00:00Hi guys. So in this
00:03lesson, we're going to look
00:03at the vector equation of
00:05a line. Now, this is
00:05one of the most important
00:06concepts in vectors. It's one
00:08of the most important applications
00:11of vectors. We can actually
00:12draw straight lines in three
00:14dimensions. And this is how
00:16we're going to do it.
00:17So this is in the
00:18formula booklet, 3 .14. That's
00:21a nice number. Vector equation
00:23of a line or equals
00:24A plus lambda B. Now,
00:28first
00:28Let's remember what a straight
00:31line is a straight line
00:33is essentially an infinite number
00:37of points. So when we
00:38have like y, let's say
00:40y equals 2x plus 5,
00:46you can sub in any
00:48x you want and you'll
00:49get a y -coordinate and
00:50this gives you a straight
00:51line in 2d. So we
00:55can get to
00:56to all the points on
00:59the line and infinite number
01:00of points just by subbing
01:02in the next five because
01:02there's an infinite number of
01:03x values I can sub
01:04in. So I get an
01:05infinite number of y values,
01:06hence an infinite number of
01:07points. So just remember that
01:09that's obviously useful to understand
01:13what we're going to do
01:14here. We're trying to find
01:15a way to get an
01:16infinite number of points that
01:18are all along a straight
01:19line. OK, and this is
01:21how we're going to do
01:22it.
01:24Starting from the origin, we
01:28are going to move along
01:32a vector to at point
01:34on the line. So we
01:36need a point on the
01:38line A. And this little
01:42A is the position vector
01:44of a point on the
01:45line. So let me write
01:46that down. That's important. A
01:48is a position vector.
01:52A is a position vector
01:55on the line. That's what
02:00A is. And then once
02:04we get to get to
02:06add point on the line
02:07and it can be any
02:07point on the line. So
02:08once we're on the, now
02:10we're on the line, we
02:12can add or subtract some
02:16amount of this B, which
02:19we're going to call it
02:20a direction vector. So if
02:22I add or subtract this
02:25direction vector, like I'll add
02:26it, I'll get to here.
02:28If I subtract it, I'll
02:28get to here. That gives
02:31me another two points on
02:32the line, because it's parallel
02:33to the line, so it's
02:34good. We go up here
02:35and along here. So that
02:36gets me to these two
02:37points. But I can add
02:42a part of this line,
02:44like half of this vector.
02:45I can add half of
02:46this vector.
02:48And I get to a
02:49point here. Or I could
02:51add a quarter of the
02:52vector. Or I could add
02:54an eighth of the vector,
02:57or one hundredth of a
02:58vector, or two of this
03:02vector, or negative two of
03:04this vector, and you get
03:05where I'm going, I can
03:07add as many as many
03:09of these vectors as I
03:10want, and any multiple of
03:13this vector, and I will
03:15get all the points
03:16the line. The way I
03:18deal with the, when I
03:21say any multiple of this
03:24direction vector, that is this
03:26lambda. So this is just
03:28a parameter, sometimes t and
03:29you're going to see it
03:30down here. It's t, two,
03:31two, one. It can be,
03:32can use any letter, but
03:33here in the formula, they
03:34use the Greek letter lambda.
03:36So we can add or
03:37subtract any amount or any
03:41multiple of this direction vector
03:43to give me all the
03:44points in the line. Now,
03:47that's confusing. The first thing
03:49here, I know, but you
03:53need to remember. We need
03:55a position vector and we
03:56need a direction vector. So
03:58I'm going to write down
03:58here B is the direction
04:01vector. You're going to hear
04:03that a lot from now
04:04on. Direction vector. Lambda.
04:12is a parameter. So that's
04:17the thing that we can
04:17change parameter. We can change
04:21that to get to all
04:22the different points on the
04:23line. And or think of
04:27or as the general vector.
04:34So are the general position
04:36vector. So what it is
04:37is or is I like
04:39to think of or as
04:40x, y, z, it is
04:42an x, y, z vector.
04:43So it's basically all the
04:45points. Okay. Assuming that makes
04:50some better sense, let's go
04:51and do some examples, which
04:53will help the whole thing
04:54make more sense. So here
04:56we have let oar equals
04:58this. So this is your
05:01position vector plus t times
05:04your direction vector. Fine. Now,
05:08what I do
05:08I want to show you
05:09guys is in geodubre. So
05:14geodubre can deal with you
05:15have to write capital X.
05:17I don't know why they
05:17make, let's use capital X.
