00:00Hi guys. So in this
00:02lesson, we're going to look
00:03at two things, the magnitude
00:05of a vector and unit
00:07vectors and they are quite
00:08very much related. So firstly,
00:11before I do this, let's
00:14just go back to like
00:15vectors in 2D. Imagine I
00:16have the vector, I don't
00:18know, let's go 5, 3,
00:23something like that. So this
00:24is the vector 5, 3.
00:27We're going along
00:285, 1, 2, 3, 4,
00:295, and up 3. What
00:31is the magnitude of this
00:34vector? Well, hopefully you can
00:36see that the way you
00:38get it would be to
00:41use Pythagoras' theorem. The magnitude
00:46of this vector and the
00:47magnitude you can see here,
00:49this is how you write
00:50the magnitude you put it
00:51in these modular signs. So
00:53the magnitude, let's say this
00:54was V, this is
00:56the vector v, v equals
00:58this. The magnitude of v
01:00is just going to be
01:02the square root of five
01:04squared plus three squared because
01:06that's Pythagoras' theorem. That's five,
01:08that's three, that's v, v
01:11squared equals this squared plus
01:12this squared. So the length
01:13of v, or the magnitude
01:15of v is just the
01:16square root of this which
01:17will be 25 plus 934.
01:20So it's root 34 and
01:21you can just leave it
01:21like that. That's fine. That
01:23is in
01:24exact form. Now when we
01:27go into three dimensions, so
01:28this is in 2D, we're
01:29going to three dimensions, it's
01:31the same thing and you
01:32may or may not remember,
01:35hopefully you do from a
01:39previous topic when we got
01:42the length between the distance
01:44between two corners of a
01:46cuboid. It was in the
01:503D trigonometry I called it.
01:52It's the same kind of
01:55thing. This is essentially Pythagoras'
01:58theorem in 3D. So to
02:01get the magnitude of a
02:02vector, all we have to
02:04do, so this is A,
02:06so the magnitude of A,
02:09is we're basically using Pythagoras'
02:11theorem in three dimensions. So
02:13let me do that again.
02:17It is the square root
02:19of,
02:20Look, the nice thing about
02:21this is because you're squaring
02:23negatives, it's all just going
02:25to, it's just going to
02:26be 1 squared plus 2
02:27squared plus 4 squared. We
02:28will put in the negative
02:304 just to be clear
02:31with 1 squared plus 2
02:33squared plus minus 4 squared.
02:36Don't please make the mistake
02:39of saying this is 1
02:41squared plus 2 squared minus
02:43minus 4 or minus 16.
02:46Obviously it's plus 16. So
02:48this is then
02:48the square root of the
02:51square root of 1 plus
02:534 plus 16, which is,
02:56I don't need such a
02:57big square root, square root
03:00of 21, 16 plus 4
03:02is 20 plus 21, and
03:04leave it like that. Of
03:04course, if they wanted given
03:07to a certain number of
03:08decimal places fine, but I
03:10prefer that that's exact form.
03:13Alright, so that's easy. That's
03:14the magnitude of a vector.
03:16Next example, position vectors a
03:19and b, sorry, points a
03:21and b have position vectors
03:22this and this find the
03:24length of a b. So
03:25I've put in this example
03:26just to kind of make
03:27sure we all understand there's
03:29a difference between a point
03:31and a position vector. So
03:33when they say find the
03:34length a b, the length
03:38of the vector a b,
03:39that's the vector that goes
03:40from point a to point
03:41b, it's neither this nor
03:43this. And I need to
03:44find
03:44Maybe so the way I
03:45find a b, again, hopefully
03:47you remember this from the
03:48previous lesson, a b is
03:51b minus a, provided these
03:52are position vectors and they
03:53are, because it says it.
03:55So a b is a
03:58b is three negative one,
04:01negative two minus one, two,
04:06negative four. So we have
04:07to get this first, which
04:09is three minus one is
04:11two.
04:12Negative 1 minus 2 is
04:13negative 3 and negative 2
04:16minus negative 4 is negative
04:172 plus 4, which is
04:202. So it's 2, negative
04:213, 2. So then the
04:24magnitude of A, B is
04:28equal to the square root,
04:30again, straight from the formula
04:32booklet, the square root of
04:342 squared plus negative 3
04:38squared. Again, you can just
04:38write plus 3 squared there,
04:40plus
04:402 squared, which is the
04:44square root of 2 squared
04:484 plus 4 is 8
04:50plus 9, 17 squared of
04:5417. That's the magnitude. Next
04:58one. Find a unit vector
05:00parallel to 2, 3, 3,
05:021. So what's a unit
05:05vector? A unit vector, quite
05:07simply,
05:08is a vector of length
05:111. So it's like 1
05:12unit. So our magnitude once.
05:14So the magnitude of a
05:15vector that has a magnitude
05:17of 1. This guy's, this,
05:19this, this vector, his magnitude
05:21is, um, okay, well, let's
05:25say let, let A equal
05:28this 2, 3, 1. So
05:33if A is equal to
05:34this, what is the length
05:36of a, the magnitude of
05:38a, the magnitude of a
05:39is square root of 2
05:44squared plus 3 squared plus
05:461 squared, which is the
05:48square root of 4, 9
05:52is 13 plus 1 is
05:5514. So it's root 14.
