:04:53.727
Yes, I am indeed in degree mode.

00:04:53.727 --> 00:04:55.724
And so let's exit that.

00:04:55.724 --> 00:04:58.975
And so I wanna calculate
the square root of

00:04:58.975 --> 00:05:06.641
736 squared, plus 915 squared,

00:05:06.641 --> 00:05:15.993
minus two times 736, times 915,

00:05:15.993 --> 00:05:19.657
times cosine of three degrees.

00:05:19.657 --> 00:05:22.474
And we deserve a drum roll now.

00:05:22.474 --> 00:05:24.552
X is 100, if we round...

00:05:24.552 --> 00:05:25.785
Let's see, what did they want us to do?

00:05:25.785 --> 00:05:27.552
Round your answer to
the nearest light years.

00:05:27.552 --> 00:05:28.460
So to the nearest light year

00:05:28.460 --> 00:05:31.932
is going to be 184 light years.

00:05:31.932 --> 00:05:40.590
So X is approximately
equal to 184 light years.

00:05:40.590 --> 00:05:43.559
So it would take light 184 years

00:05:43.559 --> 00:05:47.970
to get from Mintaka to Alnitak.

00:05:47.970 --> 00:05:49.189
And so hopefully this actually shows you

00:05:49.189 --> 00:05:51.927
if you are going to do any astronomy,

00:05:51.927 --> 00:05:54.057
the law of cosines, law of sines,

00:05:54.057 --> 00:05:55.992
in fact all of trigonometry,

00:05:55.992 --> 00:05:59.992
becomes quite, quite handy.


Law of sines

WEBVTT
Kind: captions
Language: en

00:00:00.958 --> 00:00:02.136
Voiceover:We've got a triangle here where

00:00:02.136 --> 00:00:04.844
we know two of the angles
and one of the sides.

00:00:05.750 --> 00:00:08.420
And what I claim, is that I
can figure out everything else

00:00:08.420 --> 00:00:10.359
about this triangle just
with this information.

00:00:10.359 --> 00:00:12.107
You give me two angles and a side,

00:00:12.107 --> 00:00:13.719
and I can figure out what the other

00:00:13.719 --> 00:00:14.972
two sides are going to be.

00:00:14.972 --> 00:00:17.120
And I can, of course,
figure out the third angle.

00:00:17.120 --> 00:00:19.031
So, let's try to figure that out.

00:00:19.155 --> 00:00:20.742
And the way that we're going to do it,

00:00:20.742 --> 00:00:23.236
we're going to use something
called the Law of Sines.

00:00:23.236 --> 00:00:26.776
In a future video, I will
prove the Law of Sines.

00:00:26.776 --> 00:00:29.095
But here, I am just going to show you how

00:00:29.095 --> 00:00:30.617
we can actually apply it.

00:00:30.617 --> 00:00:32.395
And it's a fairly straightforward idea.

00:00:32.395 --> 00:00:33.665
The Law of Sines just tells us

00:00:33.665 --> 00:00:37.280
that the ratio between
the sine of an angle,

00:00:37.280 --> 00:00:39.174
and the side opposite to it,

00:00:39.174 --> 00:00:42.607
is going to be constant for any
of the angles in a triangle.

00:00:42.607 --> 00:00:45.405
So for example, for this
triangle right over here.

00:00:45.405 --> 00:00:46.674
This is a 30 degree angle,

00:00:46.674 --> 00:00:48.729
This is a 45 degree angle.

00:00:48.729 --> 00:00:50.132
They have to add up to 180.

00:00:50.132 --> 00:00:54.037
So this right over here
has to be a, let's see,

00:00:54.314 --> 00:00:59.872
it's going to be 180 minus 45 minus 30.

00:00:59.872 --> 00:01:04.677
That's 180 minus 75, so
this is going to equal

00:01:04.677 --> 00:01:07.701
105 degree angle, right over here.

00:01:08.839 --> 00:01:10.522
And so applying the Law of Sines,

00:01:10.522 --> 00:01:12.261
actually let me label the different sides.

00:01:12.261 --> 00:01:14.535
Let's call this side right over here,

00:01:17.150 --> 00:01:20.770
side A or has length A.

00:01:20.955 --> 00:01:26.816
And let's call this side,
right over here, has length B.

00:01:27.320 --> 00:01:28.758
So the Law of Sines tells us

00:01:28.758 --> 00:01:31.195
that the ratio between
the sine of an angle,

00:01:31.195 --> 00:01:33.266
and that the opposite side is going to be

00:01:33.266 --> 00:01:35.639
constant through this triangle.

00:01:35.946 --> 00:01:41.806
So it tells us that sine of
this angle, sine of 30 degrees

00:01:41.806 --> 00:01:44.735
over the length of the side opposite,

00:01:44.735 --> 00:01:50.695
is going to be equal to
sine of a 105 degrees,

00:01:53.587 --> 00:01:56.628
over the length of the
side opposite to it.

00:01:56.982 --> 00:02:02.019
Which is going to be equal
to sine of 45 degrees.

00:02:04.726 --> 00:02:08.126
equal to the length of the side opposite.

00:02:08.218 --> 00:02:10.658
So sine of 45 degrees over B.

00:02:10.858 --> 00:02:12.988
And so if we wanted to figure out A,

00:02:12.988 --> 00:02:16.166
we could solve this
equation right over here.

00:02:16.166 --> 00:02:17.816
And if we wanted to solve for B,

00:02:17.816 --> 00:02:19.802
we could just set this equal
to that right over there.

00:02:19.802 --> 00:02:21.553
So let's solve each of these.

00:02:21.553 --> 00:02:24.110
So what is the sine of 30 degrees?

00:02:24.110 --> 00:02:25.855
Well, you might just remember
it from your unit circles

00:02:25.855 --> 00:02:28.795
or from even 30, 60, 90
triangles and that's 1/2.

00:02:28.795 --> 00:02:30.265
And if you don't remember it,

00:02:30.265 --> 00:02:32.725
you can use a calculator to verify that.

00:02:32.725 --> 00:02:33.742
I have already verified that

00:02:33.742 --> 00:02:36.924
this is in degree mode, so it's 0.5.

00:02:36.924 --> 00:02:42.220
So this is going to be
equal to 1/2 over two.

00:02:42.220 --> 00:02:44.195
So another way of thinking about it,

00:02:44.195 --> 00:02:45.882
that's going to be equal to 1/4,

00:02:45.882 --> 00:02:47.626
this piece is equal to 1/4

00:02:47.626 --> 00:02:53.413
is equal to sine of a 105 degrees over A.

00:02:53.905 --> 00:02:55.184
Let me write this, this is equal to

00:02:55.890 --> 00:03:00.990
sine of 105 degrees over A.

00:03:00.990 --> 00:03:02.769
And actually, we could also say,

00:03:02.769 --> 00:03:04.299
since we could actually
do both at the same time,

00:03:04.299 --> 00:03:06.358
that this is equal to that.

00:03:06.666 --> 00:03:14.888
That 1/4 is equal to sine
of 45 degrees over B.

00:03:14.888 --> 00:03:18.300
Actually, sine of 45 degrees
is another one of those

00:03:18.300 --> 00:03:20.544
that is easy to jump out of unit circles.

00:03:20.544 --> 00:03:23.168
You might remember it's
square of two over two.

00:03:23.168 --> 00:03:27.973
Let's just write, that's
square root of two over two.

00:03:27.973 --> 00:03:28.928
And you can use a calculator,

00:03:28.928 --> 00:03:31.229
but you'll get some decimal
value right over there.

00:03:31.229 --> 00:03:32.645
But either case, in
either of these equations,

00:03:32.645 --> 00:03:34.320
let's solve for A then let's solve for B.

00:03:34.536 --> 00:03:35.833
So one thing we could do is we could

00:03:35.833 --> 00:03:38.872
take the reciprocal of both
sides of this equation.

00:03:38.872 --> 00:03:41.203
The reciprocal of 1/4 is four.

00:03:41.203 --> 00:03:43.881
And the reciprocal of this right-hand side

00:03:43.881 --> 00:03:49.243
is A over the sine of 105 degrees.

00:03:49.243 --> 00:03:50.785
And then to solve for A,
we could just multiply

00:03:50.785 --> 00:03:54.790
both sides times the
sine of a 105 degrees.

00:03:54.790 --> 00:04:03.764
So we get four times the sine
of 105 degrees is equal to A.

00:04:03.764 --> 00:04:05.411
Let's get our calculator out,

00:04:05.411 --> 00:04:11.372
so four times the sine of 105 gives us,

00:04:11.372 --> 00:04:14.182
it's approximately equal to,

00:04:14.182 --> 00:04:18.034
let's just round to the
nearest 100th, 3.86.

