IB · Mathematics Analysis and Approaches (AA)
AAHL 3.15.1 Intersecting lines
Learn AAHL 3.15.1 Intersecting lines in this free IB Mathematics Analysis and Approaches (AA) video lesson for AHL 3.15—Classification of lines.
Learn AAHL 3.15.1 Intersecting lines in this free IB Mathematics Analysis and Approaches (AA) video lesson for AHL 3.15—Classification of lines.
Learn AAHL 3.15.1 Intersecting lines in this free IB Mathematics Analysis and Approaches (AA) video lesson for AHL 3.15—Classification of lines.
In this lesson, the focus is on understanding the different types of lines in geometry: coincident, parallel, intersecting, and skew lines.
The process of finding the intersection point of two lines involves equating their equations and solving for the parameters, typically denoted as and . The final intersection point is determined by substituting these values back into the line equations, resulting in a position vector that represents the point of intersection.
00:00Hi guys, so in this
00:02lesson we're going to do
00:03a few things. We're going
00:05to look at these four
00:07different types of pairs of
00:08lines and then I'm going
00:10to show you how we
00:11find the intersecting points of
00:14two lines that intersect. We're
00:15going to use this example.
00:18So firstly, let's look at
00:19what these forwards mean. So
00:21coincident, coincident means they are
00:26the same.
00:28So this line and this
00:29line and I talked about
00:30this before they are actually
00:32the same line because they
00:35both have a position vector
00:37of a point on the
00:38line so 1, 1, 2
00:39and 4 or 6 are
00:40on the line and their
00:42direction vectors are parallel. Look
00:45this times 3 gives you
00:47this. So the direction vectors
00:48they don't have to be
00:49the same but they are
00:50going they are going in
00:52the same direction. Now note
00:54my lambda and
00:56and mu are different. I
01:00haven't used lambda twice. I
01:03have here actually, I need
01:04to delete it. But the
01:07parameter is different because obviously
01:12if you put 0 in
01:14here, you get 112. And
01:16if you put 0 in
01:16here, you get 406. So
01:18I have to have a
01:18different parameter. OK, that's coincident
01:22lines.
01:24Paralleluines, you know what Paralleluice,
01:26in 2D, it's same gradient,
01:29this is Paralleluice, it's the
01:31exact same in 3D. So
01:34we know what Paralleluice. Note
01:35the way we know what
01:37Paralleluice, the direction vectors are
01:39again Paralleluice, going in the
01:41same direction. I don't even
01:42care about the position vector.
01:44So 3, negative 1, and
01:454, is Paralleluice to 6,
01:47negative 2, 8. This times
01:512, is this.
01:52Next one, intersecting lines. So
01:55they intersect simple. I mentioned
01:57in a previous lesson two
01:59lines in 2D always intersect
02:03unless they're parallel. In 3D
02:05that's just not the case.
02:07In fact, if I was
02:08to just make up two
02:10lines, in fact I did
02:11for this one, the chances
02:13are they're not going to
02:15intersect. In fact it's highly
02:16unlikely if you just made
02:18up two lines in 3D
02:19that they
02:20would actually intersect. So these
02:23ones do, but these ones
02:25don't. These are called skew
02:26lines. So skew lines are
02:28lines that don't intersect and
02:31are not parallel. So I'm
02:33just going to show you
02:34in geodebra, an example of
02:38these pairs. So the first
02:39two, these are the same
02:41lines that I've done in
02:43my example. So this is
02:45the first line. And you
02:48see your
02:48position vector 1, 1, 2
02:50there. But when I draw
02:57this line on top of
02:58it, it's actually the exact
03:01same line. That's why you
03:02didn't see anything because it's
03:03the same line. If I
03:04take that one off, it's
03:06still the same line. Okay.
03:10The next one was this
03:13one and
03:16This one's their parallel. Look,
03:20you can clearly see their
03:21parallel. They're never going to
03:22touch each other and they're
03:23going in the same direction.
03:28Next, I had my two
03:30intersecting lines. That was this
03:32one and this one. So
03:33here you see the intersect
03:34and that's the point that
03:38we are going to find
03:39in a second. Let me
03:41just zoom a little bit.
03:43Okay, so they
03:44get intersect there. And then
03:46finally my skew lines were
03:48just two, I, well I
03:49picked, I used the first
03:50one, I just made up
03:51this one. Hopefully they don't
03:53intersect. But as I said,
03:55it's highly, highly unlikely two
03:57lines were intersects. So clearly
03:58they don't intersect and clearly
04:01they're not parallel. They are
04:03skew lines. Okay, hopefully that
04:08explanation makes a bit of
04:09sense. And what I'm going
04:12to do now
04:12is using this example we
04:14are going to find that
04:16intersecting point. So to find
04:19the intersecting point what we're
04:20going to do is a
04:21bit like if we're finding
04:23where two curves meet or
04:27two straight lines meet we
04:28equate them so that's exactly
04:29what we're going to do
04:30we're going to equate this
04:32to this so I'm going
04:37to go 6, 1, 2,
04:406, 1, 2 plus lambda
04:43times negative 1, 2, 1
04:47equals 2, 1, negative 14
04:52plus mu times 2, negative
04:582, 3. So what I'm
05:01going to do is I'm
05:02going to find where this
05:06equals this, I'm going to
05:07find the value of lambda
05:08and mu for which they
05:11are equal. So it's pretty
05:12straightforward. I'm going to do
05:146 minus lambda equals 2
05:18plus 2 mu. That's my
05:21first equation. Then I have
05:22a second equation, 1 plus
05:242 lambda equals 1 minus
05:282 mu. Then I have
05:30a third equation, 2 plus
05:33lambda equals
05:36negative 14 plus 3 mu.
