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When P is the plane in R3 given in vector form byFind the shortest distance from the plane P to the origin Show all work...

Question

When P is the plane in R3 given in vector form byFind the shortest distance from the plane P to the origin Show all work

When P is the plane in R3 given in vector form by Find the shortest distance from the plane P to the origin Show all work



Answers

Find the shortest distance from the point $ (2, 0, -3) $ to the plane $ x + y + z = 1 $.

Given a plane. The explosive beaver less season is acquittal d We need to brew that wind up in the section is a dele ready by leas Were BDD right by these were see the divided by the square where the zika will do e squared B squared C squared No, on the distance at least, distance is equal to more off the developed body. So for this, let us consider Oh Boston, it's why is dead? It is through toe X squared, plus by square place they're square bill X squared plus vice Purpose X squared is equal to root on. There's a function g it. Advise it. When is the cool, too? He expressed Beaver this season minus d. So let us say this function is given by F minus Linda Jean Dilemma is and the number what multiplier. So there we have to differentiate with respect to add eggs off it. X minus. Linda and she thanks so many even differentiate its function. Respect the ways I get X divided. But Ruto and square by square because they're square minus lend outside and the transition off eight. Similarly, this is a final value off it said. Similarly, Mulligan edge live it is by divided by flu talk and swept less by square. The square minus lame. Don't be and we will get it said, which is that divided by Roto Exploit. This one is work with their squared minus Linda steep. So let us put XX music will do. And why is he going toe to toe visible zero? So we get text divided by Rudolph and square by square place. That's where minus lambda A is equal to a variety by burning photo. That's where plus vice purpose. That square minus them they'll be is it will do their developed very Ruto experiments by square, with that square minus named C music will do Tzeitel. So therefore we will get from her ex by is a good bye Bobby physical toe then. But see is equal Lambda and explain this wise worthless was excluded. So there, poor by is equal to be X by and that is equal to see its by a You know it doesn't that is, what does that use are? Buy ins and integration experts Be well, policies Any school visit the so I get a X plus the square expert, a Plessy Square X, but is going to be I get a score. Expect these for expenses for eggs is good to 80. So therefore my ex is equal to a really divided by a squared B squared C square. They poured my vibe before be derailed by a squared plus B squared plus C squared and method will be Seeley divided by a squared plus B squared C square. So they're call my P point will become a really very, very square. Must be square. Percy Square Really divided Very square. Let's be square to six. Well, I'll see Delivery Bracewell, That's B squared C school. So we have food. One of the part. Oh, this is a question. Next is to prove the distance. So that s a distance. Is it Will do X squared plus y squared plus export as beer scene. So this X is a D divided by a squared plus B squared. C square was well blessed Be derailed by ice. Where does this? Where? Presley Square. It was good. Bless CD divided by a squared plus B squared C square. Well, this way. So what we will get is Hey, square the square. Leslie square this world. Let's see squarely square under the road, divided by a squared plus B squared C square it was, but But this is we'll do the square woman into a squared plus B squared C square. The very, very square does B squared plus C squared I was with under zero. So this will be Quito on the rule these squarely by anyway. A squared plus B squared that c square. We know that a squared plus B squared nous C square is in Quito. These court and the square is given. But these well so we get this more d do any but the and it's their crude. Both the butts given a plane equation a point. We need to prove that these distance does it will do the eggs. Not this. Be by not proceed. They're not. We need to brew. Least distance is in Quito. Mordo, it's not this. Be by not see that not minus Steve do more. I didn't buy the the but he is a dog. This where it'll be square. Let us consider. There's a point even X one by one said one, which is on the plane. Backed it off it wishes every seeing, you know, minus being gone is given by ex not minus X went from over. I note minus my will Did not minus so let us substitute explored minus X Wanted to go to eggs. Why not? Minus by notable toe They're not mine. Is that when physical does it So I had the physique Will do e experts be like this season Lord divided by Bruto in squared plus B squared C square because we have least distances going through more of developed by the So this is equal toe into explored minus x one less be into by no minus by well less we have these stills is equal to Lord of the very, very devoted they will to Mordor X plus a press Is it provided by the door? You school Let's be squared. Let's see squid poor That's physical works not minus explanation by supported by not minus variable There they're not minus eight So they're all my DZ Porto a two x not minus X one plus a two By not minus violent breast Ain't they're not minus that run divided by little Do a phase where Let's be squid This So I get the X squared less airai square. This is the square one side, minus the X one minus three by minus one. You would do you feel he's missing square this c squared? No, we know that the x one plus B by one lessees then one Is it going to develop on the is? It will do Ruto X squared plus y squared less that way So begin is equal toe No, I don't e x square. Let's be right square minus foresees a square minus t divided by de as through

