1
00:00:00,000 --> 00:00:01,090
 

2
00:00:01,090 --> 00:00:03,890
I've been requested to do a
video on the cross product,

3
00:00:03,890 --> 00:00:08,390
and its special circumstances,
because I was at the point on

4
00:00:08,390 --> 00:00:11,580
the physics playlist where I had
to teach magnetism anyway,

5
00:00:11,580 --> 00:00:15,050
so this is as good a time as any
to introduce the notion of

6
00:00:15,050 --> 00:00:15,870
the cross product.

7
00:00:15,870 --> 00:00:18,170
So what's the cross product?

8
00:00:18,170 --> 00:00:21,890
Well, we know about vector
addition, vector subtraction,

9
00:00:21,890 --> 00:00:23,840
but what happens when you
multiply vectors?

10
00:00:23,840 --> 00:00:25,960
And there's actually two ways to
do it: with the dot product

11
00:00:25,960 --> 00:00:27,200
or the cross product.

12
00:00:27,200 --> 00:00:30,300
And just keep in mind these
are-- well, really, every

13
00:00:30,300 --> 00:00:33,140
operation we've learned is
defined by human beings for

14
00:00:33,140 --> 00:00:35,500
some other purpose, and there's
nothing different

15
00:00:35,500 --> 00:00:36,700
about the cross product.

16
00:00:36,700 --> 00:00:38,830
I take the time to say that
here because the cross

17
00:00:38,830 --> 00:00:41,130
product, at least when I first
learned it, seemed a little

18
00:00:41,130 --> 00:00:41,810
bit unnatural.

19
00:00:41,810 --> 00:00:42,740
Anyway, enough talk.

20
00:00:42,740 --> 00:00:44,180
Let me show you what it is.

21
00:00:44,180 --> 00:00:47,140
So the cross product of two
vectors: Let's say I have

22
00:00:47,140 --> 00:00:53,600
vector a cross vector b, and the
notation is literally like

23
00:00:53,600 --> 00:00:56,150
the times sign that you knew
before you started taking

24
00:00:56,150 --> 00:00:58,130
algebra and using dots and
parentheses, so it's

25
00:00:58,130 --> 00:00:59,770
literally just an x.

26
00:00:59,770 --> 00:01:05,400
So the cross product of vectors
a and b is equal to--

27
00:01:05,400 --> 00:01:08,130
and this is going to seem very
bizarre at first, but

28
00:01:08,130 --> 00:01:10,670
hopefully, we can get a little
bit of a visual feel of what

29
00:01:10,670 --> 00:01:12,180
this means.

30
00:01:12,180 --> 00:01:19,830
It equals the magnitude of
vector a times the magnitude

31
00:01:19,830 --> 00:01:28,350
of vector b times the sine of
the angle between them, the

32
00:01:28,350 --> 00:01:30,430
smallest angle between them.

33
00:01:30,430 --> 00:01:34,580
And now, this is the kicker, and
this quantity is not going

34
00:01:34,580 --> 00:01:36,040
to be just a scalar quantity.

35
00:01:36,040 --> 00:01:37,340
It's not just going
to have magnitude.

36
00:01:37,340 --> 00:01:41,190
It actually has direction, and
that direction we specify by

37
00:01:41,190 --> 00:01:43,760
the vector n, the
unit vector n.

38
00:01:43,760 --> 00:01:45,630
We could put a little
cap on it to show

39
00:01:45,630 --> 00:01:48,540
that it's a unit vector.

40
00:01:48,540 --> 00:01:50,790
There are a couple of things
that are special about this

41
00:01:50,790 --> 00:01:54,370
direction that's
specified by n.

42
00:01:54,370 --> 00:01:59,730
One, n is perpendicular to
both of these vectors.

43
00:01:59,730 --> 00:02:02,890
It is orthogonal to both of
these vectors, so we'll think

44
00:02:02,890 --> 00:02:04,140
about it in a second
what that implies

45
00:02:04,140 --> 00:02:05,580
about it just visually.

46
00:02:05,580 --> 00:02:09,788
And then the other thing is the
direction of this vector

47
00:02:09,788 --> 00:02:11,840
is defined by the right
hand rule, and we'll

48
00:02:11,840 --> 00:02:12,620
see that in a second.

49
00:02:12,620 --> 00:02:15,290
So let's try to think
about this visually.

50
00:02:15,290 --> 00:02:18,780
And I have to give you an
important caveat: You can only

51
00:02:18,780 --> 00:02:22,050
take a cross product when
we are dealing in three

52
00:02:22,050 --> 00:02:22,890
dimensions.

