 Hi, I'm Zor. Welcome to Unisor Education. We continue talking about certain not exactly standard parts of mathematics and illustrated with problems. Today I would like to talk about one very, very simple but very elegant and nicely-looking theorem in geometry of three-dimensional space. We are talking about a relationship between vectors and planes and vectors perpendicular to planes or normal, as they say, to planes. This lecture is part of the course called Math Plus and Problems. Plus means basically that this is like a second course after the standard course, which is prerequisite for this one. All these courses presented on Unisor.com and that's where I suggest you to watch these lectures because all lectures have textual parts like a textbook, so you can read it and watch the lecture. It definitely helps because sometimes the textual material is a little bit more detailed. There are exams in some cases. The site is totally free. No ads, no sign-on is optional unless you are supervised by somebody else. So for self-study it's absolutely free. There is also physics for teens, so math for teens, physics for teens, math plus, relativity for all. These are all courses which you can find on Unisor.com. Okay, so today we will talk about something which again I can consider very elegant and very simple theorem in geometry of three-dimensional space. Primarily we are talking about planes and vectors perpendicular to these planes and how they are related. Well, first of all about planes. How planes can be basically defined in three-dimensional space in Cartesian coordinates. Well, the standard or canonical representation of a plane is a times x, b times y, c times z plus z is equal to zero. So this equation connects together x, y and z in some kind of equation. Now, a and b and c are not simultaneously equal to zero. Obviously, because if they are, there is basically no equation. If g is equal to zero and a, b and c is equal to zero, basically any x and any y and any z would fit, which means the entire space is covered by this equation zero equals zero. And if g is not equal to zero, then there is no such x, y and z which can satisfy this, so it will be empty space. So in any case it's not a plane. But if one of these a, b and c is not equal to zero, then this is defining a plane. Now, let me just give a couple of examples. Now, before that I will really tell you the result which I'm going to prove as a theorem. If you take a vector a, b, c, these are components of the vector. It will be perpendicular to this plane. Well, when I saw it the first time, I kind of did not expect this. But then, again, it's a very simple theorem. So if you think about this, it's kind of obvious. But anyway, if you see it the first time, okay, this is a plane. So what is the vector which is perpendicular to this plane or normal to this plane as they say? Well, this defines the vector. Well, obviously if you will multiply all of these by some number, it will be the same vector, well, the same direction, but longer or shorter vector, right? k times a, k times b, k times c. But the perpendicularity is defined by direction, not by length of the vector. So it will be still perpendicular, same thing as here. The same plane will represent not only by this particular equation, but if we will multiply all the coefficients by some factor, by 2 for example, 2a, 2b, 2c and 2d, it will be the same equation because you can always cancel the 2 because this is 0, right? So in any case, the proportionality is not really important. What important is whenever you have an equation like this, this will be a perpendicular vector. And now I'm going to exemplify it first and then prove. Well, the first example is, for example, let's take the plane called z equals 1. What does it mean? It means any x, any y, but z is always 1, which means in three-dimensional space, if this is x, this is y, and this is z. Well, it will be the plane which is kind of parallel, parallel to xy plane on the distance 1 up along the z. So what will be the perpendicular vector? Well, for example, this vector which goes from 0 to 0.001, so 0, 0, 1 would be a vector perpendicular to this plane. Well, let's see if this really kind of corresponds to whatever. In this way, how this can be expressed actually. It's 0 times x plus 0 times y plus 1 times z minus 1 is equal to 0. This is canonical representation of the same thing, right? Because z is equal to 1, so I put 1 instead of z. 1 goes to the left, that's minus 1, and there is no x and y, no dependency on x and y, which means we have to put this coefficient. So this equation is a standard way or canonical way of the same as this plane. And now look, 0, 0, 1 coefficients. That's exactly what we have for this vector. It has 0 projection on x, 0 projection on y, and 1 projection on z axis. That's an example, simple example. Now I'm going to prove this thing in general, and I will use the vector algebra for this. So you have to be familiar with this vector algebra, and in particular