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
00:23analogy is open a book
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.