00:00Hi guys, so this is
00:02my operations with vectors lesson.
00:04What I'm going to do
00:05is add vectors subtract vectors
00:08and multiply and or divide
00:11vectors by a scalar. I'm
00:13not going to multiply two
00:14vectors together because that's complicated
00:17and I'm going to do
00:17it in following lessons. So
00:21to add vectors very straightforward,
00:23I just add the I
00:24components at the Jacob ones
00:25at the K component. So
00:271 minus 5 is negative
00:28four minus two plus one
00:29is negative one, three plus
00:31seven is ten. So this
00:34vector plus this vector equals
00:35this vector easy. What that
00:37means geometrically is imagine I
00:40had, imagine I had four
00:42one. I'm just gonna do
00:44it in 2d because you
00:45can see it in 2d
00:47but it's the same principle
00:48in 3d but it's easy,
00:49a lot easier for me
00:50to show you in 2d.
00:51So imagine four one plus
00:54let's say
00:56two, three. So four one
01:00looks like this, four one.
01:03This is four one like
01:05that. Like that. And two
01:09three looks like, oops, two
01:17three looks like this, two,
01:19three. So let that. Now
01:21if I add these, what
01:24This means geometrically, if I
01:25put this like so, this
01:31plus this vector, plus this
01:33vector gives me this vector.
01:40And if you look, this
01:42is one, two, three, four,
01:44five, six, one, two, three,
01:46four, six, four. And this
01:50is
01:52It's 4 plus 2 is
01:536, 1 plus 3 is
01:544. Pretty straightforward. Subtracting not
01:59really in any way more
02:01difficult, except some people are
02:03going to make mistakes here
02:04with these minus minus. So
02:05it's 1 minus negative 5,
02:06which is 6 because it's
02:07plus. Minus 2 minus 1
02:10is minus 3, and 3
02:12minus 7 is negative 4,
02:15all in a vector, done
02:16easy. Let's do, well,
02:20Let's do the same numbers,
02:21four one, but this time
02:23I'm gonna do minus two
02:26three. So I should get
02:28two over three, two three.
02:30I should get two negative
02:33two. When I'm subtracting, oh,
02:36I do the same kind
02:37of thing, I do my
02:41four one, my four one,
02:44and then instead of plus
02:46two three, I,
02:48subtract two three. So instead
02:52of going that way, I'm
02:55gonna go, I'm gonna go
03:00that way. But I don't
03:02put it there now, I
03:03put it here. So this
03:07vector minus, let me just
03:11gonna do this. So that's
03:14what two three would have
03:15looked like.
03:16If it was 4 1
03:18plus 2 3, I'd be
03:19up here, but it's 4
03:201 minus 2 3, so
03:22I go down here. And
03:23this is then, you'll see
03:262. This is 2, negative
03:302, which is what we
03:32expected. Okay, nice. Next one.
03:38If A is this, find
03:403 A. This is straightforward.
03:44If a is 2, 1,
03:47negative 3, then 3a equals
03:513 times this. Now I
03:54can write it like this.
03:55I can write, I can
03:56put the 3 outside like
03:58this. Or, which I guess
04:01he wants me to do,
04:02I can multiply inside the
04:04vector. So 3 times 2
04:05is 6. 3 times 1
04:07is 3. 3 times negative
04:093 is negative 9.
04:12like this. And then two,
04:18per Ii, a half a,
04:21a half a is a
04:23half of this. Again, sometimes
04:27when we're dealing with fractions,
04:28it's actually nicer to just
04:30leave the half outside. But
04:31if you want just to
04:32be clear, this equals is
04:34the exact same thing as
04:35one, one over two, and
04:38negative three over two. I
04:39can put the fractions
04:40into the vector if I
04:42want. And also, similarly, I
04:45can write I plus a
04:46half J minus three over
04:47two K, same thing. Now,
04:49there's an interesting property, or
04:52an important property with multiples
04:54of vectors. So, let's stick
04:58with my one, four example.
05:01Here's, let me just get
05:01rid of this stuff here.
05:03Let's say I have one,
05:06let's say I have the
05:06vector, our four one, the
05:08vector.
05:08for one like this. Now
05:11if I multiply this vector
05:12by two, what I get
05:14is eight, two. Now one,
05:18two, three, four, five, six,
05:20seven, eight, one, two. Now
05:23what can you tell me
05:24about this vector compared to
05:25this vector? Well clearly it's
05:27twice as long and it's
05:32parallel, so that's important. If
05:35a vector is a multiple
05:36of another vector, then they
05:39are parallel. So 8, 2
05:41is parallel to 4, 1.