05:19So capital X equals, now
05:23let me just use this
05:24line. It is one, two,
05:26negative three and the direction
05:27of it was two, two,
05:29one. So it is, not
05:32that, X equals one,
05:362, negative 3, and yes,
05:39I know this looks like
05:41a point, but this is
05:42how geodrude works. I'm going
05:44to do t plus t
05:45times 2, 2, 1. Press
05:49Enter. This is not what
05:51I want. I need to
05:53delete this thing here. Let
05:56me start again. I don't
06:01need to delete this. Let's
06:02try again. X equal
06:04equals 1, 2, negative 3,
06:111, 2, negative 3. Again,
06:14is that even when I
06:15said 1, 2, negative 3,
06:162, 2, 1. Okay, plus
06:20t times t times 2,
06:262, 1. Press Enter. There
06:29we go. So this is
06:30my equation of
06:33the line. Now as you
06:39see it is in three
06:40dimensions and it doesn't appear
06:45to be crossing any axes
06:47so remember in two dimensions
06:49a line has to cross
06:52it has to cross the
06:57x -axis and the y
06:58-axis unless it's parallel to
06:59them in three
07:01dimensions a line does not
07:02have to do that. Similarly
07:03with intersecting with another line,
07:05it doesn't have to intersect
07:06another line. Okay, so this
07:09is our straight line. Now,
07:12I hope you could tell
07:14me one point that is
07:16definitely on the line. I
07:19have a try right now
07:20before I tell you. Well,
07:22one point that's definitely on
07:23the line is this point
07:24one, two, negative three, because
07:26that is the position
07:29vector to this point, sorry,
07:32that is the position vector
07:36of the straight line. So
07:39one of the points in
07:41this see if we can
07:42see it here, it'd be
07:43difficult to see, but one,
07:45two, so there's one, two,
07:46and then it'll go down,
07:47it should cross it down
07:49at negative three. I'm guessing
07:52that looks something like that.
07:53So let's add it. One,
07:55two,
07:57negative three, there's the point
07:59and it's on the line.
08:02Okay, let's go back to
08:04this. So question A, it
08:07says write down, write down
08:11any point on the line.
08:13So these questions are just,
08:15these aren't typical exam type
08:16questions, they're just gonna get
08:17us thinking. So write down
08:18any point on the line.
08:20Now one, two, negative three,
08:21fine, that's a point on
08:22the line. I should have
08:23written write down a
08:25other than 1, 2, negative
08:263. So let's try and
08:27get another point. Remember there's
08:29an infinite number of points
08:31and in a class what
08:32I normally do is I
08:33get every student in the
08:34last to give me one
08:35point and they have to
08:35give me a different point.
08:38But it's quite easy because
08:39you can make t anything.
08:41So imagine t was 1.
08:42So a point on the
08:44line would be if t
08:45is 1, 1 plus 2
08:47is 3, 2 plus 2
08:52is 4.
08:53and minus three plus one
08:57is negative two. So this
09:00point is on the line
09:01three, four, negative two. Let's
09:02check it out. Three, four,
09:04negative two, three, four, negative
09:08two on the line. And
09:10then you can go, look
09:11what if t was 10?
09:13If t was 10, you'd
09:15have, and let's just do
09:17another point. If t was
09:1910, you would have
09:2120. So you do t
09:24times the 10 times 2
09:25is 20 and it's 1
09:27plus 20 is 21. So
09:30you're adding the x. So
09:32these are the x components
09:35here. So 1 plus 20.
09:39comma 2 plus 20 22
09:42and negative 3 plus 21
09:45would actually be 18. So
09:4721 22
09:4918, let's check it out.
09:5221, 22, 18, 21, 22,
09:5818. Now obviously I need
10:00to zoom out to see
10:01that point. That is not
10:03on the line white. Not
10:05did I do the wrong
10:06thing. 21, 22, 18. Let's
10:09figure it out. Yes, sorry.
10:11This is what I did
10:12wrong. Look, it's 10 times
10:14one is 10. 10, my
10:16three is seven.
10:17Oops, I should have put
10:20seven. Okay. Fine, I'm gonna
10:23leave this in because it's,
10:25those kind of mistakes are
10:26always interesting. So 21, 22
10:28and seven because no doubt
10:30if I have made them,
10:32some of you guys will
10:33make them 21, 22 and
10:34seven. There on the line.
10:43Okay.
10:45Next thing next question a
10:47note guys the difference between
10:49a point so that's a
10:50point and a vector even
10:52though it might be a
10:53position vector Even though it
10:57might be a position vector
10:58It's still a vector and
11:00it's not a point. This
11:01is a point zero right
11:02points horizontally with commas like
11:05you normally do and vectors
11:08Vertically in this vector form
11:10Okay par B it says
11:12write down a vector equation
11:13for the same line with
11:15a different position vector and
11:17different direction vector. Okay, so
11:21I want to show you.
11:22Another way you can write
11:23y equals two x plus
11:24three, y equals two x
11:26plus three, the same line,
11:28exactly the same line is
11:2910y equals 20x plus 30.
11:33That's the same line. Or
11:3620x minus 10y equals 20x
11:41minus 10y equals 20x minus
11:4110y equals 20x minus 10y
11:41equals 20x minus 10y equals
11:41minus 10x minus 10x
11:41plus 30 equals zero. They're
11:46all the same line. So
11:47I can do that with
11:48these vector equations as well.