scalar product. So the kind of very basic information about scalar product of vectors is that if you have two vectors, let's call it p and q, then their scalar product, sometimes it's really expressed as length of the p times length of the q times cosine of the angle between them. Now in coordinates, if the coordinates of this, x1, y1, z1, and coordinates of q is x2, y2, z2, then in coordinates their scalar product is x1 times x2 plus xy1 times y2 plus z1 times z2. This is their scalar product. Scalar means the result of this scalar product is a constant. One dimensional real number. And that's how we get it from coordinates of these vectors. So these are all known factors from vector algebra. If you are not comfortable with this, you have to really go back and repeat this part of the course of called mass proteins, the prerequisite course for this one. So I'm using this basically as given as known factors. And now let's just take a look. We will consider two cases. Case number one, d is equal to zero. So our equation is xx ax plus by plus cz equals to zero. Well, this is also a plane. I mean, whether d equals to zero or not equal to zero, it's just two different planes. Okay, what's interesting about this particular plane when d is equal to zero? It goes through origin of coordinates because if x and y and z are equal to zero, then the plane actually, the equation of plane is satisfied. So origin of the coordinate system is a point on the plane. So if you look graphically somehow displayed, well, this is a plane which goes through this point. This point is part of the plane, okay? Okay, great. Now let's take any point on the plane, x, y and z which satisfy this particular equation. So x, y and z point is, let's call it p. It's on the plane. Now, origin of coordinates is also on the plane, right? Because x, y and z equal to zero satisfies the equation. And x, y and z, whatever I have chosen for point p, substituted into this also satisfies this equation. So that's what it means. It lies on this plane. So what are the coordinates of the vector op? Well, this is a point x, y, z, which means, and this is a vector which goes along the plane from zero, zero, zero to x, y, z, which means coordinates of this vector. Now we are talking about vector x, y, z. So the point x, y, z coordinates and vector x, y, z. So the coefficients of this vector onto point, onto x-axis would be x projection on the y, would be y and projection on the z-axis would be z. Okay. Now, let's talk about vector a, b, c. Now what is this? It is a scalar product between this vector and this vector. This is basically a coordinate representation of scalar product between two vectors, vector which projections a, b, c, and vector with projections x, y, z, and a times x, b times y and c times z is a scalar product. In case of my example, it was x1 times x2. So this is x1, x2, y1, y2, and z1, z2. Their sum is a scalar product and is equal to zero. If scalar product between two vectors is equal to zero, they are perpendicular to each other. Again, remember if I talked about scalar product as lengths of p times lengths of q times cosine of angle, if it's equal to zero, lengths are not equal to zero, so cosine is equal to zero, angle is 90 degrees. p divided by 2, pi divided by 2 in regions, 90 degrees perpendicular. So that's it. That's basically a simple proof in case d is equal to zero. You can't really get simpler than that because the formula of the equation, which represents the plane, is basically a confirmation that vector a, b, and c is perpendicular to vector x, y, and z, which is kind of any point on the plane. So it looks like our vector a, b, c is perpendicular to vector op. Let's call this vector somehow n because it's normal, it's perpendicular. So vector n is perpendicular to vector op. Great. But now look, is anything specific about point p? I said it's any point on the surface with some coordinate x, y, z. It doesn't really matter. I can put this point and call this point x, y, z, and it will be perpendicular to this vector or this or anything. So it looks like vector n is perpendicular to any vector lying within the plane. Well, if you remember the course of geometry, if a straight line is perpendicular to at least two lines on the plane, not coinciding with each other, of course, then it's perpendicular to entire plane. And obviously to any other line on the plane. So this is the theorem of regular course of geometry of three-dimensional space. And that proves that this vector n with these coordinates taken from the coefficients of the canonical representation of the plane, that this vector is normal to the plane or perpendicular to the plane. So whenever somebody gives you the equation of the plane and said, okay, what is the vector which is perpendicular to this plane? We'll just take these coefficients and you will get it. But look, I have just proven this for one particular case. When z is equal to zero. What if g is not equal to zero? Okay, and now we will have a very simple