00:04:18.434 --> 00:04:22.630
So A is approximately equal to 3.86.

00:04:25.537 --> 00:04:27.610
Which looks about right if this is two,

00:04:27.610 --> 00:04:29.376
and I have made my angles appropriately,

00:04:29.376 --> 00:04:31.535
that looks like about 3.86.

00:04:31.535 --> 00:04:33.191
Let's figure out what B is.

00:04:33.191 --> 00:04:34.629
We could once again take the reciprocal of

00:04:34.629 --> 00:04:37.122
both sides of this and we get

00:04:37.122 --> 00:04:44.831
four is equal to B over
square root of two over two,

00:04:44.831 --> 00:04:46.639
we could multiply both sides times

00:04:46.639 --> 00:04:48.448
square root of two over two.

00:04:48.448 --> 00:04:51.379
And we would get B is equal to

00:04:51.379 --> 00:04:55.354
four times the square
root of two over two.

00:04:57.892 --> 00:04:59.923
Come to think of it, B is four times

00:04:59.923 --> 00:05:02.109
the sine of 45 degrees.

00:05:02.755 --> 00:05:04.344
Let's figure out what that is.

00:05:04.344 --> 00:05:05.900
If we wanted actual numerical value,

00:05:05.900 --> 00:05:08.267
we could just write this
as two square roots of two.

00:05:08.267 --> 00:05:09.492
But let's actually
figure out what that is.

00:05:09.492 --> 00:05:15.599
Two square roots of two is equal to 2.83.

00:05:15.876 --> 00:05:20.870
So B is approximately equal to 2.83.

00:05:20.870 --> 00:05:23.822
So [I'm] be clear, this
four divided by two is

00:05:23.822 --> 00:05:27.179
two square roots of two, which is 2.8.

00:05:27.179 --> 00:05:31.509
Which is approximately equal to 2.83

00:05:31.509 --> 00:05:34.112
if we round to the nearest 100th, 2.83,

00:05:34.112 --> 00:05:36.653
which also seems pretty reasonable here.

00:05:36.653 --> 00:05:40.750
So the key of the Law of Cosines is

00:05:40.750 --> 00:05:44.722
if you have two angles and a side,

00:05:44.722 --> 00:05:46.596
you're able to figure out
everything else about it.

00:05:46.596 --> 00:05:52.286
Or if you actually had
two sides and an angle,

00:05:52.286 --> 00:05:54.250
you also would be able to figure out

00:05:54.250 --> 00:05:56.412
everything else about the triangle.


Magnitude and direction of vector sums

WEBVTT
Kind: captions
Language: en

00:00:00.369 --> 00:00:01.368
Voiceover: In the last video,

00:00:01.368 --> 00:00:05.371
we were able to figure out
what vectors a plus b is

00:00:05.371 --> 00:00:09.700
when you view it from a
component point of view,

00:00:09.700 --> 00:00:11.270
and so if we were to visualize that,

00:00:11.608 --> 00:00:12.967
that's vector a.

00:00:12.967 --> 00:00:16.701
Then let me paste vector b here.

00:00:16.701 --> 00:00:18.166
It's a little bit messy.

00:00:18.166 --> 00:00:19.700
That is vector b there.

00:00:19.700 --> 00:00:23.134
Actually, let me see if I can
clean this up a little bit.

00:00:24.320 --> 00:00:25.700
Let me clean this up.

00:00:25.700 --> 00:00:27.466
I'll get the eraser tool.

00:00:27.500 --> 00:00:29.499
Let me clean this up a little bit

00:00:29.499 --> 00:00:32.071
so that we can see a little bit clearer.

00:00:32.369 --> 00:00:34.867
The vector that we care about,

00:00:35.183 --> 00:00:37.200
almost done cleaning,

00:00:37.613 --> 00:00:38.633
the vector,

00:00:38.633 --> 00:00:39.967
I'll get rid of this too

00:00:39.967 --> 00:00:41.767
just because we just want to visualize ...

00:00:41.767 --> 00:00:43.633
well, actually, I shouldn't
get rid of that part.

00:00:43.633 --> 00:00:45.768
I'll delete this right over here.

00:00:45.768 --> 00:00:48.633
The vector that we care about, a plus b,

00:00:48.633 --> 00:00:50.500
that is this vector.

00:00:50.500 --> 00:00:53.167
That is this vector right over here.

00:00:53.167 --> 00:00:56.567
We could start at the tail
of a, go to the head of a,

00:00:56.567 --> 00:00:58.167
and then place the tail of b there,

00:00:58.167 --> 00:00:59.567
and then where the head of a is,

00:00:59.567 --> 00:01:01.833
that's going to be the head of a plus b.

00:01:01.833 --> 00:01:04.167
That's this vector right over here.

00:01:04.167 --> 00:01:07.167
We can think of its horizontal
and vertical components.

00:01:07.167 --> 00:01:10.200
Its vertical component is this vector,

00:01:10.500 --> 00:01:14.000
is this vector right over here,

00:01:14.433 --> 00:01:17.900
and that's that vector right over there.

00:01:17.900 --> 00:01:20.600
Its horizontal component is this vector,

00:01:20.700 --> 00:01:25.000
is this vector, which is
this one right over there.

00:01:25.166 --> 00:01:27.566
What I want to do now is figure out

00:01:27.566 --> 00:01:29.134
what is the magnitude ...

00:01:29.730 --> 00:01:30.834
let's just call this vector c

00:01:30.834 --> 00:01:33.132
so I don't have to keep writing a plus b.

00:01:33.166 --> 00:01:37.267
I want to figure out what is
the magnitude of vector c,

00:01:37.300 --> 00:01:39.633
and I want to figure out the angle.

00:01:39.633 --> 00:01:41.366
I want to figure out its direction.

00:01:41.366 --> 00:01:43.433
I want to figure out this angle.

00:01:43.433 --> 00:01:46.866
I want to figure out this
angle right over there.

00:01:47.233 --> 00:01:48.515
Let's think about it.

00:01:48.515 --> 00:01:50.402
Let's think about it step by step.

00:01:50.402 --> 00:01:51.567
The magnitude is actually maybe

00:01:51.567 --> 00:01:53.167
this most straightforward one.

00:01:53.167 --> 00:01:55.333
Let me redraw vector c here.

00:01:57.449 --> 00:01:59.233
Actually, I'll use this
space right over here.

00:01:59.725 --> 00:02:01.866
Vector c looks like this.

00:02:02.596 --> 00:02:05.667
Vector c looks like that,

00:02:05.792 --> 00:02:08.000
and it has its horizontal component.

00:02:08.761 --> 00:02:10.102
You have its horizontal component

00:02:10.102 --> 00:02:12.333
and you have its vertical component.

00:02:14.647 --> 00:02:16.033
The magnitude of it,

00:02:16.033 --> 00:02:18.300
we just know from the Pythagorean theorem,

00:02:18.300 --> 00:02:23.100
the square of the magnitude
of the horizontal component

00:02:23.100 --> 00:02:26.233
plus the square of the magnitude
of the vertical component,

00:02:26.233 --> 00:02:28.300
that's going to be equal to the square of

00:02:28.300 --> 00:02:29.703
the magnitude of the vector.

00:02:29.703 --> 00:02:31.300
Or another way of thinking about it,

00:02:31.300 --> 00:02:34.500
the magnitude of the vector
is going to be equal to

00:02:34.500 --> 00:02:38.370
the square root of this business squared

00:02:38.370 --> 00:02:39.900
plus this business squared,

00:02:39.900 --> 00:02:40.967
and that's going to be ...

00:02:40.967 --> 00:02:42.700
actually, I'll just write
it just so that we ...

00:02:42.700 --> 00:02:43.500
Well, it's getting messy,

00:02:43.500 --> 00:02:44.633
so let me just get the calculator out

00:02:44.633 --> 00:02:46.666
to get a reasonable approximation.

00:02:47.363 --> 00:02:48.867
Get the calculator out.

00:02:50.690 --> 00:02:52.800
It's going to be the square root.

00:02:53.100 --> 00:02:55.669
It's going to be a little
bit of a hairy statement.

00:02:55.768 --> 00:02:57.167
It's going to be the square root of

00:02:57.167 --> 00:03:03.400
three times the square
root of three over two,

00:03:03.700 --> 00:03:06.300
over two, so that's that,

00:03:06.300 --> 00:03:11.401
minus square root of two,

00:03:11.568 --> 00:03:13.501
minus the square root of two,

00:03:13.501 --> 00:03:15.233
so it's going to be that quantity.

00:03:15.233 --> 00:03:17.100
That's what we have in
orange right over there.