53
00:02:22,890 --> 00:02:25,970
A cross product really has--
maybe you could define a use

54
00:02:25,970 --> 00:02:28,760
for it in other dimensions or a
way to take a cross product

55
00:02:28,760 --> 00:02:31,230
in other dimensions, but it
really only has a use in three

56
00:02:31,230 --> 00:02:33,820
dimensions, and that's useful,
because we live in a

57
00:02:33,820 --> 00:02:35,420
three-dimensional world.

58
00:02:35,420 --> 00:02:36,120
So let's see.

59
00:02:36,120 --> 00:02:37,280
Let's take some cross
products.

60
00:02:37,280 --> 00:02:39,300
I think when you see it
visually, it will make a

61
00:02:39,300 --> 00:02:41,490
little bit more sense,
especially once you get used

62
00:02:41,490 --> 00:02:43,332
to the right hand rule.

63
00:02:43,332 --> 00:02:49,250
So let's say that
that's vector b.

64
00:02:49,250 --> 00:02:50,550
I don't have to draw a
straight line, but it

65
00:02:50,550 --> 00:02:53,130
doesn't hurt to.

66
00:02:53,130 --> 00:02:54,610
I don't have to draw
it neatly.

67
00:02:54,610 --> 00:02:57,370
 

68
00:02:57,370 --> 00:02:59,320
OK, here we go.

69
00:02:59,320 --> 00:03:04,650
Let's say that that is vector
a, and we want to take the

70
00:03:04,650 --> 00:03:06,690
cross product of them.

71
00:03:06,690 --> 00:03:09,040
This is vector a.

72
00:03:09,040 --> 00:03:10,230
This is b.

73
00:03:10,230 --> 00:03:12,250
I'll probably just switch to one
color because it's hard to

74
00:03:12,250 --> 00:03:13,550
keep switching between them.

75
00:03:13,550 --> 00:03:18,690
And then the angle between
them is theta.

76
00:03:18,690 --> 00:03:23,350
Now, let's say the length of a
is-- I don't know, let's say

77
00:03:23,350 --> 00:03:28,060
magnitude of a is equal to
5, and let's say that the

78
00:03:28,060 --> 00:03:32,815
magnitude of b is equal to 10.

79
00:03:32,815 --> 00:03:34,310
It looks about double that.

80
00:03:34,310 --> 00:03:36,060
I'm just making up the
numbers on the fly.

81
00:03:36,060 --> 00:03:37,730
So what's the cross product?

82
00:03:37,730 --> 00:03:39,910
Well, the magnitude
part is easy.

83
00:03:39,910 --> 00:03:45,660
Let's say this angle is
equal to 30 degrees.

84
00:03:45,660 --> 00:03:48,880
30 degrees, or if we wanted
to write it in radians, I

85
00:03:48,880 --> 00:03:51,200
always-- just because we grow
up in a world of degrees, I

86
00:03:51,200 --> 00:03:53,400
always find it easier to
visualize degrees, but we

87
00:03:53,400 --> 00:03:55,780
could think about it in terms
of radians as well.

88
00:03:55,780 --> 00:04:01,030
30 degrees is-- let's see,
there's 3, 6-- it's pi over 6,

89
00:04:01,030 --> 00:04:04,670
so we could also write
pi over 6 radians.

90
00:04:04,670 --> 00:04:07,280
But anyway, this is a 30-degree
angle, so what will

91
00:04:07,280 --> 00:04:09,810
be a cross b?

92
00:04:09,810 --> 00:04:16,120
a cross b is going to equal
the magnitude of a for the

93
00:04:16,120 --> 00:04:21,000
length of this vector, so it's
going to be equal to 5 times

94
00:04:21,000 --> 00:04:26,950
the length of this b vector, so
times 10, times the sine of

95
00:04:26,950 --> 00:04:28,150
the angle between them.

96
00:04:28,150 --> 00:04:29,690
And, of course, you
could've taken the

97
00:04:29,690 --> 00:04:31,100
larger, the obtuse angle.

98
00:04:31,100 --> 00:04:33,970
You could have said this was the
angle between them, but I

99
00:04:33,970 --> 00:04:36,780
said earlier that it was the
smaller, the acute, angle

100
00:04:36,780 --> 00:04:38,505
between them up to 90 degrees.

101
00:04:38,505 --> 00:04:46,900
This is going to be sine of 30
degrees times this vector n.

102
00:04:46,900 --> 00:04:50,690
And it's a unit vector, so I'll
go over what direction

103
00:04:50,690 --> 00:04:51,770
it's actually pointing
in a second.

104
00:04:51,770 --> 00:04:54,100
Let's just figure out
its magnitude.

105
00:04:54,100 --> 00:04:57,910
So this is equal to 50, and
what's sine of 30 degrees?

106
00:04:57,910 --> 00:04:59,930
Sine of 30 degrees is 1/2.

107
00:04:59,930 --> 00:05:02,520
You could type it in your
calculator if you're not sure.

