00:00Hi guys, so in this
00:02video I want to introduce
00:03you to the vector product
00:05sometimes called the cross product
00:06So I told you in
00:08previous lessons there's kind of
00:10two ways that we look
00:10at two ways at multiplying
00:12vectors the first one's the
00:14scalar product and when we
00:15do that we get a
00:16scalar we just get a
00:17number and When we do
00:19the vector product we get
00:21you guessed it a vector
00:23and the way we do
00:25it is This I'm gonna
00:27do an example
00:28it looks confusing and complicated
00:33but it's not, it's fairly
00:35straightforward but it is actually
00:36easy to make mistakes and
00:37students often do make mistakes
00:38so we gotta be very
00:39very careful. Okay this is
00:44the vector product and this
00:45is a vector, this is
00:46a vector. The magnitude of
00:50the vector product is equal
00:52to, or the magnitude of
00:54V cross W is equal
00:56to
00:56the magnitude of V times,
00:57the magnitude of W times
00:58the sine of the angle
00:59between them. I'm going to
01:01talk about this formula in
01:02more detail in a following
01:04lesson. This lesson I just
01:06want us to find the
01:10vector product of two vectors
01:11and understand in which direction
01:17that vector goes and that
01:19leads me to this right
01:21hand rule. So the right
01:23hand rule
01:24This is kind of difficult
01:25for me to explain without
01:27literally showing you my hand.
01:28But here I have a
01:29hand and it has to
01:31be a right hand. So
01:33essentially v cross w does
01:36not equal w cross v.
01:38They're different. So you have
01:39to be very careful. They're
01:41the scalar product. Yes, a
01:43dot b equals b dot
01:44a, but that's not true
01:44for the vector product. So
01:46here essentially you're getting your
01:48first finger, your index finger
01:50points at the first vector.
01:52It's the first vector. In
01:55this case, V points at
01:56the first vector. And then
01:58your second finger points at
02:01the second vector, B, or
02:04W in this case. And
02:06then the cross product is
02:08your thumb when your thumb
02:09is going straight up or
02:11perpendicular to both fingers. And
02:13that's important because the cross
02:15product, and you'll see now
02:17when you see in the
02:18next lessons where we use
02:19the cross product a little
02:20bit,
02:20lot in when we're finding
02:22equations of planes, the cross
02:24product is perpendicular to both
02:27vectors. Now the cross product
02:30we can only get it
02:31or the vector product. I
02:32will use sometimes I'll say
02:34vector product sometimes I'll say
02:35cross product is the same
02:37thing. So yeah, we're going
02:41to get the cross product
02:43of two vectors and it
02:45is perpendicular to both vectors
02:47and it is only
02:48it's only defined for vectors
02:51in three dimensions so we
02:52can't get it into dimensions.
02:54You have to have a
02:55x, y and z component.
02:58Okay, so let me try
03:00to show you how this
03:03relates to this. So imagine
03:04your first and I actually
03:06want you to get out
03:06your right hand and try
03:08and do this. So obviously
03:10this isn't too deep but
03:11get your first finger and
03:14point it in the direction
03:15of a
03:16Then your second finger points
03:19in the direction of B
03:20and notice how your thumb
03:22just kind of naturally points
03:25straight up into the sky.
03:27That's the vector product. That's
03:28the direction of the vector
03:29product of A and B.
03:30But if I have them
03:31this way, now try and
03:35put your index finger pointing
03:38at A, but put your
03:40second finger pointing at B
03:42and you'll notice just naturally
03:44your
03:44thumb, you have to turn
03:46your hand upside down to
03:47get that second finger pointing
03:48at B and naturally your
03:50thumb points straight down to
03:52the ground. Okay, I want
03:54to show you that on
03:56geodruder. So I'll do it
03:57using these vectors. I have
03:58two negative three, one and
03:59four, one negative two. So
04:02here I do vector and
04:05I just do it's a
04:06point here. So it goes
04:07from, so what did I
04:09say two, two, negative three,
04:121, 2 negative 3, 1,
04:16and 4, 1 negative 2,
04:19and 4, 4, 1 negative
04:252. So they both go,
04:28when this is how geodia
04:29works, it just puts them
04:30from 0, 0, but that's
04:31fine for me now, just
04:33to show you how this
04:33works. So there is u,
04:36this one's u, and this
04:37one's v. So again, try
04:39and point here
04:40point your index finger at
04:45u, your second finger at
04:46v, and you'll see the
04:48cross product will kind of
04:50come straight out of it
04:51there, straight up, well not
04:53straight up, but perpendicular to
04:55both of those vectors can
04:56out that way. Look at
04:58this, I'll do cross, cross
05:01vector u and v, and
05:05there you go, it's coming
05:06straight out of there, and
05:06it's perpendicular
05:08two both vectors and maybe
05:11you can kind of see
05:12now how this is going
05:14to be useful for a
05:15plane because if these if
05:16you and V are you
05:18and V are on a
05:19plane this is perpendicular to
05:21the plane but anyway I'll
05:21talk about that in later
05:23videos okay so that's u
05:25cross v. Now imagine doing
05:27v cross u get your
05:28first finger your index finger
05:30pointing in v and your
05:31second finger pointing at u
05:33and what happens is you'll
05:34end up again turning your
05:36your
05:36and upside down. So I
05:39do cross v cross u.