00:03:17.100 --> 00:03:18.633
We're going to square that,

00:03:18.633 --> 00:03:21.766
plus three over two,

00:03:21.766 --> 00:03:23.766
three divided by two,

00:03:23.766 --> 00:03:25.967
plus the square root of two,

00:03:25.967 --> 00:03:30.200
plus the square root of two,
so that quantity squared,

00:03:30.500 --> 00:03:33.007
and we're of course taking the
square root of all of that.

00:03:33.233 --> 00:03:35.100
That's going to be equal to,

00:03:35.100 --> 00:03:37.766
we deserve a little
bit of a drum roll now,

00:03:37.766 --> 00:03:40.000
3.14 ...

00:03:40.368 --> 00:03:42.900
it felt like it might have
been close to pi, but it's not.

00:03:42.900 --> 00:03:45.171
Pi would be 3.1415 and I can keep going,

00:03:45.171 --> 00:03:47.567
but it's 3.145.

00:03:47.567 --> 00:03:52.000
I'll just write this approximately 3.146.

00:03:52.833 --> 00:03:53.500
Let me write this.

00:03:53.500 --> 00:04:06.666
This is approximately 3.146
is the magnitude here.

00:04:06.767 --> 00:04:07.833
That makes sense.

00:04:07.833 --> 00:04:10.834
We saw vector a right over here

00:04:10.834 --> 00:04:13.433
had a magnitude of 3,

00:04:13.433 --> 00:04:14.301
and as we can see,

00:04:14.301 --> 00:04:16.833
this one looks a little
bit longer than that,

00:04:16.833 --> 00:04:20.133
even when we look at it visually.

00:04:20.167 --> 00:04:24.142
That actually is consistent
with what we see here.

00:04:24.236 --> 00:04:26.066
Now let's think about the direction.

00:04:26.100 --> 00:04:27.733
Let's call this angle ...

00:04:29.638 --> 00:04:32.134
I don't want to do it in that dark color.

00:04:32.166 --> 00:04:35.984
Let's call this angle
right over here theta.

00:04:36.497 --> 00:04:37.433
What do we know?

00:04:37.433 --> 00:04:39.034
We know that, well,

00:04:39.034 --> 00:04:42.033
we know what the length
of the opposite side is,

00:04:42.033 --> 00:04:44.767
and we know what the length
of the adjacent side is,

00:04:44.767 --> 00:04:46.434
and actually, now we know
the length of the hypotenuse,

00:04:46.434 --> 00:04:47.700
but let's just say we didn't even know

00:04:47.700 --> 00:04:49.301
the length of the hypotenuse.

00:04:49.301 --> 00:04:51.367
If we know the opposite and the adjacent,

00:04:51.367 --> 00:04:52.566
you can use tangent.

00:04:52.566 --> 00:04:54.766
We know that the tangent of theta,

00:04:54.766 --> 00:04:56.767
tangent of theta,

00:04:56.767 --> 00:04:59.233
is going to be equal to the
length of the opposite side

00:04:59.233 --> 00:05:01.033
over the length of the hypotenuse.

00:05:01.033 --> 00:05:03.800
That would be equal to this,

00:05:04.233 --> 00:05:07.233
so copy and paste.

00:05:07.233 --> 00:05:11.433
It's going to be equal to that over this,

00:05:11.433 --> 00:05:15.033
over this right over here,

00:05:15.033 --> 00:05:17.473
so copy and paste.

00:05:17.700 --> 00:05:18.634
All right.

00:05:18.634 --> 00:05:21.066
It's going to be equal to that.

00:05:22.295 --> 00:05:24.200
Let me draw a little line here.

00:05:24.567 --> 00:05:28.434
Or we could say that theta is equal to

00:05:28.434 --> 00:05:29.967
the inverse tangent,

00:05:29.967 --> 00:05:31.900
sometimes called the arc tangent,

00:05:31.900 --> 00:05:36.234
the inverse tangent of
all of this business,

00:05:36.234 --> 00:05:38.666
of all of this.

00:05:39.033 --> 00:05:40.967
Let me just copy and paste,

00:05:40.967 --> 00:05:44.667
copy and paste right over there.

00:05:44.766 --> 00:05:48.867
Let's put that into our
calculator and see what we get.

00:05:50.023 --> 00:05:50.566
Let's see.

00:05:50.566 --> 00:05:54.033
We're going to take the inverse tangent,

00:05:54.033 --> 00:05:55.966
and I've already verified
that I'm in degree mode

00:05:55.966 --> 00:05:57.433
in this calculator,

00:05:57.433 --> 00:05:58.966
and we saw that over there,

00:05:58.966 --> 00:06:01.567
that it wasn't interpreting
this correctly as 30 degrees,

00:06:01.567 --> 00:06:02.433
not 30 radians,

00:06:02.433 --> 00:06:04.768
so inverse tangent of.

00:06:04.768 --> 00:06:05.666
All right.

00:06:06.033 --> 00:06:13.653
Once again, three divided by
two plus square root of two,

00:06:13.653 --> 00:06:17.433
plus the square root of two divided by,

00:06:17.433 --> 00:06:22.100
divided by this, which is three times

00:06:22.100 --> 00:06:26.633
the square root of three divided by two

00:06:26.633 --> 00:06:29.633
minus the square root of two,

00:06:29.633 --> 00:06:31.300
and the order of operations
with a calculator

00:06:31.300 --> 00:06:33.501
will interpret this, it should be correct,

00:06:33.501 --> 00:06:35.900
and so close the parenthesis there.

00:06:35.900 --> 00:06:37.234
That closes that parenthesis,

00:06:37.234 --> 00:06:39.501
and then we have to close
that parenthesis there,

00:06:39.501 --> 00:06:42.400
and we deserve another mini drum roll.

00:06:42.433 --> 00:06:44.900
We get 67 point ...

00:06:44.900 --> 00:06:48.634
Let's just say it's roughly 67.89 degrees.

00:06:48.634 --> 00:06:51.524
So theta,

00:06:52.340 --> 00:06:53.771
let me write this here.

00:06:53.771 --> 00:06:58.320
Theta is approximately 67.89 degrees.

00:06:58.320 --> 00:06:59.253
Did I get that right?

00:06:59.253 --> 00:07:01.101
Yup, 67.89 degrees,

00:07:01.101 --> 00:07:02.866
and that also looks about right.

00:07:02.967 --> 00:07:04.962
If you look at this angle right over here,

00:07:05.961 --> 00:07:06.900
if we eyeball it,

00:07:06.900 --> 00:07:09.901
it looks like a little
bit more than 60 degrees.

00:07:09.901 --> 00:07:11.701
Just like that, we were able to figure out

00:07:11.701 --> 00:07:15.000
the magnitude of the
sum and the direction.

00:07:15.301 --> 00:07:16.767
Now there's one thing that I think

00:07:16.767 --> 00:07:18.533
it's interesting to keep in mind.

00:07:18.566 --> 00:07:21.035
Notice that the magnitude,

00:07:21.035 --> 00:07:23.233
the magnitude of this vector

00:07:23.233 --> 00:07:25.444
is less than the sum of the magnitudes.

00:07:25.444 --> 00:07:27.435
The magnitude of vector a was three.

00:07:27.435 --> 00:07:30.966
The magnitude of vector b is two.

00:07:30.966 --> 00:07:33.033
And so three plus two
would have been five,

00:07:33.033 --> 00:07:35.433
but this has a smaller magnitude.

00:07:35.433 --> 00:07:38.833
The only way that the magnitude of a sum

00:07:38.833 --> 00:07:41.367
is going to be the same as
the sum of the magnitudes

00:07:41.367 --> 00:07:43.433
is if both of these vectors are going

00:07:43.433 --> 00:07:45.301
in the exact same direction,

00:07:45.301 --> 00:07:46.967
then they would be completely additive.

00:07:46.967 --> 00:07:49.633
But if they're going in even
slightly different directions,

00:07:49.633 --> 00:07:53.700
then you're not going to
have the magnitude of the sum

00:07:53.700 --> 00:07:56.701
being the same as the magnitude.

00:07:56.701 --> 00:08:00.235
It's always going to be
less than the magnitude of

00:08:00.235 --> 00:08:02.234
or the sum of the magnitudes.

00:08:02.234 --> 00:08:03.967
In the next video, I'll talk about that

00:08:03.967 --> 00:08:05.846
in a little bit more depth.


Breaking down vectors into components

WEBVTT
Kind: captions
Language: en

00:00:00.148 --> 00:00:02.091
Voiceover:We have two vectors here.

00:00:02.091 --> 00:00:04.268
Vector A, it has a magnitude of three

00:00:04.268 --> 00:00:06.106
so the length of this blue arrow is three.

