 Hi, I'm Zor. Welcome to a new Zor education. I would like to present you another lecture which basically is supposed to demonstrate a relatively rigorous way to solve construction problems. The problem itself is very simple. In a way this particular problem is kind of reverse the problem which I was talking about in the previous lecture. In the previous lecture I was constructing a plane perpendicular to a line and contains a given point. Now I will construct a line which is perpendicular to plane. Now this lecture is presented as any other actually on Unizor.com and it's also accompanied by detailed notes. So basically all these steps which I'm describing right now as part of the relatively rigorous solution to this problem are described in details in the notes for this lecture. Which I think is very good if you will just read it before or after the lecture. Just to compliment whatever the material is. Okay, so let's just state the problem and try to solve it. What we have is, we have a plane, let's call it sigma. We have a point outside of this plane. And what we need is we have to build a perpendicular line from this point to the plane. Okay, now again I will present a very simple solution and then I will go into the details to basically have some more details about it. The solution is very simple. When I was talking about perpendicularity between the plane and the line I suggested the following simple problem. If you have a plane and a point on this plane, how to build a line which is perpendicular to this plane at this particular point. So you can actually go to this lecture, basically find out how it's done, if you can do it yourself, yourself is actually better. But anyway, this is a known problem which has already been solved in the course. So what remains right now is very simple. Pick any point, have a perpendicular and then draw a line from M parallel to this line which you have already built. Now if you have two parallel lines, one of them is perpendicular to a plane, another would be perpendicular to a plane. And the story. Well, is it a solution which you should be satisfied? Well, definitely not. I spent probably a minute just to explain it. However, it's not the purpose to solve this particular problem, to construct this particular perpendicular which I consider to be a purpose of this lecture. The purpose is to train your mind in developing certain analytical skills. Because it's not this problem which you will be solving in your real life, but some other problems and how to approach them. So you need to have this set of mind which analytically approaches the problem. And this is just an example of a simple problem where your analysis can be very, very useful and beneficial. And I hope it will develop your mind sufficiently enough to approach other problems more practical which you will meet in your real life. And you will be able to solve them because your mind will already be tuned to this analytical approach. So this is just a training, mind training if you wish. Alright, so how to do it more analytically? Well, let's just think about it. As far as analysis of this problem is concerned, all you have to do is basically to remember, okay, now this problem I did actually solve before and the problem was to put the perpendicular to a plane if you have a point on the plane, the base of this perpendicular. Well, you can actually recall how it's done and we probably will do just that. But then you can say, hmm, I cannot easily solve this problem, but I can solve this problem. So if M is outside of the plane, I don't remember, I don't know, we didn't solve it, etc., how to solve it. But I solved a different problem and I remember we did it and the point was on the plane itself. That's the base point. Okay, so you can say to yourself, hmm, if I can build this perpendicular, all I need to do is to build the perpendicular which is parallel to this line and it will be perpendicular. So that's your analytical thinking, that's it, end of story. Now you can actually chart your way through these step-by-step instructions how to construct. So step number one, you construct, you pick a point, any point on the plane. Now, next is you build a perpendicular to this point. That's the step number two, this one. Step number three, but if this particular line does not, by sheer coincidence, hit the point M because if it does, your problem has already been finished. You have already built a perpendicular through this point M. So if it's outside of the M, or M is outside of this line, then the next thing is to consider a plane which is defined by the point and the line, this one, and just draw a parallel in this particular plane which will hit the sigma plane at point P and Mp will be parallel because we already know, sorry, perpendicular, which we already know would be because of some theorem which we have already proven before. Let me return to basically the base of this problem, the most important part of this problem, how to build a perpendicular to a plane if you have the point. Well, let's concentrate on this problem right now. Again, if you remember, I did this based on yet another problem. If you have a line and a point on the line, how to build a plane which is perpendicular to a line at this particular point. This is the easiest kind of a construction problem because all you have to do is if you have the line and the point, all you need is have one plane and in that plane have a perpendicular to this line, have another plane and have another perpendicular to this line. And then using these two perpendiculars, using this and this, you basically define a plane which goes through this. So the plane would be in this particular direction. Now let me just draw it differently, it might be a little bit more... So if you have a vertical line and a point on it, let's say this is a point. So you do, you have one plane and then another plane and you draw perpendicular to this line in this plane which is, let's say it's this one and you in this plane you draw a perpendicular line which is this one and now these two lines define, now it looks like horizontal plane which is perpendicular to this one. So we know that. Does it help us in this particular case? Well it does because what we do is when we pick any point we have two lines, any two lines on the plane sigma. Now at this point we build a perpendicular plane to this line which would probably look something like this and perpendicular to this line which probably would look something like this and the intersection of these two planes would obviously be perpendicular to both and since this is perpendicular to this and this it's perpendicular to entire plane. So that's how I unravel it. This problem I reduce to this and this I reduce to that. Now as far as the construction going we basically do the opposite. So step number one you do this, you step number two you do this and then step number three you do that or whatever number of steps every one of them actually is subdivided into. Now that's as far as the construction is going. Using these construction steps we have basically proven that there is a perpendicular from a point outside of a plane which can be dropped onto the plane. So this is an existence. We have proven by actual construction that this perpendicular exists and the second as you remember from the previous lecture the second point is to prove that this is a unique element. There is no other but this one perpendicular because you see if you have constructed certain things which satisfy the conditions maybe if you will use some other algorithm of construction some other steps maybe you will come up with a different result different line which is perpendicular. So for this we have to prove the uniqueness of the perpendicular from a point dropped to a plane. And that's how we can very easily prove it. Let's say that using some other technique we drop the perpendicular and it hit the plane at point P prime. Well, if it hits the same point P it's the same perpendicular because two points determine the line. So if line is different, if perpendicular is different it must be a different base. But now let's consider this line. So P prime and M define some kind of a plane and in that plane now this is perpendicular to any line which goes through base which means this is perpendicular and this is perpendicular to any line which is going through the base which is this is perpendicular and you cannot actually have two perpendicular to the same line in that plane. That's the Euclidean plane geometry. So that basically completes the proof that not only we can construct it but no matter how we construct the result will be one and the same one and the same perpendicular. That actually completes in a relatively concise and rigorous form the construction problem. So you have to specify the steps how to construct and you have to really specify the uniqueness of this because if somebody else uses other steps you have to prove that the result will be exactly the same. Otherwise the problem doesn't make sense. Now that's not exactly the same in practical life. I mean in the practical life if you are facing a problem to let's say design certain assembly which is supposed to do certain things well there are many ways to accomplish it and I'm pretty sure that different people might come up with different results. Now this situation is more complex, more difficult because out of all these different design ideas somebody has to choose the best one based on whatever criteria is. Math is simpler and that's why I think it's much more important actually to train your mind on simpler tasks and then you can go to the practical problems and do something about the real life problems. What I also suggest you to do and again that's from purely educational standpoint try to write down in a piece of paper everything which is related to this particular construction problem after you have listened to this lecture after you have read whatever notes are on Unizor.com try to just close the computer, open the sheet of paper and write down with letters, with mathematical symbols, etc. as rigorously as possible steps and the proof that this is uniqueness more or less in the same way as I am writing this in the notes for this lecture because I was trying to present a sample of how the thought should actually be conducted through all these rigorous steps. I think it would be extremely beneficial for you to put in writing your own thoughts after you have basically familiarized yourself with the problem. Okay, that's it for today. Thank you very much and good luck.