00:00Hi guys, so the next
00:03equation of a plane that
00:04we're going to look at
00:04is the Cartesian equation of
00:06a plane. Now these two
00:08basically go hand in hand.
00:09They're kind of like the
00:10same thing. I'll show you
00:11what I mean by that
00:12in a second. But yeah,
00:16I want to mainly focus
00:17on this, the Cartesian equation
00:18of plane. That's what we're
00:19going to find. So it's
00:24that when someone asks me
00:25the equation of a plane,
00:27that's the first thing I'm
00:28going
00:28going to try and give
00:28because it's very nice. It
00:30connects x, y and z
00:32together in one equation. And
00:34I'm actually going to show
00:34you how A, B, C
00:37very neatly gives you the
00:39normal vector. That leads me
00:41on to what is a
00:43normal vector. So the normal
00:45to the plane, hopefully you
00:46remember, remember the the the
00:49the the term, remember the
00:50when we were getting the
00:51tangent to the curve. And
00:52we said, if this is
00:53the tangent, this is the
00:55normal, it's
00:56perpendicular. This is the normal
00:57perpendicular to the tangent. So
00:59the normal is perpendicular. The
01:02normal to a plane is
01:03perpendicular to the plane. So
01:06it's a vector perpendicular to
01:09a plane. And it can
01:09be, I mean, a plane
01:12could have many normal vectors
01:14just of different lengths. But
01:18they'll all be going in
01:19the same direction. So like
01:21if I don't know, let's
01:22say three negative five, two
01:24is
01:24a normal to the plane,
01:26then negative 30, 50, and
01:28negative 20 is also a
01:30normal to the plane. It's
01:32just a bigger vector and
01:35it's going in the opposite
01:36direction. So where does this
01:38whole thing come from? So
01:41I'm going to explain to
01:42you how to get how
01:42we get this and then
01:43how this turns into this
01:44for any key. So imagine
01:47we have a point on
01:49the plane. So this example
01:51that I'm going to show
01:52you we need a normal
01:54vector and a normal a
01:57vector normal to the plane
01:58and a point a. So
01:59imagine this is point a.
02:01So this is point a
02:02and here we have some
02:08general point let's just call
02:09it p. So this is
02:12a p. So this is
02:14a vector on the line
02:16that joins a point that
02:18you know to some
02:20general point. So we know
02:22AP is equal to P
02:27minus A and P is
02:30the kind of general, it's
02:33the general point. It's x,
02:39y, z are the coordinates.
02:40So the position vector is
02:43actually going to be, well,
02:45let me just write it
02:46as R. So it's R.
02:48minus A. So that or
02:51is the some point x,
02:54y, z. And the, the,
02:56well, sorry, the, the or
02:57is the position vector x,
02:59y, z, this, this is
03:00what or is x, y,
03:02z. So we've AP is
03:05or minus A. Fine. Now,
03:08because of by definition, what,
03:10what the normal is, the
03:12normal and this vector, i
03:15.e. any vector that's on
03:16the plane,
03:16are perpendicular. So I can
03:18say AP dot n, so
03:23n is our n is
03:24the letter used for normal
03:25vector. This has to equal
03:28zero because perpendicular vectors, the
03:30dot product equal zero. But
03:32this is or minus a
03:34or minus a dot n
03:36equal zero. And then multiply
03:39this out or dot n
03:41minus a dot n equal
03:43zero. And then I
03:44have or dot n equals
03:47a dot n. This is
03:51my, this is this formula.
03:55This is where it comes
03:56from. Do you need to
03:58know what I've just done
03:59there? Not really. You wouldn't
04:02be asked to show that,
04:03but it's certainly nice to
04:05understand where this comes from.
04:07So this is the formula
04:08ordered and it was added
04:09in. Now the way that
04:10turns into the Cartesian, the
04:12Cartesian equation
04:12is or is this x,
04:16y, z, dot n. Now
04:21just imagine, well I'm going
04:24to do it in this
04:25example here, but you're going
04:26to have some vector n
04:29dot a point, the point
04:32that you know, sorry, not
04:33dot, this equals some a
04:37dot n, a dot n,
04:40we'll do
04:40the example in a second.