00:00:06.106 --> 00:00:09.629
Its direction, it forms a 33
degree angle with the positive,

00:00:09.629 --> 00:00:11.646
I guess you could say the positive x axis.

00:00:11.646 --> 00:00:13.518
I haven't drawn that here

00:00:13.518 --> 00:00:16.253
and vector B has a magnitude of two,

00:00:16.253 --> 00:00:17.629
the length of this arrow is two

00:00:17.629 --> 00:00:20.807
and it forms a 135 degree
angle with a positive x axis.

00:00:20.807 --> 00:00:22.588
What I want to think about in this video

00:00:22.588 --> 00:00:25.819
and maybe the next one depends
on how we do with the time

00:00:25.819 --> 00:00:29.328
is what is the magnitude
and direction of the sum

00:00:29.328 --> 00:00:31.237
of these two vectors.

00:00:31.237 --> 00:00:34.942
What is the magnitude and
direction of the vector A plus B?

00:00:34.942 --> 00:00:36.222
I encourage you to pause this video

00:00:36.222 --> 00:00:37.770
and try to work this through on your own

00:00:37.770 --> 00:00:39.360
before I work through it.

00:00:40.467 --> 00:00:41.696
Well to think about this,

00:00:41.696 --> 00:00:43.586
I'm going to decompose
each of these vectors

00:00:43.586 --> 00:00:46.151
into their horizontal and
their vertical components.

00:00:46.151 --> 00:00:48.247
For example vector A could be viewed

00:00:48.247 --> 00:00:52.386
as the sum of this
horizontal pointing vector

00:00:52.386 --> 00:00:56.993
plus this vertical pointing vector.

00:00:56.993 --> 00:00:59.358
Another way of thinking
about this horizontal vector,

00:00:59.358 --> 00:01:00.233
this right over here,

00:01:00.233 --> 00:01:01.673
this is going to be a scaled up version

00:01:01.673 --> 00:01:04.110
of the unit vector I

00:01:04.110 --> 00:01:09.447
and so this is going
to be something times I

00:01:09.447 --> 00:01:12.280
and this vector right over
here is going to be something

00:01:12.280 --> 00:01:15.107
times the unit vector in
the vertical direction.

00:01:15.107 --> 00:01:22.773
The vertical direction might
look something like that

00:01:22.773 --> 00:01:25.211
or let's see if this is
three then the unit vector,

00:01:25.211 --> 00:01:28.046
yeah, actually would
look something like that.

00:01:28.046 --> 00:01:30.008
That's what the J unit vector looks like

00:01:30.008 --> 00:01:35.947
and this is what the I unit
vector would look like.

00:01:37.639 --> 00:01:39.778
This is a scaled up version of it.

00:01:39.778 --> 00:01:44.006
It's something times I and
this is something times J.

00:01:44.006 --> 00:01:48.539
The same exact argument right over here.

00:01:48.539 --> 00:01:51.680
This is going to be something
times the I unit vector,

00:01:51.680 --> 00:01:52.943
its horizontal component

00:01:52.943 --> 00:01:55.442
and its vertical component
is going to be something

00:01:55.442 --> 00:01:57.889
times the J unit vector.

00:01:57.889 --> 00:01:59.400
What we really need to do

00:01:59.400 --> 00:02:02.038
is figure out the magnitudes
of these horizontal

00:02:02.038 --> 00:02:03.297
and vertical components

00:02:03.297 --> 00:02:05.014
and then we know how much to scale up I

00:02:05.014 --> 00:02:07.031
and how much to scale up J.

00:02:07.031 --> 00:02:09.363
Let's think through this a little bit.

00:02:09.363 --> 00:02:11.134
This one might jump out at you immediately

00:02:11.134 --> 00:02:13.582
is this is a 30, 60, 90 triangle.

00:02:13.582 --> 00:02:17.704
This is a 30, 60, 90 triangle
then this side right over here

00:02:17.704 --> 00:02:21.708
is going to be half the
length of the hypotenuse.

00:02:21.708 --> 00:02:25.743
This is going to be half of three,

00:02:25.743 --> 00:02:28.946
so it's three over two and
this one right over here,

00:02:28.946 --> 00:02:30.603
let me do that in same color.

00:02:30.603 --> 00:02:34.500
This is going to be three,
that's not the same,

00:02:34.500 --> 00:02:35.516
that's a different color.

00:02:35.516 --> 00:02:38.639
This is going to be three over two

00:02:38.639 --> 00:02:40.815
and then this length right over here.

00:02:40.815 --> 00:02:42.580
This is going to be square root of three

00:02:42.580 --> 00:02:43.982
times the shorter side.

00:02:43.982 --> 00:02:48.793
This is three times the
square root of three over two.

00:02:48.793 --> 00:02:50.687
I just took this and multiplied
it by the square root of three

00:02:50.687 --> 00:02:54.907
and once again that just came
out of 30, 60, 90 triangles.

00:02:54.907 --> 00:02:57.356
Now another way that you could tackle this

00:02:57.356 --> 00:02:59.453
is to use your, what would you know

00:02:59.453 --> 00:03:00.658
about your trig functions?

00:03:00.658 --> 00:03:02.506
You say, "Okay I know
this angle right here"

00:03:02.506 --> 00:03:05.573
"is a 30 degree angle,
this is the opposite side."

00:03:05.573 --> 00:03:08.837
You could say, "Well,
the opposite over three"

00:03:08.837 --> 00:03:11.824
"is going to be equal to
the sine of 30 degrees,"

00:03:11.824 --> 00:03:13.057
let me right this down.

00:03:13.057 --> 00:03:16.343
Soh cah toa.

00:03:16.343 --> 00:03:24.245
Sine of 30 degrees is going to
be equal to the opposite side

00:03:24.245 --> 00:03:26.601
over the length of the hypotenuse.

00:03:26.601 --> 00:03:30.705
Over three or we could
say that the opposite side

00:03:30.705 --> 00:03:41.757
is equal to three times
the sine of 30 degrees.

00:03:41.757 --> 00:03:43.664
If you put this in your calculator,

00:03:43.664 --> 00:03:46.255
sine of 30 degrees is one half

00:03:46.255 --> 00:03:48.628
and so you're going to
get three halves here.

00:03:48.628 --> 00:03:50.585
Similarly for this side,

00:03:50.585 --> 00:03:53.636
this side is adjacent
to the 30 degree angle.

00:03:53.636 --> 00:03:58.150
You could say cosine, cosine
is adjacent over hypotenuse.

00:03:58.150 --> 00:04:02.354
You could say that cosine of 30 degrees

00:04:02.354 --> 00:04:06.088
is equal to adjacent over the hypotenuse

00:04:06.088 --> 00:04:09.157
or multiplying both sides by three.

00:04:09.157 --> 00:04:14.935
The adjacent is going to be equal to three

00:04:14.935 --> 00:04:19.224
times the cosine of 30 degrees

00:04:19.224 --> 00:04:22.242
and the cosine of 30 degrees
if you type in your calculator,

00:04:22.242 --> 00:04:23.845
you're actually going to
get some type of a decimal

00:04:23.845 --> 00:04:26.551
but it's square root of three over two.

00:04:26.551 --> 00:04:28.469
Three square root of three over two

00:04:28.469 --> 00:04:31.607
and you can verify this if you like.

00:04:31.607 --> 00:04:37.897
We see here, sine of 30 is indeed one half

00:04:37.897 --> 00:04:40.047
and cosine of 30

00:04:40.047 --> 00:04:42.271
well you get this kind of crazy decimal

00:04:42.271 --> 00:04:43.999
but notice that is the same thing

00:04:43.999 --> 00:04:47.751
as square root of three divided by two.

00:04:47.751 --> 00:04:50.037
The exact same value right over here

00:04:50.037 --> 00:04:51.190
but we were able to figure that out

00:04:51.190 --> 00:04:53.582
just using what we know
about 30, 60, 90 triangles.

00:04:53.582 --> 00:04:56.758
Now this triangle, you might
say, "Well, what's the angle?"

00:04:56.758 --> 00:04:58.134
What angles do we know about it?

00:04:58.134 --> 00:05:02.865
Well this angle right over here

00:05:02.865 --> 00:05:06.502
is this is supplementary
to this 135 degree angle.

00:05:06.502 --> 00:05:08.287
This is a 45 degree angle.

00:05:08.287 --> 00:05:10.466
This is 90 so this is 45.

00:05:10.466 --> 00:05:14.361
This is a 45, 45, 90 degree triangle.

