IB · Mathematics Analysis and Approaches (AA)
AAHL 3.18.5 Angle between two planes
Learn AAHL 3.18.5 Angle between two planes in this free IB Mathematics Analysis and Approaches (AA) video lesson for AHL 3.18—Intersections of lines & planes.
Learn AAHL 3.18.5 Angle between two planes in this free IB Mathematics Analysis and Approaches (AA) video lesson for AHL 3.18—Intersections of lines & planes.
Learn AAHL 3.18.5 Angle between two planes in this free IB Mathematics Analysis and Approaches (AA) video lesson for AHL 3.18—Intersections of lines & planes.
The video discusses how to find the angle between two planes by examining the normals of those planes. The angle between the two normals is equivalent to the angle between the planes themselves, which can be calculated using the formula:
where and are the normal vectors of the planes.
Key points include:
00:00Hi guys, so this is
00:02I think my last lesson
00:03on Vectors and what we're
00:06going to do is find
00:07the angle between two planes.
00:09So firstly, what are we
00:10even talking about? Well, imagine,
00:15well these are, these look
00:16like two straight lines, but
00:17imagine they're planes from the
00:19side. That's what I've tried
00:20to do here with Jojo
00:21and I'll show you that
00:21in a second. A simple
00:23
00:25and two pages and the
00:28angle between the two pages
00:29that's the angle we're looking
00:30at. Or if you have
00:33a laptop in front of
00:34you just open the screen
00:35a bit and the angle
00:36between the screen and the
00:38kind of base where the
00:39keyboard is, that's the angle.
00:41Let me show you on
00:42geodebar. So here I have
00:44two planes. So they meet
00:47at a straight line. So
00:48there's a straight line there
00:49where they meet, where they
00:50meet and I'm going to
00:51try and find the angle.
00:53That's the kind of what
00:55I'm looking at.
00:56they disappear once you can
00:57go in there, but you
01:00see what I mean? So
01:01it's this angle here that
01:03we're looking for. Or the
01:04up -to -sangle depending on
01:06which they are asking for
01:08in the question. So just
01:09be aware there are two.
01:12And obviously this is equal
01:13to this, and this is
01:13equal to this because of
01:14opposite angles. Okay, so how
01:18do we do it? Well,
01:20imagine these are my two
01:22planes from the side view,
01:23but like this.
01:24Well, remember what I said
01:28in the previous videos that
01:30when it comes to directions
01:32of planes and angles and
01:34planes, the most important thing
01:35we're dealing with is the
01:37normals. So the normal, so
01:39this is the normal of,
01:40let's say this guy's normal,
01:42is normal, is this, it's
01:44coming out at 90 degrees,
01:47this is, let's call this
01:50n1. And then
01:52This guy's normal, let's choose
01:55a different color. This guy's
01:58normal is this, let's call
02:03him and actually that's not
02:06a right angle. So he's
02:08coming out like this. I'd
02:12love to do this is
02:14N2. Now let's say this
02:18is the, well let's
02:20the angle between the two
02:21normals is theta. And this
02:25is obviously a right angle
02:26because it's normal and this
02:27is obviously a right angle
02:28because it's normal because it's
02:29normal. So theta plus this
02:32thing, let's call this thing,
02:33let's call it alpha. So
02:35this angle alpha plus theta
02:38alpha plus theta has to
02:40equal 180 degrees because this
02:43is a quadrilateral. This is
02:4519, this is 90 and
02:46the four angles together have
02:47to add up to one
02:48So 180 plus 90 plus
02:5090 is sorry, I said
02:52that around the four angles
02:53have to add up to
02:54360. So this plus this
02:57has to equal 180. So
02:58if alpha plus theta equals
03:00180 and alpha plus this
03:05angle has to equal 180
03:07because it's a straight line.
03:08This angle here has to
03:10be theta. And guess what?
03:12This angle here is the
03:13angle between the two planes.
03:15So the angle
03:16between the two normals is
03:19the same as the angle
03:21between the two planes. So
03:22it's actually quite a nice
03:24easy way to do it.
03:26Definitely I want you to
03:27understand why that is the
03:28case and it's to do
03:29with this diagram here. And
03:33again be careful if they're
03:34looking for if they are
03:38looking for the up to
03:41angle or the acute angle.