logic which proves it in that case as well. Let's take our proven case and this one. Let's compare these two things. And let's talk about case when g is not equal to zero. Now I'm not going to draw anything or derive any formulas. I'll just logically talk. Now, these two where g is not equal to zero are two different equations of the planes which cannot coincide, obviously. Why? Because if x belongs to, x, y, z belongs to this plane, it cannot satisfy this. So at the same point, x, y, z, if it satisfies this plane, it cannot satisfy this. Or vice versa, if it satisfies this, it will not satisfy that. Because if it satisfies this, that would be zero. So I can substitute this with zero. It will be zero plus d which is not equal to zero equals zero, which is wrong. And vice versa. If it satisfies this equation, it means x plus by plus z plus tz is equal to minus d. If I will substitute it here, it will be minus d is equal to zero. Again, for d not equal to zero, that's not correct. So these two planes cannot have common points. What does it mean? From, you know, different perspective. It means they are parallel. Because only parallel planes do not have any points in common. Now, if planes are parallel and there is a perpendicular to one of them, it will be perpendicular to another. So this vector, obviously, is perpendicular to this one as we have just proven. That means it should be perpendicular to this one because this is parallel to this. That's it. Okay? Okay. So this is a very simple theorem. I called it Geo-Theorem 1 in this course of Mass Plus. No, it's simple, but it's, again, kind of unexpected. A little bit unexpected. Because one thing is to know the equation of the plane. In this case, I'm talking about canonical equation of the plane. And another thing is how come from this equation immediately follows the representation of the vector perpendicular to this particular plane. Now, this vector is just anywhere in the space can be. Not necessarily the vector which goes from the origin of co-ordinate, but usually vectors are drawn as going from the origin to some point ABC. In this case, ABC are projections on the axis of co-ordinates. All right. Just as a simple illustration, again, let me just talk about one particular plane and what will be the perpendicular to this plane. Consider the following plane. If you have a plane which goes through the z-axis y and along the bisector of x, y at 45 degrees here and here. So that will be the plane. This will be on this plane and the bisector between x and y will be. What will be equation of this particular plane? Well, equation will be x is equal to y. Because any point on this, if you will project it down, you will have point which is equal distance to x and y, and z can be anything. This equation is non-canonical equation of this plane. Canonical equation will be 1 times x minus 1 times y plus 0 times z equals to 0. By the way, we were talking about it's equal to 0. It goes through the origin of co-ordinates because point 0, 0, 0 belongs to this plane. So, which means that vector 1 minus 1 and 0 should be perpendicular. Well, let's think about what will be the perpendicular in this case. Well, vector which goes along this plane, along x, y plane perpendicular to bisector. If you will go to the top view, so that would be our bisector between x and y and that would be perpendicular vector. Now, this perpendicular vector should have 1 and minus 1 co-ordinates. And we will see x co-ordinate is 1, so vector, and y co-ordinate is minus 1. So, basically it's just an illustration of the same theorem again. Okay, all right. So, that's it for today. Again, I was just trying to convey something which seemed to be unexpected, but quite frankly, really kind of, I don't know, should I say beautiful? It depends on your, obviously your understanding of the word beauty in mathematics, but there is a beauty in mathematics. Something simple is always kind of attracts the eyes. It's like in physics E is equal to mc2, Einstein's formula for energy and mass. It's simple and it's beautiful. Well, how true it is is a different question, but sometimes beautiful is really true, sometimes might not for us. Well, in this case, as I said, it's very simple and that attracts actually this particular thing to the theory. I suggest you to read the notes for this book, for this lecture. Notes and lecture and notes. If you will go to Unisor.com, the main menu will contain mass plus plus and problems course. Go to this menu item, open it, and find geometry among different topics. And then next menu will give you Geotherm 1, which is this particular lecture. On the same screen you will have the video and the textual part. Read the textual part. It's really important that you not only watch whatever I'm talking, but also go through the formulas more accurately, maybe because in some cases I'm more accurate in writing than whenever I'm talking in front of the board. That's very useful and well, enjoy it. Thank you very much and good luck.