00:05:14.361 --> 00:05:18.418
Now you could say that
using the trig identities

00:05:18.418 --> 00:05:22.063
and what we did right
over here that this length

00:05:22.063 --> 00:05:24.199
is going to be the hypotenuse

00:05:24.199 --> 00:05:30.689
times the cosine of 45 degrees

00:05:30.689 --> 00:05:32.653
and you could say that
this length right over here

00:05:32.653 --> 00:05:40.177
is going to be the hypotenuse
times the sine of 45 degrees.

00:05:40.177 --> 00:05:41.968
Using the exact same logic here

00:05:41.968 --> 00:05:44.327
and as you get more practice
with finding the components,

00:05:44.327 --> 00:05:45.939
you'll realize, okay,
you take the hypotenuse

00:05:45.939 --> 00:05:48.231
times the cosine of this angle.

00:05:48.231 --> 00:05:51.084
You're going to get the adjacent side.

00:05:51.084 --> 00:05:54.079
If you do the hypotenuse
times the sine of this angle,

00:05:54.079 --> 00:05:55.806
you're going to get the opposite side

00:05:55.806 --> 00:05:58.416
but we could use this sine of 45 degrees.

00:05:58.416 --> 00:06:00.118
Once again if you put in your calculator,

00:06:00.118 --> 00:06:01.269
you get a crazy decimal

00:06:01.269 --> 00:06:04.770
but we can figure that out,
we know 45, 45, 90 triangles.

00:06:04.770 --> 00:06:08.801
Sine of 45 degrees is
square root of two over two.

00:06:08.801 --> 00:06:12.149
Cosine of 45 degrees is also
square root of two over two.

00:06:12.149 --> 00:06:15.726
I know my writing is
getting a little bit messy

00:06:15.726 --> 00:06:17.592
and actually I should have
done all of these in green,

00:06:17.592 --> 00:06:19.870
square root of two over two.

00:06:19.870 --> 00:06:22.884
What's the length of that green vector?

00:06:22.884 --> 00:06:24.917
What's two times square
root of two over two,

00:06:24.917 --> 00:06:26.431
that's going to be square root of two.

00:06:26.431 --> 00:06:29.444
What's the length of this orange vector?

00:06:29.444 --> 00:06:31.055
It's two times square root of two over two

00:06:31.055 --> 00:06:32.575
so it's going to be square root of two.

00:06:32.575 --> 00:06:35.293
Now we know the magnitudes
of the component vector

00:06:35.293 --> 00:06:37.713
is a horizontal and vertical
components of each of these

00:06:37.713 --> 00:06:40.674
so now we can write these out as the sum

00:06:40.674 --> 00:06:42.769
of these horizontal and vertical vectors.

00:06:42.769 --> 00:06:49.825
Vector A, we can write
as square root of three

00:06:49.825 --> 00:06:54.387
or three square roots of
three over two times I.

00:06:54.387 --> 00:06:56.431
That's this vector right over here,

00:06:56.431 --> 00:06:58.149
we scaled up the I unit vector

00:06:58.149 --> 00:06:59.865
by three squares root of three over two

00:06:59.865 --> 00:07:04.826
plus three halves times J.

00:07:04.826 --> 00:07:10.206
This right over here is
this vector right over here.

00:07:10.206 --> 00:07:12.608
Scaled up version of the J unit vector

00:07:12.608 --> 00:07:15.101
and if you add this orange
vector to this green vector,

00:07:15.101 --> 00:07:16.794
you get vector A.

00:07:16.794 --> 00:07:23.758
Similarly, vector B is equal to the length

00:07:23.758 --> 00:07:25.036
of the horizontal component

00:07:25.036 --> 00:07:26.448
and we got to be very careful.

00:07:26.448 --> 00:07:28.167
It's length is square root of two

00:07:28.167 --> 00:07:30.956
but it's going in the leftward direction.

00:07:30.956 --> 00:07:35.158
We're going to put a
negative on it times I.

00:07:35.158 --> 00:07:37.733
If we just squared it, I is
doing something like this,

00:07:37.733 --> 00:07:39.919
square root of two times
I would look like that.

00:07:39.919 --> 00:07:44.732
Negative square root of two
would point it to the left.

00:07:44.732 --> 00:07:47.621
This is negative square
root of two times I

00:07:47.621 --> 00:07:48.962
and then we're going to have

00:07:48.962 --> 00:07:54.712
plus square root of two times J.

00:07:54.712 --> 00:07:57.645
Now that we've broken them
up in their components,

00:07:57.645 --> 00:07:59.876
we're ready to figure out what,

00:07:59.876 --> 00:08:01.577
at least broken up into its components

00:08:01.577 --> 00:08:03.412
what A plus B is.

00:08:03.412 --> 00:08:12.969
A plus B is equal to, well
it's going to be the sum

00:08:12.969 --> 00:08:14.266
of all of these things.

00:08:14.266 --> 00:08:15.696
Let me just write that down.

00:08:15.696 --> 00:08:19.757
It's going to be that, copy and paste.

00:08:19.757 --> 00:08:27.958
That plus this, let me
just copy and paste it

00:08:27.958 --> 00:08:30.173
and you're going to get that

00:08:30.173 --> 00:08:31.780
but of course we can simplify this.

00:08:31.780 --> 00:08:35.281
We can add the I unit
vectors to each other.

00:08:35.281 --> 00:08:36.947
If I have three times
the square root of three

00:08:36.947 --> 00:08:38.862
over two Is and then I have another

00:08:38.862 --> 00:08:40.142
negative square root of two I,

00:08:40.142 --> 00:08:41.715
I can add that together.

00:08:41.715 --> 00:08:44.640
This is going to be equal to three

00:08:44.640 --> 00:08:46.980
times the square root of three over two

00:08:46.980 --> 00:08:50.525
minus square root of two times I

00:08:50.525 --> 00:08:53.296
and then I can add this to this

00:08:53.296 --> 00:08:56.229
and I'm going to get plus three halves

00:08:56.229 --> 00:08:59.938
plus square root of two times J.

00:08:59.938 --> 00:09:01.798
It looks a little bit complicated

00:09:01.798 --> 00:09:03.203
but we could type it in to our calculator

00:09:03.203 --> 00:09:06.272
and get approximations of
each of these two values

00:09:06.272 --> 00:09:10.517
and we essentially have
a at least a broken down

00:09:10.517 --> 00:09:12.995
into its components
representation of A plus B.

00:09:12.995 --> 00:09:15.819
In the next video, we're
now going to take this

00:09:15.819 --> 00:09:18.536
and figure out the actual
magnitude and direction

00:09:18.536 --> 00:09:20.498
of A plus B.


Magnitude of vector sums

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:02.270
Sal:Let's say that we have three vectors,

00:00:02.521 --> 00:00:04.171
vectors A, B and C, and we know that

00:00:04.171 --> 00:00:13.108
vector A plus vector B
is equal to vector C.

00:00:13.703 --> 00:00:17.203
Now given this I have some
interesting questions.

00:00:17.906 --> 00:00:21.250
Can you construct a scenario where the

00:00:21.250 --> 00:00:26.530
magnitude of vector C is equal to

00:00:26.938 --> 00:00:33.437
the magnitude of vector A plus
the magnitude of vector B?

00:00:36.404 --> 00:00:40.888
And can you also, using potentially
different vectors A and B,

00:00:40.888 --> 00:00:42.638
construct a scenario where the

00:00:42.638 --> 00:00:46.202
magnitude of vector C is greater than

00:00:46.560 --> 00:00:54.779
the magnitude of vector A plus
the magnitude of vector B?

00:00:55.217 --> 00:00:56.435
I encourage you to pause this

00:00:56.435 --> 00:00:58.248
video right now and try to do that.

00:00:58.248 --> 00:01:01.575
Try to come up with some
vectors A and B so that

00:01:01.575 --> 00:01:04.093
when you take their sum,
the magnitude of the

00:01:04.093 --> 00:01:07.157
sum is equal to the sum of the magnitudes.

00:01:07.157 --> 00:01:10.076
And also see if you can come
up with some vectors A and B so

00:01:10.076 --> 00:01:14.810
that if you take the sum of
the vectors that the magnitude

00:01:14.810 --> 00:01:18.460
of the sum is actually greater
than the sum of the magnitude.

00:01:18.460 --> 00:01:20.001
see if you can come up with that.

00:01:22.294 --> 00:01:24.234
I'm assuming you've given a go at it,

00:01:24.234 --> 00:01:26.184
and potentially you've gotten
a little bit frustrated,

00:01:26.184 --> 00:01:27.955
especially with the second one.

00:01:28.345 --> 00:01:31.186
The only way- Let's actually
just draw some vectors.

00:01:31.866 --> 00:01:34.716
If you have vector A like this,

00:01:34.716 --> 00:01:40.158
and let's say vector B
looks something like that,

00:01:40.498 --> 00:01:41.719
then A plus B ...

