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Find the point on the plane 4r + 3y + 2 = 1 that is closest to (1,-1,1)_ (Recall: the distance between two points (T1, Y1, 21) and (12, Y2, 22) in space is V(c1 T2)...

Question

Find the point on the plane 4r + 3y + 2 = 1 that is closest to (1,-1,1)_ (Recall: the distance between two points (T1, Y1, 21) and (12, Y2, 22) in space is V(c1 T2)2 + (y1 = 92)2 + (21 22)2 .) [4 marks]

Find the point on the plane 4r + 3y + 2 = 1 that is closest to (1,-1,1)_ (Recall: the distance between two points (T1, Y1, 21) and (12, Y2, 22) in space is V(c1 T2)2 + (y1 = 92)2 + (21 22)2 .) [4 marks]



Answers

Find the distance between the point and the plane. $$ (1,-2,3) ; 2 x-2 y+z=4 $$

Yeah, going to fight the point on this plane closest to this pathetic lapointe that we are interested in one minus one and one. Now, before we start on the like process to fi a solution, I would like to say that this question is you post them or is a trick question. Because if you look at this 0.1 minus 11 and you pluck them in the plane, you will fi that you get the well Ooh, for minus three plus one, which is exactly two. It means the Pipi is on. The plane is on this plane. And so the closest point to this would be itself right, Because this time would just be zero. And that is exactly what we are going to five from the process we are about to do. But just be aware that some question give days like long question just to trick you into doing the whole process. But instead of you, just pay Abby like, pay some attention to these new ones, you can just find an answer right away. But it's not very good for an exercise question. So we just going to ignore that and do it the usual way. So we interpret this question, we will have that the plane itself is a restriction function, right? And that this tense function is going to be what we want to minimize is that this 10 because, like did this 10 is there would be some choir room off if but to simplify things up, we just look at what's inside, which is ever. And here we go we're using, like, ranch multiplier. So we have a great day in situation like this. I have calculated this out and it's very simple. They are just Polina meals. Um, comparing the coordinates. I jen k, you're gonna help this system. Lambda equals this tree. So when you get to this step, what you want to do is choose one variable in this case, I choose Why. So I'm going to fi eggs and see in terms of why, from this, uh, system, once I have those too. I just put these like substitute is in our restriction function so that everything turns into why, right is gonna be it's gonna become something off. Why? Said going to be something off? Why and then this gonna become equation based on Why alone and which can be so pretty easily. They are Lumia, and you're gonna find out why. Use indeed minus one. When we put why is minus one into these? We have that X is one and said he's one. So is Indian D course. This point is itself. So again, the Soviet question. Ah, and that is it. Thank you.

So the formula in our three for the distance between a point in the plane very similar to the distance, um, formula and R two for the distance between a point and a line. Right? But now we're in in three space. So now we have a point with three corn itself from a point X not Why not? Ex not. Why not? Zeenat? Um and a plane is given by this formula where the distance d is equal to the absolute value off a times x dot plus b times. Why not? Plus, But now it's plus C times the knot plus d so really very similar. We just add on extra component here, and then we'll This is then all over the square root instead of the square root of this age. Group of speed squared. Now if the square root of a squared plus B squared plus C squared. Okay, So our given point is the 0.0.31 negative too. Okay. And our plane is X plus two. Why, minus choosy is equal to four. Okay, so therefore our distance to convert the standard form right, and then we have that are distance D is equal to where all the absolute value off while one times three. So that's three. And then plus two times one. So plus two. And then, um, we had minus two times negative, too. So that's plus four. And then we have minus four. So an absolute value over one of the square root of one squared plus two squared plus negative two squared. That's one plus four plus four. So the square root of nine. So therefore this is just equal, while to the absolute value of the numerator. Just becomes five right over the square to nine. That's despite over three, so the distance here is equal to five or three or five thirds. All right, take care.

We're looking for the distance from some point to negative 34 to the plain X plus two y plus two z equals 13 so that the picture that we're looking at is that we have some plain and then we have some point up here and we're looking for this distance down to this plane. Well, this distance D is a distance along the normal. So if you if this perpendicular to the plane is what we're using to find this point So this is yeah, that just some portion of the normal. And because we know the equation where the plane we know that this normal is one too two. Hey, if we have some point and we'll just But the point p one down here Hi, We we can extend a vector up to there that I would call Vector V. And so, Victor, the will be this vector that started here and ended here that it's ending at the point to negative 34 and beginning at the point. What? That's our question. How can we find another point here in order to get this vector? Well, fortunately, any point that satisfies this equation will be a point on the plane so we can arbitrarily choose values of X and Y and sulfur Z or any combination. So, for example, if I put X equals one and why equals five that will give me 11? That means he must be one that gives me 13. So I can have a 130.1 five one. And so this vector V is one negative. Eight, three. All right, So what are we up to? If I have some vector the down here and some vector you up here. There's an angle fate of between here that this distance here is going to be you times the co sign of that angle. And this length down here is the projection of the vector u on the plane. The so this projection is going to be you go sign data. Also remember, dead the dot product you don't be is equal to the magnitude of you. That's the magnitude of B times that co sign of data and so combining these two things we see that this projection of you after you on V is going to be This part is you co sign data. So this will be you God the provided by. See. Thank you, Toby. All right. And and so if we know this vector the and we know a vector you, in this case, the normal my project. This onto here, This portion, this projection will be the distance. So the distance is the projection of onto the normal of the vector. The and that is equal to the dot product of the normal who's and the provided by the magnitude of the normal. So in our case, the normal is 122 and D is one activity three. So this dot product will be one times one plus two times negative. Eight US two times three. It will be divided by the magnitude. Oh, and one squared plus two squared. Plus you're squared. So this is one minus 16 15 plus six negative nine divided by the square root of for he nine, which is three. It is negative three, because this thing is a distance, right? The negative has no meaning. Tow us so this distance would in fact be the absolute value of this projection. The absolute value of this and the absolute value of this. So the distance that we calculate is three

Find, uh, distance between the won't be equal. Ju ju Ministry farp and Plan Express Teoh pressed to Z equals 13 We know record in the distant region. Upon to the plan de we go Jew the Dr that have to be asked I would no more better the inviting bind And no, I'm the better. And here we have the beast are given to us here on for the ask and we need upon on the plan so it can from the equation on the planet Can fix the XY coaches zero the white coaches Oh, on dizzy now ico Judah 13 devoting. But you would be the point s here and now the normal it will be those follow be in front of the X Quincy Therefore we have the 12 and chew on We have everything hassle from here We will be able to find out what that be asked within the ask you Minister be Then we get Nico do months to three And the thing that in other to Manus far Then we should get equal Ju five outer. Uh do on Dan we have 30 minus eight. Should be fine. Yeah, and now from here we should be able to compute a distant Now we coach you the top it up between the two months. 23 Fire to chew! We don't win the one Ju Ju Do you think the absolute blue the fighting by then no one of it no more you? Could you one square plus juice square Best you square. Then we should get equal Jew Here we have no absolutely of the Manus to the six and then plus five be inviting by the square Root off the have here will be nine And here we will have equal Jew 11 minus Jew would be nine I was good at night and we go to screw it up Ni and that's a distant we're looking for.


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