05:44And there it is. It
05:45goes down like this. Now
05:47the magnitude, as I said,
05:50kind of very briefly, the
05:51magnitude of these vectors, let's
05:53just look at the first
05:54one, the magnitude of this
05:55vector is the magnitude of
05:57u times the magnitude of
05:58v times the sign of
06:00the angle between them. And
06:04again,
06:04I will do a lesson
06:07explaining that in more detail.
06:09Okay, let's go back to
06:11this here and we are
06:13going to actually do it.
06:15So it's fairly straightforward. We
06:18don't thankfully have to prove
06:20this. Again, I encourage you
06:21to just kind of Google
06:22it. You'll find out how
06:24the cross product equals this.
06:27I like how this with
06:30a unit vector equals
06:33equals this, please by all
06:35means go and find that
06:36out, but I'm not going
06:36to show it to you
06:37because we don't need to
06:38know how to do it.
06:39So this cross this is
06:41equal. So it, we're simply
06:44exchanging these letters, these components
06:47for these numbers, these components
06:49for these numbers and just
06:50filling them in there. You
06:51can imagine, you can see
06:53hopefully that this can cause,
06:56um, caustrine to make small
06:57mistakes, but we hopefully are
06:59going to be careful and
07:00that's not
07:01That happened. One little thing,
07:02let me show you this
07:03one little thing. If you,
07:07I'm just going to get
07:10a rectangle here and fill
07:12it in. So the first,
07:15the X component, what we
07:17do is we cover up
07:19the first two and it
07:24is, I think that's the
07:29second.
07:29and it is this v2
07:32times w3. So this times
07:34this minus this times this.
07:37So let's write that. Negative
07:39three times negative two minus
07:43this times this minus this
07:45times this, which is just
07:46one times one, one times
07:47one. Then the second of
07:51the y component. So now
07:53I'm going to pick this
07:54up and move it down
07:55here and cover the middle
07:57one.
07:57Now this one's a little
07:59bit different. It's not actually
08:01this times this minus this
08:03times this. It's this times
08:05this. So V3 times W1.
08:08There it is V3 times
08:08W1. So it's 1 times
08:104, 1 times 4, minus
08:13V1 times W3, 2 times
08:16negative 2, minus 2 times
08:18negative 2. And then finally,
08:21cover this. Now obviously you
08:22don't have to draw a
08:23rectangle, just put your finger
08:24in top of them.
08:25If you want, I think
08:26it helps. You don't even
08:27have to do that, so
08:28long as you can clearly
08:29follow which is which. Finally,
08:32then, it's just this times
08:34this, minus this times this.
08:36So two times one minus
08:38four times negative three. And
08:41this equals, and six minus
08:45one is five, four plus
08:49four is eight, and two
08:53plus 12, again, be very,
08:55very careful there, guys, is
08:5714. So the vector product,
09:00the vector product, I'm going
09:04to get rid of this,
09:05the vector product, of this
09:07times this is 5, 8,
09:0914, and it is a
09:11vector. And if you want
09:12to try and see is
09:13the magnitude of this, times
09:17the sine of the angle
09:19between them equal to the
09:20magnitude of this, you will
09:21find that it is. Okay,
09:24that's all I want to
09:24show you in this video.
09:26I'm certainly certainly not finished
09:27with the vector product. But
09:30hopefully you've got some kind
09:32of understanding of what it
09:33is. In the next videos,
09:34I'll talk about the properties
09:37of the vector product and
09:38I'll go into more detail
09:39about what it means geometrically.
09:42Okay, hope that makes sense.
09:43See you then.