00:01:42.549 --> 00:01:46.222
We can just shift this
over, copy and paste.

00:01:46.622 --> 00:01:54.079
A plus B is going to look like this.

00:01:55.020 --> 00:01:56.670
A plus B, or vector C I guess we

00:01:56.670 --> 00:02:00.951
could say, is going to look like that.

00:02:01.741 --> 00:02:04.922
And notice, these three
vectors always form a triangle.

00:02:05.332 --> 00:02:10.921
If you have a triangle, one side cannot be

00:02:10.921 --> 00:02:15.717
longer than the sum of
the other two sides.

00:02:15.717 --> 00:02:16.500
Think about it.

00:02:16.500 --> 00:02:20.296
If you wanted this to
be longer what you could

00:02:20.296 --> 00:02:25.341
try to do is maybe change
vector B in a way so

00:02:25.341 --> 00:02:27.372
you're pushing it further and further out.

00:02:29.874 --> 00:02:32.482
Maybe if you change your
vector B a little bit you

00:02:32.732 --> 00:02:35.498
could get this vector C
to be longer and longer.

00:02:35.498 --> 00:02:37.717
Maybe if you made your vector B like this.

00:02:38.140 --> 00:02:42.050
Maybe your vector B would
look something like this.

00:02:42.279 --> 00:02:45.951
Now your vector C is getting
pretty long, but it's

00:02:45.951 --> 00:02:49.827
still shorter than the
sum of these two sides.

00:02:49.827 --> 00:02:51.405
To make it equal to the sum of these two

00:02:51.405 --> 00:02:53.640
sides you essentially have to make these

00:02:53.640 --> 00:02:56.250
two vectors go in the
exact same direction.

00:02:56.670 --> 00:03:00.768
To make it equal you have to
have vector A looking like this.

00:03:01.123 --> 00:03:03.264
You need to change the direction of B,

00:03:03.264 --> 00:03:04.761
or essentially construct a vector B

00:03:04.761 --> 00:03:07.422
that's going in the exact same direction.

00:03:09.790 --> 00:03:14.836
Only in this circumstance
will you get this scenario,

00:03:14.836 --> 00:03:18.746
where the magnitude of
vector C is equal to.

00:03:19.306 --> 00:03:23.657
Really the largest that
the magnitude of the sum

00:03:23.657 --> 00:03:26.277
can be is the sum of the
magnitudes, and that only

00:03:26.277 --> 00:03:29.809
happens when these two are
going in the same direction.

00:03:30.279 --> 00:03:36.232
These are going in the same,
in the exact same direction.

00:03:36.714 --> 00:03:38.935
This right over here is impossible.

00:03:39.465 --> 00:03:44.008
You could never get one side
of a triangle being longer than

00:03:44.008 --> 00:03:47.290
the sum of the other two sides,
based on what we just saw.

00:03:47.871 --> 00:03:49.671
You're probably saying, "What
about the circumstance where

00:03:49.671 --> 00:03:59.778
"the magnitude of our sum is
less than the magnitude of ..."

00:04:01.639 --> 00:04:04.039
Than the sum of the magnitudes,
I guess I could say.

00:04:05.510 --> 00:04:06.993
This is fairly typical.

00:04:06.993 --> 00:04:09.024
This is pretty much always the scenario,

00:04:09.024 --> 00:04:11.045
this is what's always going to be the case

00:04:15.049 --> 00:04:17.240
when the vectors are not
in the same direction.

00:04:18.236 --> 00:04:20.408
If someone drew a vector like this-

00:04:21.268 --> 00:04:22.842
Let me draw that a little bit straighter.

00:04:23.422 --> 00:04:26.566
If someone drew a vector
like this and a vector like

00:04:26.566 --> 00:04:30.147
this these clearly are not
going in the same direction,

00:04:30.606 --> 00:04:32.258
so the sum of these two vectors,

00:04:32.499 --> 00:04:35.056
the magnitude of that is going to be less

00:04:35.056 --> 00:04:36.898
than the sum of these two magnitudes.

00:04:36.898 --> 00:04:40.509
For example if the
magnitude here is five and

00:04:40.759 --> 00:04:44.479
the magnitude here is three then we know

00:04:44.479 --> 00:04:46.322
that if we were to add
these two things ...

00:04:46.822 --> 00:04:48.087
Let me just show you.

00:04:49.948 --> 00:04:52.061
Copy and paste.

00:04:52.916 --> 00:04:54.900
Actually let me just cut and paste,

00:04:55.821 --> 00:04:57.726
so that we can clean
things up a little bit.

00:04:58.837 --> 00:05:00.790
Cut and paste.

00:05:01.322 --> 00:05:03.151
Let's add these two vectors.

00:05:04.009 --> 00:05:07.134
So we know that the sum of these two,

00:05:07.681 --> 00:05:09.608
which is going to be this
vector right over here,

00:05:09.608 --> 00:05:16.493
its magnitude is going to be
less than five plus three.

00:05:16.714 --> 00:05:18.306
It is going to be less than eight.

00:05:18.571 --> 00:05:21.481
The only way that this magnitude
could even get to eight

00:05:21.481 --> 00:05:28.292
is if these two vectors went
in the exact same direction.


Scaling up a unit vector

WEBVTT
Kind: captions
Language: en

00:00:00.433 --> 00:00:04.100
Voiceover:Let's say that
I have the unit vector U

00:00:04.100 --> 00:00:06.300
and in the horizontal direction

00:00:06.300 --> 00:00:09.633
for every 1/3rd it goes,
it goes square root

00:00:09.633 --> 00:00:12.933
of eight over three in
the vertical direction.

00:00:13.300 --> 00:00:16.567
And we can verify that this
is indeed a unit vector.

00:00:16.567 --> 00:00:20.433
The magnitude of our vector U

00:00:20.433 --> 00:00:23.233
is going to be equal to the square root of

00:00:23.233 --> 00:00:25.566
the sum of the squares of the components.

00:00:25.566 --> 00:00:26.900
And this, once again, just comes

00:00:26.900 --> 00:00:28.666
straight out of the Pythagorean Theorem.

00:00:28.900 --> 00:00:30.233
So it's going to be equal to the

00:00:30.233 --> 00:00:33.133
square root of 1/3rd squared plus

00:00:36.100 --> 00:00:39.200
square root of eight over three squared.

00:00:42.500 --> 00:00:44.066
And what's that going to be?

00:00:44.100 --> 00:00:45.833
Well that's going to
be equal to the square

00:00:45.833 --> 00:00:51.733
root of one over nine
plus eight over nine.

00:00:54.300 --> 00:00:57.200
Which is equal to, and I
think you see this coming,

00:00:57.433 --> 00:01:01.366
square root of nine over
nine, which is equal to one.

00:01:01.366 --> 00:01:04.133
So this is indeed a unit vector.

00:01:04.566 --> 00:01:06.766
Let's say someone says,
"Well I like this direction,

00:01:06.766 --> 00:01:09.966
"but I don't want the
magnitude to just be one.

00:01:09.966 --> 00:01:14.733
"I want to find some vector V

00:01:15.433 --> 00:01:18.100
"that has the same
direction, so it has same

00:01:18.100 --> 00:01:24.827
"direction as U, but has
a magnitude of eleven."

00:01:27.011 --> 00:01:31.966
So we want the vector V to
have a magnitude of eleven.

00:01:31.966 --> 00:01:33.800
So how could I define vector V?

00:01:34.433 --> 00:01:36.566
One way to think about
it is if we just scale

00:01:36.566 --> 00:01:40.133
each of the components of U up by eleven,

00:01:40.300 --> 00:01:43.700
then we will have gone
in the same direction,

00:01:43.700 --> 00:01:47.233
but now we will have a
magnitude eleven times as big.

00:01:47.233 --> 00:01:48.500
And if we start with the magnitude of one

00:01:48.500 --> 00:01:50.133
we'll now have a magnitude of eleven.

00:01:50.366 --> 00:01:55.166
So we could say that vector
V could be eleven times

00:01:55.166 --> 00:02:01.933
one third, comma, square
root of eight over three.

00:02:02.433 --> 00:02:04.100
And so what is this going to be equal to?

00:02:04.100 --> 00:02:07.133
This is going to be equal
to eleven over three,

00:02:10.566 --> 00:02:19.600
comma, eleven square roots
of eight, over three.

00:02:20.100 --> 00:02:22.433
And I encourage you, if
you don't believe me,

00:02:22.433 --> 00:02:25.233
I guess on one level it should make sense

00:02:25.233 --> 00:02:27.166
that when you multiply a vector times

00:02:27.166 --> 00:02:32.966
a [scalerite] like this, it just
scales it in that direction by

00:02:32.966 --> 00:02:35.633
this factor, so if your magnitude was one,

00:02:35.633 --> 00:02:38.866
your magnitude will now be
eleven in that direction.