05:44Um, 40, 10 is parallel
05:46to 4, 1. And as
05:47well, you can say exactly
05:48how much bigger it is
05:49by whatever the multiple is,
05:51the skater multiple. Next bit.
05:54If points A and B
05:55have position vectors A and
05:57B, now this, if you
05:58only learn anything from this
05:59lesson, it's this point here.
06:01This is really, really, really
06:02important. It comes up all
06:04the time once we're still
06:04vector equations of lines, you'll
06:06see this happening all the
06:07time. If points a and
06:08b of position vectors a
06:09and b, then a b
06:11equals b minus a. Now
06:14remember, position vectors, if we
06:16have zero here, now let's
06:17say I have some point
06:19here, a, and I have
06:20some point here, b. Position
06:24vectors mean that from the
06:27origin, this vector is a,
06:32and from the origin,
06:33this vector is B. So
06:37how do I get from
06:39A to B? Well, I
06:41have to go minus A
06:44or negative A plus B.
06:47So A, B, let's just
06:49change it to red color
06:51here. Obviously, you don't have
06:52to change colors. A, B
06:56is equal to minus A,
07:01plus B, because this should
07:05have put this is B,
07:07little B. So, B is
07:09the point, little A is
07:11the position vector, B is
07:13the point, little B is
07:14the position vector, get use
07:16to those. And sometimes you'll
07:20see points and position vectors
07:22being used kind of synonymously,
07:25but be clear, be careful
07:26that you know which is
07:27which, there's a difference
07:29Green a point and a
07:30position vector. One's a point,
07:31one's a vector. So here
07:33anyway, I have a, b
07:34is negative a plus b,
07:37because I'm going to go
07:37this way, which is negative
07:38a plus b, which is
07:40obviously equal to b minus
07:42a. So that's why this
07:43is the rule, but you're
07:44going to use this rule
07:45all the time, b minus
07:46a to get a, b,
07:48it's b minus a, provided,
07:49these are the position vectors.
07:51Okay, so those are the
07:51kind of three main points
07:54I wanted to talk about.
07:57I now want to do
07:58two examples that hopefully aren't
08:00that difficult. Okay. A, B,
08:04C, D, A, B, C,
08:06D is a trapezium. This
08:07is a trapezium. A, D
08:09equals two, B, C. So
08:11that means, so the trapezium,
08:13just to be clear, this
08:15and this are the two
08:16parallel sides, obviously. Because you
08:21need two parallel sides in
08:23a trapezium. So this and
08:24this are parallel.
08:25He says AD is 2BC.
08:28So this is half the
08:29size of this long. This
08:31length here AD is twice
08:32as big as BC. P
08:35is the point on AD
08:36for which AP, so I
08:39have done, I can call
08:40that R. I should have
08:42called that P once again.
08:44That is point P. AP
08:50to PD is 3.
08:53to 1. So this is
08:55like the ratio is 3
08:57to 1. The common mistake
09:01is to say that this
09:02is 1 third of this,
09:06it's not, or sorry it's
09:081 third of the full
09:10thing, it's not, it's a
09:11quarter of the full thing
09:13because this is 1 quarter
09:14and this is 3 quarters,
09:15I'll get to that in
09:16a second. Then it says
09:18given A, B equals V.
09:20So the vector
09:21A B equals V. Now
09:23notice, you're going to start
09:25getting used to having your
09:26capital letters as the points
09:29and small letters as the
09:31vector. So this vector is
09:32just V. And B C
09:35is equal to W. Fine.
09:38Express in terms of V
09:39and W, A C. So
09:41what's A C? So to
09:42get from A, so I
09:44should put an arrow there.
09:45To get from A to
09:46C, I go V plus
09:49W.
09:49So you just kind of
09:50walk along the lines, walk
09:51along the lines you know.
09:53So I think these, if
09:54you want to get from
09:55A to C, walk along
09:56the path that you know.
09:57So look, you go up
09:58V plus W. So AC
10:00is V plus W. Easy.
10:05How do I get from
10:05B to D? Well, B
10:07to D is negative V
10:09or minus V, because I
10:11have to go down against
10:12the arrow. So minus V
10:14plus, I remember because AD
10:16is two of these.
