WEBVTT
1
00:00:00.540 --> 00:00:02.980 A:middle L:90%
Welcome to our review video where we're gonna look at
2
00:00:02.980 --> 00:00:06.019 A:middle L:90%
dot products and cross products in the later videos will
3
00:00:06.019 --> 00:00:09.230 A:middle L:90%
look at some calculus operations before I do this.
4
00:00:09.230 --> 00:00:11.109 A:middle L:90%
I want to do Ah, quick review of vector
5
00:00:11.109 --> 00:00:14.539 A:middle L:90%
notation here. So remember, uh, if you
6
00:00:14.539 --> 00:00:16.559 A:middle L:90%
watch the last video then you saw that I wrote
7
00:00:16.559 --> 00:00:23.050 A:middle L:90%
vectors like this X ex hat, plus a y
8
00:00:23.839 --> 00:00:29.410 A:middle L:90%
. Why Hat plus a see. See hat.
9
00:00:29.420 --> 00:00:32.670 A:middle L:90%
So what this means is that I have a vector
10
00:00:32.670 --> 00:00:35.820 A:middle L:90%
that has an X component, a y component and
11
00:00:35.820 --> 00:00:37.679 A:middle L:90%
a Z component. Now, if I were to
12
00:00:37.679 --> 00:00:40.789 A:middle L:90%
do this in the two dimensional direction in two dimensional
13
00:00:40.799 --> 00:00:46.380 A:middle L:90%
plane, I'd say x and a y. And
14
00:00:46.380 --> 00:00:50.200 A:middle L:90%
then this would be my total vector A where X
15
00:00:50.200 --> 00:00:52.320 A:middle L:90%
hat means thistles, the amount I'm going in the
16
00:00:52.320 --> 00:00:55.159 A:middle L:90%
X. Why hat means this is the amount I'm
17
00:00:55.159 --> 00:00:57.310 A:middle L:90%
going in. The Y and Z hat would mean
18
00:00:57.310 --> 00:00:59.619 A:middle L:90%
that's the amount I'm going in the sea. The
19
00:00:59.619 --> 00:01:03.270 A:middle L:90%
magnitude of such a vector would be equal to the
20
00:01:03.270 --> 00:01:07.459 A:middle L:90%
square of a X squared, plus a y squared
21
00:01:07.239 --> 00:01:11.819 A:middle L:90%
, plus a Z squared. Now let's give ourselves
22
00:01:11.819 --> 00:01:15.390 A:middle L:90%
a second vector again exactly as we had before.
23
00:01:15.390 --> 00:01:21.120 A:middle L:90%
B x X hat plus B y. Why Hat
24
00:01:21.120 --> 00:01:26.879 A:middle L:90%
plus Easy Z had now we already discussed dot products
25
00:01:26.890 --> 00:01:30.519 A:middle L:90%
and it was pretty simple affair. Where we have
26
00:01:30.519 --> 00:01:33.510 A:middle L:90%
a dot product gives us a scaler quantity, a
27
00:01:33.510 --> 00:01:36.629 A:middle L:90%
quantity that is not a vector anymore, and we
28
00:01:36.629 --> 00:01:38.409 A:middle L:90%
can write it in two ways. One is we
29
00:01:38.409 --> 00:01:42.230 A:middle L:90%
could have the product of the two magnitudes of the
30
00:01:42.230 --> 00:01:46.659 A:middle L:90%
vectors multiplied by the cosine of the angle between them
31
00:01:47.140 --> 00:01:52.019 A:middle L:90%
or identically. We can have the some of the
32
00:01:52.019 --> 00:01:53.209 A:middle L:90%
products of the similar components, so we have X
33
00:01:53.209 --> 00:01:59.500 A:middle L:90%
times, bx plus a Y Times B y plus
34
00:01:59.510 --> 00:02:04.930 A:middle L:90%
a Z times BZ. Okay, And what the
35
00:02:04.930 --> 00:02:07.560 A:middle L:90%
dot products doing is it's giving us some evaluation of
36
00:02:07.939 --> 00:02:10.659 A:middle L:90%
of how these things are multipliers, taking an amount
37
00:02:10.669 --> 00:02:13.979 A:middle L:90%
of B that's in the direction of a and then
38
00:02:13.979 --> 00:02:17.090 A:middle L:90%
multiplying those two quantities together. This is very helpful
39
00:02:17.090 --> 00:02:20.629 A:middle L:90%
when we come across things like work later on and
40
00:02:20.629 --> 00:02:23.069 A:middle L:90%
energy, um, and it shows up a couple
41
00:02:23.069 --> 00:02:27.159 A:middle L:90%
other places in the introductory physics classes as well.