11:51Remember, I just need app
11:53position vector of a point
11:55on the line. So let's
11:56pick one like this. Let's
11:58just pick this right now.
11:59To write down a vector
12:00equation, you have to write
12:01R equals, so R equals,
12:05I've seen people do things
12:06like equation equals this, but
12:08you have to do it,
12:08you have to write R.
12:09So R equals
12:09equals, let's do this point
12:1121, 22 and seven. And
12:15then I'll do plus t
12:17times, and I'm going to
12:19pick a different direction vector.
12:20So a different direction vector.
12:22I can pick any vector
12:23that is parallel to two
12:24two two two two one.
12:27So basically multiply those numbers
12:29by anything and you'll get
12:31the same direction vector. So
12:33I'm going to go with,
12:34let's multiply by a negative
12:3510. So I have negative
12:3720.
12:38negative 20, negative 10. So
12:42this is the same, this
12:45vector is in the same
12:47direction as this vector, or
12:49it's at least, it's actually
12:50not in the same direction,
12:51because it's going negative, whereas
12:55this is going positive. But
12:57by changing T to positive
12:59or negative, you get to
13:00all the points on the
13:02line, the same way as
13:03you get to all the
13:03points on this line. So
13:05when I draw this,
13:06It looks very different to
13:07this. It will give me
13:08exactly the same equation. So
13:11let's try it. Hopefully I
13:14haven't made another mistake. So
13:15x equals 21, 22, 7,
13:2221, 22, 7, plus t
13:27times, he actually changed him
13:30to lambda for me. So
13:32t times,
13:34Negative 20, negative 20, negative
13:4110. That's what I said.
13:43And you'll see, well, it's
13:45just replace this one, because
13:47I made it x. But
13:48as you saw there, it's
13:50exactly the same line. It
13:52still goes through those three
13:54points. So yeah, it's the
13:58same line. Okay, back to
14:01this next
14:02Next question. Hopefully this is
14:04helped with your understanding of
14:07how lines work, position vector
14:09and direction vector with the
14:11parameter gets you to all
14:12the points, right? Let's go
14:14to how many examples do
14:15I have here? I have
14:16just one. Just one more.
14:19Example two. Find the equation
14:22of the line that passes
14:23through the points p and
14:24q and p is this
14:25point and q is this
14:26point. Okay, to find the
14:29vector equation
14:30of a line, we need
14:31a point and we need
14:32a direction vector. So imagine
14:33here we have a point
14:35P and here we have
14:38a point Q. Now to
14:42find the equation of a
14:43line that goes through them,
14:48I'm just going to move
14:49Q. That's going to be
14:50easier. So to find the
14:53equation of this line that
14:54goes through these two points,
14:56remember zeros down here.
14:58So zero's down here somewhere
15:01could be anywhere but it's
15:02not on the line I'm
15:04assuming. The position vector little
15:11p is 5 negative 2,
15:193 and the position vector
15:22little q is
15:261, 1, 4. So neither
15:31of these vectors are a
15:34direction vector, so it would
15:35be very, very clear. We
15:36need to find the direction
15:38vector. How do I find
15:39the direction vector? Well, when
15:40you're given two points, the
15:42obvious one to find, well,
15:44the only one that you
15:45can find, is the vector
15:49pq. Because if you find
15:51that vector that goes from
15:52p to q, vector pq,
15:54you
15:54have a direction vector, remember
15:56a direction vector can be
15:57any vector that's in the
15:58direction of the line, or
16:00just parallel to the line.
16:03Okay, so that's the first
16:03thing I'm going to do.
16:04I'm going to do p,
16:06q, what is p, q.
16:07Remember again, I told you
16:09this rule is like always
16:11super, super important, always comes
16:13up. p, q is q
16:15minus p. And q is
16:20114,
16:22p is 5 negative 2,
16:263, q minus p equals
16:30negative 4, 1 plus 2
16:33is 3 and 4 minus
16:363 is 1. That is
16:37my direction vector. So my
16:40equation, therefore my equation, or
16:43equals a point. And again,
16:46I could choose any of
16:47these. So let's do, let's
16:50just
16:50choose the first one, 5
16:52negative 2, 3 plus lambda
16:57times, or t times whatever
16:58you want to choose for
16:59your parameter plus lambda times
17:02negative 4, 3, 1 because
17:06this is my position vector
17:07and this is my direction
17:09vector. Okay, that's it. That's
17:13the lesson. That's my explanation
17:15of vector equations of
17:18of a line in this
17:19form, we're going to come
17:20across two more forms of
17:22the vector equation of a
17:24line. But yeah, just remember
17:26position vector, direction vector, and
17:30parameter. And what I find
17:31helps, what really, really helps
17:34to understand this is that
17:35OR is x, y, z,
17:38it is kind of like
17:38all the points or all
17:40the points on the line.
17:43OK, that's it.
17:46the next lesson where we're
17:48going to look at different
17:52forms of equations of align
17:55in 3D. See you then.