04:43So if this dot, this,
04:44this, this, this, this, all
04:45you're going to get is
04:46something x plus something y
04:47plus something z, something x
04:49plus something y plus something
04:49z, equals a scalar, which
04:52is d and that's your
04:53a, that's your point dot
04:56normal vector. So what does
04:58the d tell as well?
05:00Not much really on its
05:01own. It's the dot product
05:03of the point of some
05:04point on the vector with
05:06the normal.
05:08But yeah, as I said,
05:11that doesn't tell you that
05:12much. But the ABC tells
05:14you a lot. The ABC
05:15is the normal vector. So
05:18when you see an equation
05:19written like this, the normal
05:21is ABC, the normal vector.
05:24OK, so let's go down
05:26and do this. And this
05:28is why I said this
05:29kind of turns into this.
05:32So I'm going to say,
05:34or dot n equals
05:36A dot n, or is
05:39my x, y, z, n
05:42is my three negative five,
05:45two. So maybe, sorry, I
05:47should maybe have read this.
05:49Find the equation of the
05:50plane with normal vector n
05:52equals this, and which contains
05:55the point A, three, three,
05:57one. So again, remember guys,
05:59you can kind of use
06:01them synonymously, but just be
06:02clear. This is not the
06:04point
06:04And A, this is the
06:06position vector A, but it's
06:07kind of the same thing.
06:09So three, three, one, dot,
06:12n, which is three, negative
06:14five, two. So this becomes
06:16three x minus five y
06:21plus two z, plus two
06:25z equals, and then we'll
06:28just do this, nine minus
06:3015 plus two,
06:33Oof, three x minus five
06:35y plus two z equals
06:38nine minus 15 is negative,
06:39six plus two is negative.
06:41Four, there we have it.
06:43That is the equation of
06:45a plane. Okay, great. Next
06:49example. By the Cartesian equation
06:53of the plane, pi, are
06:56pi one. So I just
06:58wanted to introduce this to
06:59you. This is a capital
07:00pi.
07:01So often we use the
07:03letter capital pi to define
07:08a plane. So anyway, capital
07:10pi containing points p, q,
07:12and r. So again, let's
07:16be clear what's going on
07:17here. This is some plane.
07:21Now p is on the
07:23plane, q is on the
07:24plane, and r is on
07:25the plane. p, q, or.
07:29I need to get the
07:31normal, well remember I need
07:33a point and I need
07:35a normal. So how do
07:38I get the normal to
07:41a plane? I need to
07:42get a vector that is
07:45perpendicular, a vector perpendicular to
07:48the plane. Well what I
07:49can do is, if I
07:51get this vector, p over
07:53and this vector,
07:57PR and this vector pq
08:00and I'm just looking at
08:01these numbers. I think I
08:02prefer to use, I like
08:06to use PR and let's
08:08do pq. Let's do pq
08:11and let's do qp and
08:17qp and qp and q
08:21or okay like that. So
08:25obviously
08:25the guys, you're probably wondering,
08:28where am I doing? But
08:30I can choose any of
08:32the three vectors. So there's
08:34three vectors and I just
08:35need to choose two because
08:37if I want to get
08:37the cross, if I want
08:38to get the vector normal
08:41to the plane, if I
08:42get the cross product of
08:43these two vectors, or any
08:44of the two vectors, I'm
08:46choosing these two because these
08:47numbers are smaller. If I
08:49get the cross product of
08:51those two vectors, what I
08:52get is
08:53the normal. So this, and
08:57again, you can imagine this
08:58in 3D, this is the
09:00normal that is perpendicular. It's
09:02coming like straight out of
09:03the table and these are
09:04like two pencils on the
09:06table and this guy's coming
09:07straight out of the table.