42
00:02:27.169 --> 00:02:30.949 A:middle L:90%
Now the cross product, which again is written like
43
00:02:30.949 --> 00:02:32.219 A:middle L:90%
this, is a little more finicky. We can
44
00:02:32.219 --> 00:02:38.699 A:middle L:90%
quickly come up with the magnitude with a very similar
45
00:02:38.699 --> 00:02:43.139 A:middle L:90%
equation, except we're going to use the sine of
46
00:02:43.139 --> 00:02:46.020 A:middle L:90%
the angle between them but coming up with the direction
47
00:02:46.020 --> 00:02:49.539 A:middle L:90%
because, ah, cross product vector product actually produces
48
00:02:49.550 --> 00:02:53.189 A:middle L:90%
a vector as the result, Um, coming up
49
00:02:53.189 --> 00:02:54.199 A:middle L:90%
with the direction is a little more difficult. Generally
50
00:02:54.199 --> 00:02:58.009 A:middle L:90%
speaking will use what's called the right hand rule,
51
00:02:58.020 --> 00:03:00.949 A:middle L:90%
which is where you're going to curl your fingers and
52
00:03:00.949 --> 00:03:02.229 A:middle L:90%
have your thumb pointed. Now it's a little difficult
53
00:03:02.229 --> 00:03:06.990 A:middle L:90%
to illustrate this without having video, so I definitely
54
00:03:06.990 --> 00:03:08.009 A:middle L:90%
recommend if you want to see the right hand video
55
00:03:08.009 --> 00:03:10.039 A:middle L:90%
, you can look it up on YouTube or I'll
56
00:03:10.039 --> 00:03:13.669 A:middle L:90%
have a video later where I'll illustrate it with my
57
00:03:13.669 --> 00:03:16.580 A:middle L:90%
own hand. Um, but personally, the way
58
00:03:16.580 --> 00:03:19.139 A:middle L:90%
I like to think about it, when all I'm
59
00:03:19.139 --> 00:03:25.030 A:middle L:90%
doing is is calculating across product is with the Matrix
60
00:03:25.030 --> 00:03:27.800 A:middle L:90%
that I showed before. In fact, let's get
61
00:03:27.800 --> 00:03:32.460 A:middle L:90%
a clean sheet here and well, do the determinant
62
00:03:32.469 --> 00:03:35.620 A:middle L:90%
of this matrix very quickly. So if I have
63
00:03:35.629 --> 00:03:38.310 A:middle L:90%
a cross be that's going to be equal to X
64
00:03:38.310 --> 00:03:43.680 A:middle L:90%
hat. Why Hat Z hat? You have a
65
00:03:43.680 --> 00:03:51.560 A:middle L:90%
X a y and a Z B x b y
66
00:03:52.139 --> 00:03:53.669 A:middle L:90%
, and busy, and I'm going to take the
67
00:03:53.669 --> 00:03:58.000 A:middle L:90%
determinant of this matrix. Okay, so there's two
68
00:03:58.000 --> 00:04:00.030 A:middle L:90%
ways Thio to write this out. The simplest one
69
00:04:00.030 --> 00:04:02.830 A:middle L:90%
to remember might be this that you're going to say
70
00:04:02.830 --> 00:04:04.909 A:middle L:90%
. Okay, Um Well, first of all,
71
00:04:04.909 --> 00:04:08.889 A:middle L:90%
I'm gonna have the X hat direction that's gonna be
72
00:04:08.889 --> 00:04:13.860 A:middle L:90%
multiplied by this determinant, which is just a two
73
00:04:13.860 --> 00:04:16.360 A:middle L:90%
by two determinant, which is a little easier to
74
00:04:16.360 --> 00:04:21.310 A:middle L:90%
calculate. The most challenging thing about this is remembering
75
00:04:21.310 --> 00:04:25.449 A:middle L:90%
to put the negative in here. So now the
76
00:04:25.449 --> 00:04:28.149 A:middle L:90%
determined for this matrix will be the X's and disease
77
00:04:28.149 --> 00:04:31.560 A:middle L:90%
. So I'll have a X b X and a
78
00:04:31.939 --> 00:04:42.160 A:middle L:90%
see busy plus z hat times the times, the
79
00:04:42.160 --> 00:04:48.829 A:middle L:90%
determinant of X a Y over bx e y.
80
00:04:49.180 --> 00:04:50.579 A:middle L:90%
Now, if you remember, a two by two
81
00:04:50.579 --> 00:04:58.209 A:middle L:90%
matrix can be calculated like this. And as long
82
00:04:58.209 --> 00:05:00.240 A:middle L:90%
as you plug all those in, you'll get exactly
83
00:05:00.240 --> 00:05:01.819 A:middle L:90%
the right answer again. We'll review this when we
84
00:05:01.819 --> 00:05:04.569 A:middle L:90%
come to it. It's not gonna happen until physics
85
00:05:04.569 --> 00:05:08.730 A:middle L:90%
102 Really? And with a very small application here
86
00:05:08.730 --> 00:05:09.649 A:middle L:90%
in physics 101