00:02:38.966 --> 00:02:42.100
But if you want, you could actually verify

00:02:42.100 --> 00:02:44.100
mathematically that the magnitude here,

00:02:44.100 --> 00:02:47.366
if you were to calculate
it, is going to be eleven

00:02:47.366 --> 00:02:51.866
instead of one, as in the
case of unit vector U.


Unit vector in same direction

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:01.826
Voiceover:We have the vector A,

00:00:01.826 --> 00:00:04.173
being equal to five, negative six.

00:00:04.173 --> 00:00:05.647
If we wanted to visualize that,

00:00:05.647 --> 00:00:07.306
we could start at the tail of vector A,

00:00:07.536 --> 00:00:09.445
and we'd wanna move five,

00:00:09.445 --> 00:00:11.507
positive five in the X directions.

00:00:11.507 --> 00:00:14.000
That'd be five to the right
based on our conventions.

00:00:14.000 --> 00:00:17.058
We'd wanna move negative
six in the Y directions,

00:00:17.058 --> 00:00:21.226
and based on our conventions
that would be six down.

00:00:21.226 --> 00:00:23.896
This length would be six.

00:00:23.896 --> 00:00:24.973
If this was a positive six,

00:00:24.973 --> 00:00:26.706
we would have moved six up, over here.

00:00:26.998 --> 00:00:29.664
This length over here is five.

00:00:30.725 --> 00:00:32.354
Going five to the right,

00:00:32.354 --> 00:00:35.660
and then six down would take us

00:00:35.660 --> 00:00:37.656
from the tail to the head.

00:00:37.871 --> 00:00:39.851
Now we are at the head,

00:00:39.851 --> 00:00:45.304
or tip, or arrow side of the vector.

00:00:45.304 --> 00:00:48.022
Let's draw it just like that.

00:00:48.022 --> 00:00:50.829
That's a reasonable
representation for our vector A.

00:00:51.029 --> 00:00:52.807
What else can we think about?

00:00:52.807 --> 00:00:55.120
How else could we think about this vector?

00:00:55.304 --> 00:00:58.595
What if we wanted to
figure out its magnitude?

00:00:58.595 --> 00:01:00.312
The magnitude of this vector

00:01:00.312 --> 00:01:02.254
is just the length of this arrow

00:01:02.254 --> 00:01:03.311
if we think of it visually,

00:01:03.311 --> 00:01:05.541
and this is a right triangle.

00:01:05.802 --> 00:01:07.763
We just need to figure out the length

00:01:07.763 --> 00:01:09.491
of the hypotenuse of this right triangle.

00:01:09.491 --> 00:01:11.776
Well we know the lengths
of the two shorter sides.

00:01:11.776 --> 00:01:14.711
The magnitude of vector A

00:01:14.711 --> 00:01:17.039
is just going to be
equal to the square root.

00:01:17.039 --> 00:01:17.977
This is coming straight out

00:01:17.977 --> 00:01:19.021
of the Pythagorean theorem.

00:01:19.021 --> 00:01:20.631
Square root of five squared.

00:01:21.323 --> 00:01:26.477
Five squared plus six squared.

00:01:26.908 --> 00:01:29.322
Which is equal to the square root of,

00:01:29.552 --> 00:01:35.047
25 plus 36 is 61.

00:01:35.047 --> 00:01:37.037
The length of this arrow, right over here,

00:01:37.037 --> 00:01:38.205
is square root of 61,

00:01:38.205 --> 00:01:39.931
which is a little bit under eight,

00:01:39.931 --> 00:01:42.207
which actually just makes visual sense.

00:01:42.207 --> 00:01:43.729
That was interesting.

00:01:43.729 --> 00:01:44.729
Let's say we wanna do something else.

00:01:44.729 --> 00:01:46.917
Let's say we wanted to
figure out some unit vector

00:01:46.917 --> 00:01:50.081
that goes in the same direction as A.

00:01:50.081 --> 00:01:51.530
It's a unit vector.

00:01:51.530 --> 00:01:53.276
That means that its length is one,

00:01:53.276 --> 00:01:55.113
but it goes in the same direction as A.

00:01:55.113 --> 00:01:56.787
We've seen this before.

00:01:56.787 --> 00:01:59.313
If we call this unit vector U,

00:01:59.313 --> 00:02:00.681
and I'll put the little hat on top of it

00:02:00.681 --> 00:02:01.915
to show that it's a unit vector.

00:02:01.915 --> 00:02:06.913
What we could do is,
we could take vector A,

00:02:06.913 --> 00:02:10.428
and divide it by,
essentially scale it down,

00:02:10.428 --> 00:02:15.070
by the magnitude, by the
magnitude of vector A.

00:02:15.070 --> 00:02:16.573
Another way of thinking about it,

00:02:16.573 --> 00:02:17.931
this would be equal to.

00:02:18.223 --> 00:02:20.024
This part right over here,

00:02:22.177 --> 00:02:26.002
we could write it like this.

00:02:26.371 --> 00:02:29.216
We could say one over the magnitude

00:02:29.216 --> 00:02:32.337
of vector A, times vector A.

00:02:32.721 --> 00:02:34.290
This makes it pretty clear

00:02:34.290 --> 00:02:36.018
we're just scaling vector A.

00:02:36.018 --> 00:02:37.745
This part, right over here,

00:02:37.929 --> 00:02:41.775
that's going to be equal to
one over square root of 61,

00:02:41.775 --> 00:02:42.629
we already figured out that

00:02:42.629 --> 00:02:44.713
this was already the square root of 61.

00:02:44.713 --> 00:02:46.249
We're gonna multiply it times vector A,

00:02:46.541 --> 00:02:50.378
which is five, negative six.

00:02:51.285 --> 00:02:55.202
Five, negative 6.

00:02:55.202 --> 00:02:59.225
Now we can scale it up, or scale it down,

00:02:59.225 --> 00:03:01.025
in this case, by just multiplying

00:03:01.025 --> 00:03:03.148
the scaler times each of these components.

00:03:03.148 --> 00:03:15.008
U is equal to five over
the square root of 61,

00:03:16.115 --> 00:03:18.824
all I'm doing is I'm scaling
each of the components

00:03:18.824 --> 00:03:20.041
in the X direction and the Y direction

00:03:20.041 --> 00:03:22.912
by one over the square root of 61,

00:03:22.912 --> 00:03:26.525
negative six over the square root of 61.

00:03:26.940 --> 00:03:28.441
That was interesting.

00:03:29.087 --> 00:03:30.431
Let's ask ourselves another question.

00:03:30.431 --> 00:03:35.461
What would we have to
multiply U by to get A?

00:03:35.461 --> 00:03:38.143
What scaler could we
multiply it by to get A?

00:03:38.358 --> 00:03:41.236
Our answer is actually right over here.

00:03:41.236 --> 00:03:43.927
We see that the vector U
is equal to the vector A,

00:03:43.927 --> 00:03:47.240
essentially divided by this scaler.

00:03:47.240 --> 00:03:49.247
If you multiply both
sides times the scaler,

00:03:49.247 --> 00:03:52.808
you get the magnitude of A,

00:03:52.808 --> 00:03:54.141
which is just a number,

00:03:54.141 --> 00:03:55.479
it's just a length of the side,

00:03:55.479 --> 00:03:59.551
times our unit vector U,

00:03:59.551 --> 00:04:04.381
is going to be equal
to our original vector.

00:04:04.381 --> 00:04:06.433
If we wanted to write our original vector,

00:04:06.433 --> 00:04:08.164
kinda broken up this way,

00:04:08.164 --> 00:04:09.896
we could say that our original vector,

00:04:09.896 --> 00:04:11.823
is equal to its magnitude,

00:04:11.823 --> 00:04:15.949
is equal to the square root of 61,

00:04:15.949 --> 00:04:20.894
times our unit vector, that makes sense,

00:04:20.894 --> 00:04:22.655
the length of this unit vector

00:04:22.655 --> 00:04:27.921
is one square root of 61th
of this entire length,

00:04:27.921 --> 00:04:29.938
right over here, and
this of course is going

00:04:29.938 --> 00:04:31.987
to be the same thing as the square root

00:04:31.987 --> 00:04:37.054
of 61 times all of this business.

00:04:38.376 --> 00:04:42.524
Copy and paste times all that business,

00:04:42.524 --> 00:04:44.800
and I think it might jump out at you

00:04:44.800 --> 00:04:46.071
if you multiply the scaler,

00:04:46.071 --> 00:04:47.751
you're gonna multiply it times
each of these components,

00:04:47.751 --> 00:04:51.287
you're going to be left
with five, negative six.