09:08So if I get the
09:08cross product of those two,
09:10I'll get the normal. Okay,
09:11now first what I actually
09:12need to do is find
09:14p, qp and qr. So
09:16let's do that first. qp,
09:20qp equals
09:21P minus Q, which is
09:231, 6 minus 7 minus
09:26Q, which is 0, 1,
09:291. This minus this is
09:315. This minus this is
09:34negative 8. And then I
09:37need Q or Q or
09:41Q or Q or equals
09:42or minus Q. 2, 0,
09:46negative 4, minus Q, same
09:48thing.
09:491 equals 2 negative 1
09:54and negative 5. Now, I
10:00need to get the cross
10:01product of Qp and Q
10:03are, and that will give
10:04me the normal. So Qp,
10:07Qp cross Qr is equal
10:13to 1, 5, negative 8
10:17cross
10:172 minus 1 minus 5
10:22equals. So obviously guys take
10:24out your formula booklet if
10:26you're not sure how to
10:26do this but remember cross
10:29out the first two we'll
10:30put it put your hand
10:31over the top two and
10:32it's this times this negative
10:3425 minus this times this
10:37which is actually minus 8.
10:40Then it's this times this
10:42minus 16. This times this
10:45minus this times this, which
10:47would be plus five. And
10:49then finally, this times this
10:53minus one, like one times
10:55negative one, minus two times
10:57five, minus 10. And this
11:00equals minus 25, minus eight
11:02is minus 33, minus 5,
11:0516 plus five is negative
11:0711 and negative 11. Again,
11:13So I have a normal
11:17vector of negative 33, negative
11:2011, and negative 11. So
11:25this is where this, this
11:28is where it gets interesting.
11:29This equals, this actually equals,
11:32I can take out a
11:33negative three. This equals negative
11:34three. I'm sorry, not negative
11:38three, negative 11. I can
11:39take out negative 11, and
11:41I get negative
11:413, sorry, I got 3,
11:441, 1. So a normal
11:47vector, normal vector, is, because
11:55remember what I said, you
11:55can use any normal vector
11:56you want, so I could
11:57use negative 33, negative 11,
11:59negative 11, but wouldn't it
12:01be nicer to just use
12:023, 1, 1? It's a
12:05much nicer vector to use.
12:07So to get the equation
12:08of the plane.
12:09So equation of plane is
12:17so back to this formula
12:20guys it is or dot
12:23n equals a dot n
12:24or dot n equals a
12:25dot n so I want
12:26you to kind of remember
12:27that or dot n equals
12:29a dot n or dot
12:31n equals a dot n
12:33now you say well you
12:34don't need to remember because
12:35it's in the form of
12:36a click but yeah fine
12:37but remember to use
12:37So R is your XYZ.
12:42N is 311 equals A
12:46dot N so I can
12:47use any point here. I'm
12:49going to use Q. I
12:50think it's the nicest, easiest
12:52point to use. So I'm
12:55going to use Q011. So
12:57the point is 011. I
13:02should really put a dot
13:03here. This dot this equals
13:05this.
13:06dot three one one multiply
13:11it out and you can
13:12see well I'll show you
13:13in this in a second
13:14three x plus y plus
13:17z equals zero plus one
13:21plus one which is two
13:23that's just leave like that
13:24perfect and you can see
13:26now guys if you put
13:27in negative 33 negative 11
13:29negative 11 you'd have it
13:32on
13:34It would be here on
13:36both sides, so what would
13:37happen is it would just
13:38cancel out anyway and you'd
13:40end up with you would
13:43end up with the exact
13:47same equation. Okay, that's it.
13:52That's the equation of a
13:53plane. That is the equation
13:56of a plane at the
13:58Cartesian equation of a plane.
14:00Hope that all made sense,
14:01as I said, that is
14:02by far my favorite equation
14:04of the plane. So get
14:06used to it and be
14:08happy dealing with it. Okay,
14:09that's it. See you in
14:10the next video.