03:43And as well if you
03:44find the angle
03:44angle between two normals, and
03:48it's an up -to -sangle,
03:50then you found this angle.
03:52And this angle, for the
03:54same reason, is going to
03:55be the same as this
03:56angle. So either way, you'll
03:58know what to do, because
03:59if they say the acute
04:00angle, you just give the
04:01acute angle. If they say
04:02the up -to -sangle, you
04:03just give the up -to
04:04-sangle. OK, let's do an
04:06example. So consider the planes,
04:09this, and this, find the
04:11acute angle between the two
04:12planes.
04:12So I want the two
04:15normals. So the normal, let's
04:17call it m1 and 1
04:20equals, this is the normal
04:21of pi 1, it's 1
04:24negative 2 negative 3. Again,
04:28I definitely hope at this
04:29stage you understand how to
04:31do that. It's just 1
04:32negative 2 negative 3, that's
04:34the normal. And 1 and
04:352 then the normal of
04:39this plane is 2.
04:40negative 1, negative 1, negative
04:431, negative 1. Now this
04:45guy equals 2 and this
04:46guy equals k. And you
04:48could ask good question, we
04:50will, how am I going
04:51to deal with the k?
04:52But the thing is, and
04:53this is actually a pass
04:54paper question. I'm sure this
04:55did cause a lot of
04:57panic when students saw that
04:58k. The truth is, you
05:00don't need it. I'm not
05:01using the 2 and I'm
05:02not using the k. All
05:04I'm using is the normal,
05:05which is here and here.
05:07So now I have n1
05:08and then two. You might
05:11as well draw a little
05:13diagram to show the examiner
05:16that you know what you're
05:17doing and to help you
05:19remind yourself what you're doing.
05:22So this is Pi 1
05:23Pi 2, this is n1
05:29and this is, let me
05:33just put, not go too
05:34far, this is n1.
05:36and two. And then this
05:40is 90. And then this
05:42is 90. The angle between
05:43the two normals is theta.
05:46So you can even write
05:46it. Angle between, this just
05:49shows the examiner. You know
05:51what you're doing. Angle between
05:52normals is the same as
06:01angle between the two normals.
06:04planes, now do you need
06:07to write this out like
06:08this? No, but I think
06:11a diagram like that looks
06:12good for sure. So now
06:13I'm just going to find
06:14the angle between these two
06:16vectors which I know how
06:17to do and I learned
06:18this one of my first
06:19vectors, well you learned it
06:21in one of my first
06:22vectors, so cos of theta
06:25is equal to n1 dot
06:30n2 over
06:33the magnitude of n1 times
06:36the magnitude of n2 which
06:38is equal to this dot
06:41this is 1 times 2
06:43is 2 minus 2 times
06:46minus 1 is plus 2
06:48and minus 3 times minus
06:501 is plus 3 so
06:522 plus 2 plus 3
06:54which is obviously 7 all
06:56over the square root of
07:011 squared plus 2 squared
07:04plus 3 squared times the
07:09square root of 2 squared
07:13plus 1 squared plus 1
07:15squared. And this, this is
07:19cos of theta. Therefore, theta,
07:22therefore theta equals, and I
07:24just need to do the
07:25inverse cos of this. So
07:29I'm going
07:29go trig inverse costs put
07:33in my brackets and I'll
07:37do 7, 2 plus 2
07:38plus 3 is 7 of
07:40course all over the square
07:42root of you can write
07:46this out or I'm going
07:46to do it in my
07:47head this is 9, 10,
07:4914 and then multiply by
07:53the square root of
07:574, 5, 6 inverse cost
08:01of this, I'm in degrees,
08:03press enter, 40 .203 degrees
08:06goes 40 .203 degrees, approximately
08:11equal to 40 .2 degrees,
08:14and three significant figures. And
08:17that's it. That's how you
08:18find the angle between two
08:22vectors. The angle between two
08:23vectors is the same
08:25name is the angle. Sorry,
08:27that is how you find
08:28the angle between two planes.
08:29The angle between two planes
08:31is the same as the
08:33angle between the two normals
08:36to those planes. Hope that
08:39makes sense. Hope you enjoyed
08:41all these vectors lessons and
08:46depending on what you're going
08:47on to next, I will
08:49see you in the next
08:50video.