05:44Now we could, well what
05:50I'm going to do is
05:50we're going to solve this
05:52with the calculator, but I'm
05:55just going to solve for,
05:57I'm just going to use,
05:58I only have two unknowns,
05:59I only need two equations.
06:02So I'm going to solve
06:03it for lambda and mu
06:04using the first two equations.
06:06Let's call this equation one,
06:08this equation two, and this
06:09equation three. So obviously if
06:13it was paper one, you
06:14have to do this out
06:15by hand, but I'm not
06:17going to go to that
06:18much effort. I am going
06:19to go to algebra, solve
06:21system of linear equations, two
06:23equations. I'm not going to
06:24use x and y. Let's
06:26use, so it's lambda and
06:27mu. Let's use l and
06:30m for lambda and mu.
06:33So it's 6 minus lambda
06:39equals 2 plus 2m over,
06:47what about over? Next equation,
06:491 plus 2 lambda equals
06:521 minus 2 mu. Press
06:56Enter and I get negative
06:584, negative 4. So I'm
07:00going to say from
07:01GDC. And I'm going to
07:05write and equations, equations 1
07:08and 2. Lambda equals negative
07:144 and mu equals 4.
07:19Now, if you solve these
07:21two equations for two unknowns,
07:23you are going to get
07:26a solution.
07:29because these two, these are
07:33just two kind of straight
07:34lines in two dimensions. So
07:37you're gonna get a solution.
07:40But you have to check
07:42for the third equation because
07:44like, let me try to
07:46explain this. If you have
07:48two, imagine you have two
07:49straight lines like, well let's
07:52go back to my two
07:54skew lines. Well look at
07:55these are my two
07:57Now if I just look
07:59at these in 2d, this
08:03is 2d, this is without
08:04my, if I, this is
08:05me looking straight down, they
08:06intersect. So I'm going to
08:08find, I am going to
08:10find a solution. If they
08:13intersect there, it's only when
08:15I go into my third
08:17dimension like this, that I'm
08:21actually able to have a
08:22look to see if they
08:23actually intersect or not.
08:25It's very, very important that
08:27you check, you have to
08:29check if that third equation
08:32is satisfied. So I'm going
08:34to sub into sub into
08:37equation three to see if
08:42it is satisfied. So that's
08:43two minus four equals negative
08:4714 plus three times four,
08:52which is negative two.
08:53equals negative two correct lines
09:00intersect. So the first thing
09:03I need to check is
09:04if they intersect now I
09:06said yes they if you're
09:07told they intersect then yeah
09:10you can just you can
09:11go straight to lambda because
09:12minus negative foreign and mu
09:13equals 4 but sometimes you
09:17might have to you might
09:18be asked do they intersect
09:20and the
09:21if they intersect to find
09:22the point of intersection, so
09:23you have to go and
09:24find that. If you don't
09:25know the intersection, you have
09:26to go and just show
09:27that the third equation is
09:28satisfied. Then, once I have
09:32my lambda and my mu,
09:35I'm going to sub it.
09:36I can sub it into
09:37either of these. I can
09:38either sub lambda in here
09:39or mu in here. I'll
09:40get the same answer, and
09:41that will give me the
09:44position vector of the point
09:45of intersection, and then hence
09:46I have the point.
09:496, 1, 2, I'm going
09:52to sub into line. Let's
09:58just call it L1. So
10:01L1 is 6, 1, 2,
10:046, 1, 2, plus lambda,
10:08lambda is negative 4, negative
10:114 times negative 1, 2,
10:151, negative 1, 2, 1.
10:17And this gives me six
10:21plus four is ten. One
10:25minus eight is negative seven,
10:27two minus four is negative
10:29two. Therefore, point of intersection
10:35is, because remember, this is
10:37a position vector. They wanted
10:38a point, it's ten negative
10:42seven, negative two.
10:45And if you sub 4
10:46in here, guess what you're
10:48gonna get? You're gonna get
10:4910, negative 7, and negative
10:542. So that's the point
10:56of intersection. Okay, that's it.
10:59You need to be able
10:59to know how to find
11:00the point of intersection between
11:02two lines, and you need
11:03to know what coincident lines
11:06are. Parallel lines are, and
11:09skew lines are. Hope that
11:11was all understood,
11:13see you in the next
11:15video.