108
00:05:02,520 --> 00:05:11,780
So it's 5 times 10 times 1/2
times the unit vector, so that

109
00:05:11,780 --> 00:05:14,870
equals 25 times the
unit vector.

110
00:05:14,870 --> 00:05:17,310
Now, this is where it gets,
depending on your point of

111
00:05:17,310 --> 00:05:20,250
view, either interesting
or confusing.

112
00:05:20,250 --> 00:05:24,090
So what direction is this
unit vector pointing in?

113
00:05:24,090 --> 00:05:25,310
So what I said earlier is, it's

114
00:05:25,310 --> 00:05:27,140
perpendicular to both of these.

115
00:05:27,140 --> 00:05:28,300
So how can something be

116
00:05:28,300 --> 00:05:30,710
perpendicular to both of these?

117
00:05:30,710 --> 00:05:32,480
It seems like I can't
draw one.

118
00:05:32,480 --> 00:05:35,240
Well, that's because right here,
where I drew a and b,

119
00:05:35,240 --> 00:05:37,340
I'm operating in
two dimensions.

120
00:05:37,340 --> 00:05:40,270
But if I have a third dimension,
if I could go in or

121
00:05:40,270 --> 00:05:45,540
out of my writing pad or, from
your point of view, your

122
00:05:45,540 --> 00:05:50,350
screen, then I have a vector
that is perpendicular to both.

123
00:05:50,350 --> 00:05:53,190
So imagine of vector that's-- I
wish I could draw it-- that

124
00:05:53,190 --> 00:05:56,620
is literally going straight in
at this point or straight out

125
00:05:56,620 --> 00:05:57,290
at this point.

126
00:05:57,290 --> 00:05:58,350
Hopefully, you're seeing it.

127
00:05:58,350 --> 00:05:59,590
Let me show you the
notation for that.

128
00:05:59,590 --> 00:06:04,010
So if I draw a vector like this,
if I draw a circle with

129
00:06:04,010 --> 00:06:09,670
an x in it like that, that is a
vector that's going into the

130
00:06:09,670 --> 00:06:12,290
page or into the screen.

131
00:06:12,290 --> 00:06:17,690
And if I draw this, that is a
vector that's popping out of

132
00:06:17,690 --> 00:06:18,190
the screen.

133
00:06:18,190 --> 00:06:20,106
And where does that convention
come from?

134
00:06:20,106 --> 00:06:21,920
It's from an arrowhead,
because what does

135
00:06:21,920 --> 00:06:23,810
an arrow look like?

136
00:06:23,810 --> 00:06:27,900
An arrow, which is our
convention for drawing

137
00:06:27,900 --> 00:06:34,410
vectors, looks something like
this: The tip of an arrow is

138
00:06:34,410 --> 00:06:37,680
circular and it comes to a
point, so that's the tip, if

139
00:06:37,680 --> 00:06:40,520
you look at it head-on, if it
was popping out of the video.

140
00:06:40,520 --> 00:06:41,980
And what does the tail of
an arrow look like?

141
00:06:41,980 --> 00:06:43,742
It has fins, right?

142
00:06:43,742 --> 00:06:46,020
There would be one fin here
and there'd be another fin

143
00:06:46,020 --> 00:06:47,680
right there.

144
00:06:47,680 --> 00:06:51,840
And so if you took this arrow
and you were to go into the

145
00:06:51,840 --> 00:06:55,070
page and just see the back of
the arrow or the behind of the

146
00:06:55,070 --> 00:06:57,050
arrow, it would look
like that.

147
00:06:57,050 --> 00:06:59,140
So this is a vector that's going
into the page and this

148
00:06:59,140 --> 00:07:01,650
is a vector that's going
out of the page.

149
00:07:01,650 --> 00:07:06,390
So we know that n is
perpendicular to both a and b,

150
00:07:06,390 --> 00:07:07,890
and so the only way you can
get a vector that's

151
00:07:07,890 --> 00:07:11,110
perpendicular to both of these,
it essentially has to

152
00:07:11,110 --> 00:07:16,750
be perpendicular, or normal,
or orthogonal to the plane

153
00:07:16,750 --> 00:07:19,260
that's your computer screen.

154
00:07:19,260 --> 00:07:23,020
But how do we know if it's going
into the screen or how

155
00:07:23,020 --> 00:07:26,130
do we know if it's coming out of
the screen, this vector n?

156
00:07:26,130 --> 00:07:28,950
And this is where the right hand
rule-- I know this is a

157
00:07:28,950 --> 00:07:29,680
little bit overwhelming.

158
00:07:29,680 --> 00:07:32,550
We'll do a bunch of example
problems. But the right hand

159
00:07:32,550 --> 00:07:37,620
rule, what you do is you take
your right hand-- that's why

160
00:07:37,620 --> 00:07:40,560
it's called the right hand
rule-- and you take your index

161
00:07:40,560 --> 00:07:44,010
finger and you point it in the
direction of the first vector

162
00:07:44,010 --> 00:07:45,960
in your cross product,
and order matters.