00:04:51.287 --> 00:04:52.600
The whole reason why I did this

00:04:52.600 --> 00:04:54.321
is just to make you a
little bit more comfortable

00:04:54.321 --> 00:04:55.527
with the different ways

00:04:55.527 --> 00:04:57.137
of manipulating these notations.

00:04:57.137 --> 00:04:58.792
Sometimes when you see this type of thing

00:04:58.792 --> 00:05:00.977
with these double lines on either side,

00:05:00.977 --> 00:05:02.275
it looks like something very fancy,

00:05:02.275 --> 00:05:04.056
but it's really just
talking about the magnitude

00:05:04.056 --> 00:05:06.344
of the vector, the length of the vector.

00:05:06.344 --> 00:05:08.538
The whole point of this is just to relate

00:05:08.538 --> 00:05:10.770
the vector itself to the magnitude

00:05:10.770 --> 00:05:13.266
of the vector to a unit vector

00:05:13.266 --> 00:05:15.620
going in the same direction.


Unit vectors

WEBVTT
Kind: captions
Language: en

00:00:00.752 --> 00:00:02.252
What I want to do in this video is

00:00:02.252 --> 00:00:06.510
explore the idea of a unit vector.

00:00:07.183 --> 00:00:09.413
A unit vector is just
a vector that goes in a

00:00:09.413 --> 00:00:16.454
particular direction that
has a magnitude of one.

00:00:17.885 --> 00:00:19.345
Let's take an example.

00:00:19.785 --> 00:00:23.536
Let's say that I have the
vector, let's say the vector A,

00:00:24.126 --> 00:00:28.066
and in the horizontal
direction for every three that

00:00:28.066 --> 00:00:31.897
it moves in the vertical
direction it moves up four.

00:00:34.072 --> 00:00:35.835
What else do we know about this?

00:00:36.066 --> 00:00:37.726
We could figure out A's magnitude,

00:00:38.276 --> 00:00:39.417
we can denote it like this.

00:00:40.257 --> 00:00:42.157
The magnitude of vector A,

00:00:42.368 --> 00:00:44.488
well this would just be the length of it.

00:00:44.758 --> 00:00:46.258
Let's try to visualize A.

00:00:46.770 --> 00:00:49.959
For every three that we go
in the horizontal direction

00:00:50.297 --> 00:00:52.699
we're going to go four in
the vertical direction.

00:00:57.495 --> 00:00:59.650
Vector A would look something like this.

00:01:02.250 --> 00:01:04.101
It would look like that.

00:01:04.771 --> 00:01:08.602
That is vector A.

00:01:09.148 --> 00:01:10.422
What's its magnitude?

00:01:10.463 --> 00:01:12.013
The magnitude is just the length of this

00:01:12.013 --> 00:01:13.820
vector right over here and we can use the

00:01:13.820 --> 00:01:16.523
Pythagorean theorem to figure this out.

00:01:17.003 --> 00:01:20.004
This length is going to
be the square root of

00:01:20.004 --> 00:01:22.254
the sum of the squares
of the other two sides.

00:01:22.945 --> 00:01:25.445
This is just the hypotenuse
of this right triangle.

00:01:25.805 --> 00:01:29.647
This is going to be three
squared plus four squared

00:01:33.205 --> 00:01:37.558
or this is going to be the
square root of nine plus 16,

00:01:37.808 --> 00:01:39.428
square root of 25,

00:01:41.603 --> 00:01:43.399
or it's going to be equal to five.

00:01:43.399 --> 00:01:44.477
You might have just recognized that this

00:01:44.477 --> 00:01:46.648
would be a three, four,
five right triangle.

00:01:47.058 --> 00:01:49.839
The length of this side
right over here is five

00:01:49.839 --> 00:01:54.320
so I could say the magnitude
of A is equal to five.

00:01:54.890 --> 00:01:58.980
This is clearly not a unit
vector. It has a length,

00:01:58.980 --> 00:02:02.041
it has a magnitude of
something other than one.

00:02:02.481 --> 00:02:07.038
Let's say we wanted to
construct a unit vector that has

00:02:07.038 --> 00:02:11.492
the same direction as A but
has a length of only one.

00:02:11.902 --> 00:02:13.102
Another way of thinking about it,

00:02:13.102 --> 00:02:16.383
let's say we wanted to figure out a

00:02:16.383 --> 00:02:19.323
vector that goes in the
exact same direction

00:02:19.323 --> 00:02:21.274
but it has one fifth the magnitude,

00:02:21.494 --> 00:02:23.274
it only has a magnitude of one.

00:02:23.274 --> 00:02:24.862
What could we do?

00:02:25.000 --> 00:02:28.055
If we scale everything down by a fifth,

00:02:28.331 --> 00:02:31.826
if we were to multiply each
of these components of our

00:02:31.826 --> 00:02:35.157
vector A by a fifth, or another
way of thinking about it,

00:02:35.157 --> 00:02:38.198
if we divide each of these
by the magnitude of A

00:02:38.198 --> 00:02:40.539
then we can construct this unit vector.

00:02:41.009 --> 00:02:43.619
I'm going to call that unit
vector, I'll call it A but

00:02:43.619 --> 00:02:47.460
instead of putting an arrow
on top I'm going to put --

00:02:48.429 --> 00:02:50.281
Actually just to not confuse ourselves

00:02:50.281 --> 00:02:52.671
let's call it U for a unit vector.

00:02:52.992 --> 00:02:55.322
To also make it clear it's
a unit vector and not just

00:02:55.322 --> 00:02:57.713
a normal vector I'm going
to put this little hat.

00:02:58.023 --> 00:03:00.214
Instead of this little
arrow when you put this hat

00:03:00.214 --> 00:03:02.544
this denotes that you're
dealing with a unit vector,

00:03:02.854 --> 00:03:05.384
a vector with magnitude of one.

00:03:05.884 --> 00:03:08.735
The unit vector, we could write it down.

00:03:08.935 --> 00:03:12.796
It's going to be equal to
each of these components,

00:03:12.796 --> 00:03:17.556
for A we just divide
by the magnitude of A.

00:03:22.374 --> 00:03:24.367
It's three in the horizontal direction,

00:03:24.367 --> 00:03:26.527
four in the vertical direction.

00:03:26.527 --> 00:03:28.698
Once again we just
divide by the magnitude,

00:03:29.908 --> 00:03:31.698
magnitude of our vector.

00:03:32.118 --> 00:03:35.219
This is going to be equal
to, the magnitude we

00:03:35.219 --> 00:03:36.869
already figured out is five so it's going

00:03:36.869 --> 00:03:41.751
to be three fifths in
the horizontal direction

00:03:42.151 --> 00:03:46.353
and four fifths in the vertical direction.

00:03:47.734 --> 00:03:50.547
You can verify. This is
going to have, one way to

00:03:50.547 --> 00:03:52.207
think about it the ratio between these two

00:03:52.207 --> 00:03:54.138
numbers is the exact
same thing as the ratio

00:03:54.138 --> 00:03:56.559
up here so we're going
in the same direction but

00:03:56.559 --> 00:04:00.260
the magnitude here is now
going to be equal to one.

00:04:00.260 --> 00:04:01.710
You can verify that, let's do that.

00:04:02.180 --> 00:04:05.681
What's the magnitude
now of our unit vector?

00:04:06.211 --> 00:04:07.512
Let me write that hat a little bit,

00:04:07.512 --> 00:04:08.742
that hat got a little crooked.

00:04:08.993 --> 00:04:11.013
What's the magnitude of unit vector U?

00:04:11.463 --> 00:04:14.024
It's going to be the square root of the

00:04:14.024 --> 00:04:16.136
sum of the squares of
these two components.

00:04:16.546 --> 00:04:18.507
It's going to be the,

00:04:18.507 --> 00:04:21.749
the square of three fifths is nine over 25

00:04:24.309 --> 00:04:29.901
plus the square of four
fifths is 16 over 25.

00:04:32.591 --> 00:04:33.321
What's that going to be?

00:04:33.761 --> 00:04:36.312
Well nine plus 16 is 25 so it's

00:04:36.312 --> 00:04:39.923
going to be the square root of 25.

00:04:40.203 --> 00:04:42.433
That plus that is 25.

00:04:42.633 --> 00:04:46.994
25 over 25 which is just
going to be equal to

00:04:46.994 --> 00:04:49.466
the square root, or I guess
we're doing the principal root,

00:04:49.466 --> 00:04:51.744
the positive square root of one,

00:04:51.860 --> 00:04:55.244
which is just equal to one,
which is exactly what we wanted.

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It goes in the same direction
but magnitude is one,

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that's why it's a unit vector.