05:57So that is not, that
05:59is a magnitude, sorry, that
06:01is a vector not a
06:03magnitude 1, it is a
06:04magnitude 14.
06:04But if I have a
06:08vector, if its magnitude is
06:1014, how can I find
06:11a vector that's parallel to
06:14that? So going the same
06:15direction, but has only a
06:17magnitude of magnitude 1? Well,
06:19I simply, I divide the
06:23vector by root 14, because
06:26if it has a length
06:27of root 14, and I
06:30divide by root 14, now
06:31it has a length of
06:321, a bit like
06:33like a bit like this.
06:37Well, let's say, okay, let
06:39me give you a more
06:39simple example. Imagine I have
06:41a vector, 1, 0, 0.
06:43So this guy has, let
06:45me write that again. This
06:47guy, 1, 0, 0, clearly
06:49has length 1. But imagine
06:54I had 6, 0, 0.
06:58So this guy's length is
07:01This is, there is magnitude
07:02is 6. So how can
07:04I find a magnitude of,
07:08I'll say a vector of
07:09magnitude 1? Well, I just
07:10divide it by 6. So
07:12I say 1 6 of
07:14this, which is this divided
07:16by 6, which is 1,
07:17this divided by 6 is
07:170, 0. This is a
07:20vector in the same direction
07:22as this. So parallel to
07:24this, but now it has
07:26a magnitude 1. So hopefully
07:28that makes sense.
07:29If I want to get
07:29a unit vector, all I
07:31do is divide the vector
07:34by its magnitude. There's a
07:39little formula for this that's
07:40not in the formula booklet.
07:42Let me write this in
07:43red. It's a hat, so
07:46hat for unit vector is
07:48equal to one over the
07:51magnitude of a times a,
07:56obviously you could just
07:57but the A there, but
07:58we write it like that
07:59just so it's very clear
08:00what you're doing. And when
08:04you write out, when you
08:05do a scalar, multiply by
08:07a vector, and this is
08:08a scalar, the magnitude is
08:11a scalar. It looks, well,
08:14it works out better like
08:14this. So the unit vector,
08:18a hat, let's call it,
08:21for this situation is one
08:24over root 14,
08:25times two, three, one. I
08:29can actually leave it like
08:30that. If you want to
08:31write it like this, two
08:33over root 14, three over
08:37root 14, and one over
08:39root 14, that is absolutely
08:42fine. Do a big bracket
08:44here. Now, if you were
08:47to get the, and try
08:49it actually if you want
08:49to get the magnitude of
08:50this vector, you'll see you
08:53get
08:531 and you can actually
08:56see it here because it's
08:57going to be this square
08:58plus this square plus this
08:59square and which is just
09:02going to be this square
09:05plus this square plus this
09:06square is going to be
09:074 over 14 plus 9
09:08over 14 plus 1 over
09:1014 which gives you 14
09:11over 14 and that's that's
09:13that's 1 the square root
09:14of it is 1. Okay,
09:17final one. A fine a
09:19vector of length 6 you
09:21And it's parallel to this.
09:24So let's go with A
09:26again, why not? Let A
09:28equal. Now, I said to
09:30you in the past, I
09:31prefer, and I think everyone
09:34prefers dealing with vectors in
09:35this form rather than a
09:37Ijk form. Ijk form just
09:39simply makes, it just looks
09:41more difficult because you got
09:42three letters in there. We're
09:43here. I don't actually have,
09:45I don't have three letters.
09:48I don't have any letters.
09:49So this is a, the
09:50magnitude of a is the
09:58square root of 2 squared.
10:00I'm going to just go
10:00plus 3 squared plus 5
10:03squared. And this is equal
10:04to 4 plus 9 plus
10:109 plus 25, which is
10:1334 or 38. So this
10:15is root of 38.
10:17And is that right? 25
10:20plus 9 is 34 plus
10:234 is 38. So this
10:25is by into 38. So
10:27unit vector parallel to this,
10:30a unit vector parallel to
10:31this is, let's go with
10:36a hat is equal to
10:401 over 38, 1 over
10:4238 times
10:452 negative 3, 5. So
10:49this is a unit vector.
10:51Vector of length 6. Let's
10:57go magnitude. Same thing, magnitude.
11:02Vector of magnitude 6. Parallel
11:06to this will be just
11:08this multiply by 6. And
11:10I can write that as
11:116 over root 38.
11:136 over root 38 times
11:172 negative 3 and 5.
11:24And again, if you want,
11:27you can multiply this in
11:28to give you 12 over
11:31root 38, 12 over root
11:3438, negative 18 over root
11:3838. And
11:4130 over root 38. And
11:47remember guys, this, I should
11:51draw that again. This is
11:55exactly the same as this.
11:57You can write a scalar
12:00times a vector is the
12:01same as the scalar times
12:03a vector multiplied multiplied it
12:05by each component inside like
12:07this. Okay, that's
12:09the lesson on magnitude, hopefully,
12:13I think the magnitude part
12:14is straightforward. Often the unit
12:17vector stuff people find a
12:21little bit challenging at the
12:22start to kind of understand
12:23what's going on. Just remember
12:24you're looking for a vector
12:26of magnitude one, so you
12:28just divide by its magnitude
12:30and that gives you magnitude
12:31one. That's it. Hope it
12:34all made sense, and I'll
12:35see you in the next
12:36lesson.
12:37you