163
00:07:45,960 --> 00:07:47,410
So let's do that.

164
00:07:47,410 --> 00:07:49,940
So you have to take your finger
and put it in the

165
00:07:49,940 --> 00:07:53,330
direction of the first arrow,
which is a, and then you have

166
00:07:53,330 --> 00:07:56,800
to take your middle finger and
point it in that direction of

167
00:07:56,800 --> 00:07:58,290
the second arrow, b.

168
00:07:58,290 --> 00:08:00,770
So in this case, your
hand would look

169
00:08:00,770 --> 00:08:01,850
something like this.

170
00:08:01,850 --> 00:08:03,110
I'm going to try to draw it.

171
00:08:03,110 --> 00:08:11,760
 

172
00:08:11,760 --> 00:08:15,490
This is pushing the abilities
of my art skills.

173
00:08:15,490 --> 00:08:19,950
 

174
00:08:19,950 --> 00:08:21,100
So that's my right hand.

175
00:08:21,100 --> 00:08:24,510
My thumb is going to be
coming down, right?

176
00:08:24,510 --> 00:08:26,530
That is my right hand
that I drew.

177
00:08:26,530 --> 00:08:29,510
This is my index finger, and
I'm pointing it in the

178
00:08:29,510 --> 00:08:30,200
direction of a.

179
00:08:30,200 --> 00:08:34,760
Maybe it goes a little bit more
in this direction, right?

180
00:08:34,760 --> 00:08:37,850
Then I put my middle finger, and
I kind of make an L with

181
00:08:37,850 --> 00:08:40,169
it, or you could kind of say
it almost looks like you're

182
00:08:40,169 --> 00:08:42,150
shooting a gun.

183
00:08:42,150 --> 00:08:44,520
And I point that in the
direction of b, and then

184
00:08:44,520 --> 00:08:47,540
whichever direction that your
thumb faces in, so in this

185
00:08:47,540 --> 00:08:50,220
case, your thumb is going
into the page, right?

186
00:08:50,220 --> 00:08:53,040
Your thumb would be going down
if you took your right hand

187
00:08:53,040 --> 00:08:55,150
into this configuration.

188
00:08:55,150 --> 00:08:58,700
So that tells us that the vector
n points into the page.

189
00:08:58,700 --> 00:09:03,110
So the vector n has magnitude
25, and it points into the

190
00:09:03,110 --> 00:09:05,810
page, so we could draw it
like that with an x.

191
00:09:05,810 --> 00:09:07,720
If I were to attempt to draw
it in three dimensions, it

192
00:09:07,720 --> 00:09:09,890
would look something
like this.

193
00:09:09,890 --> 00:09:11,250
Vector a.

194
00:09:11,250 --> 00:09:13,150
Let me see if I can give
some perspective.

195
00:09:13,150 --> 00:09:16,030
 

196
00:09:16,030 --> 00:09:22,060
If this was straight down, if
that's vector n, then a could

197
00:09:22,060 --> 00:09:23,085
look something like that.

198
00:09:23,085 --> 00:09:25,610
Let me draw it in the
same color as a.

199
00:09:25,610 --> 00:09:29,670
a could look something like
that, and then b would look

200
00:09:29,670 --> 00:09:30,230
something like that.

201
00:09:30,230 --> 00:09:32,170
I'm trying to draw a
three-dimensional figure on

202
00:09:32,170 --> 00:09:34,320
two dimensions, so it might look
a little different, but I

203
00:09:34,320 --> 00:09:34,900
think you get the point.

204
00:09:34,900 --> 00:09:36,630
Here I drew a and
b on the plane.

205
00:09:36,630 --> 00:09:38,300
Here I have perspective
where I was able to

206
00:09:38,300 --> 00:09:40,770
draw n going down.

207
00:09:40,770 --> 00:09:44,940
But this is the definition
of a cross product.

208
00:09:44,940 --> 00:09:46,910
Now, I'm going to leave it
there, just because for some

209
00:09:46,910 --> 00:09:49,020
reason, YouTube hasn't been
letting me go over the limit

210
00:09:49,020 --> 00:09:51,490
as much, and I will do another
video where I do several

211
00:09:51,490 --> 00:09:53,880
problems, and actually, in the
process, I'm going to explain

212
00:09:53,880 --> 00:09:55,600
a little bit about magnetism.

213
00:09:55,600 --> 00:09:57,960
And we'll take the cross product
of several things, and

214
00:09:57,960 --> 00:09:59,810
hopefully, you'll get a little
bit better intuition.

215
00:09:59,810 --> 00:10:01,760
See you soon.

216
00:10:01,760 --> 00:00:00,000