10:17to BC, this is just
10:192W, so it's minus V
10:21plus 2W. Okay, there are
10:26the easy ones. Or C,
10:30okay, how do I get
10:32from or to C? And
10:34again, why do I keep
10:35pointing that or I'm meant
10:37to call that P? I
10:38think I changed the point
10:39from the question that I
10:41found, so that is
10:45P, Pc. Okay, how do
10:51I get from P to
10:52C? So I'm gonna have
10:53to go this way. I'm
10:56gonna go have to go
10:57this way, plus this way,
10:58plus this way. Okay. I
11:03can write it like that.
11:04Pc is equal to Pa,
11:07Pa, plus, Ab, plus, Bc.
11:13Now A, B plus B,
11:16C is easy as just
11:17V plus W, but this
11:18bits the tricky bit. So
11:19what I'm actually going to
11:20do first is I'm going
11:21to find AD. So I
11:24want to find AD first.
11:25That's the first kind of
11:26bit of working. And I
11:27need to actually move this
11:28down because this requires more
11:31space. Okay, so first I'm
11:35going to get AD. So
11:39AD, sorry, AP. First I'm
11:41going
11:41going to get a p.
11:50So to get a p,
11:52I need to find what
11:54fraction of of a a
11:58d is it. So remember
11:59what I said? If this
12:01is in the ratio three
12:02to one, then a p
12:04is equal to three quarters,
12:09three
12:09quarters of a d and
12:13we write it like that
12:14three quarters of a d
12:15which is actually equal to
12:17three quarters of 2w because
12:22we know 2w is a
12:24d from here to two
12:28of these. So this equals,
12:30we can cancel, we can
12:31cancel the two and the
12:32four, this equals three over
12:37two
12:37So that's AP. So PA,
12:43if AP is this, PA,
12:46and yes, this is deliberately
12:48a difficult confusing question, PA
12:50has to be negative three
12:53over two W, which finally
12:56gives me, I'm now able
12:57to deal with this PC,
12:59it's PA, which is minus
13:02three over two W, plus
13:05AB, which is
13:06plus V plus W. And
13:11yes, you do need to
13:13simplify this. I have minus
13:14three over two W plus
13:16one W, which would be
13:18minus one over two W.
13:20Let's put the V first.
13:21It's going to be V.
13:23It's going to be V
13:24minus a half W. And
13:30what I always like to
13:31do is kind of just
13:34Look at it. Does that
13:35make sense that if I
13:36went up V, up V,
13:39that's something like that. And
13:41then over a half of
13:44this over that way, does
13:46that make sense? Well, yeah,
13:48it actually does. That looks
13:50like a probability is the
13:52right answer. Maybe that doesn't
13:53look quite like half, but
13:54maybe my drawing is not
13:55perfect. So there you go.
13:57First two easy, last one,
14:00not easy. Hopefully it makes
14:02sense.
14:02That's a sense. Last question.
14:04This is easy, but again,
14:05I really, really wanted to
14:06make it clear that this
14:09is such a common question.
14:13Points A and B have
14:14position vectors, or equals this,
14:16and OB is this. Fine,
14:18I didn't even write the
14:19question here. I wanted to
14:20write, find A, B. So
14:25this is that rule that
14:27I said, this one. A,
14:30B is
14:30B minus A, if these
14:31are position vectors. So, A,
14:34B is just equal to
14:37B minus A, little B
14:39minus little A. So it's
14:40this, minus this. B minus
14:44one, four, three, minus three,
14:49one, minus two. The common
14:51mistake people make is they
14:52go, this minus this. It's
14:53always the second one, minus
14:55the first one. So if
14:56they want to B, A,
14:57you do this minus
14:58minus 1 minus 3 is
15:00negative 4, 4 minus 1
15:02is 3, and 3 minus
15:03negative 2 is 5. Let
15:06me make absolutely sure that
15:07that right, minus 4, 3
15:10negative, or sorry, plus 5,
15:11that is it. So that
15:14is finding a, b when
15:16you have position vectors a
15:17and b. Okay, that's it.
15:21Fairly long video. I don't
15:22think anything was overly overly
15:24complicated, but lots of important
15:25stuff.
15:26inside it and it's kind
15:28of just building up your
15:30kind of basic understanding and
15:33language, vector language. So you'll
15:37be able to deal with
15:40the more complicated lessons and
15:42problems that we're going to
15:42encounter. Okay, I'll see you
15:44in the next video. More
15:46vectors.