You're given a plane. The X place Be very reckless thing that is being point be lying were given a plain X plus B Y place sees that is gonna be be on the plane. We're going to find that be Cuba. Cuba's any off the point into and divided by more in physical too These distance O b que. This is a but notice right expression off being director on to they normal vector. And so we're and is equal to a B C. So be director is equal to thank you into end in to end the writing in. It's good. So therefore projection all the director in tow end is equal to be corrected into anything more in the end, more and doing more. This is Yvonne Paterson, so we get the least distance as be corrected when by more than fruit B, but says there's a few point given our turned the plane going to minus four on a plane is given less to works minus by last. See, there is 1/4 but so point on the plane assumed that is a point to mo zero minus one, which is on this plane. So be collector is given. But que rector minus B Victor Physical 11 to minus four minus 20 On this one, it is minus one. Come to going minus three. So we need to find and director, which is that? The Obama minus one From a tree in a bar. We prove that these distance is given by the director in tow. Inventors, the road a bit and Lord, So this is equal to be director is minus one from one to minus. Tree multiply right and Richter is two minus one from a tweet divided by more open is giving birth to a square that's minus one holds well the rest of these guys this is it Will jewel minus one into two, which is minus two plus two into minus monitors again. Minus two plus minus three industries minus nine. Divided by you told Worthless man less. Nine. So this is acquittal and says more. This is a little toady divided by what

Welcome back to another cross product problem. This time we're looking at what happens if we want to find the distance from a point. Let's call it P to a plane. We know we can define a plane by three different points. Let's call these Q, R and S. And so if we define a couple of vectors, let's say a between Q and r. Be between Q and S. And say see between Q and r. Point. Then what we can do is try and figure out the distance from the point to the plane. Using some things that we know about. Triple products. Specifically the volume of the triple product of the parallel pipe ed between a B and C is defined as the magnitude of across B times the magnitude of C. Cosign theta. And so rearranging this a little bit. If we write C dot a cross B divided by the magnitude of a cross B, then this is going to be the magnitude of C. It's the length of that vector C times Cosine Theta where theta is the angle between our vector and perpendicular. This is theater right here. Now, since cosign Theta is adjacent over hypotenuse, that means the length of this perpendicular lines. The distance from the point of the plane, going to be the magnitude to see times. Cosine Theta. We can use this if we want to calculate the distance from say point P 214 to the plane divided by Q, R and S. In order to do that first, we need to calculate projectors defining the plane. So A is Q. R. AR -Q Sierra -1 To -00 0 and Q. S. That's s minus cues. Again zero minus one, zero minus zero, three minus zero. We'll also need to see that's the vector QP p minus q is two minus one. One minus zero, four minus zero. So if we want to calculate the magnitude of c dot a cross B, we can do that in one step using our triple product. If we plug in C. A and B into our matrix here, I'll point out this isn't the only way we can do this. Um But we'll talk about another way in just a second. So let's plug in 114 -1-0 And -103. And using the formula from our textbook, remember we ignore the first column And look at two times 3 minus zero times zero, Going to be 6 0. And normally we would multiply by I We're actually gonna multiply by one minus and ignore the second column -1 times three zero times negative one. Eight of three minus zero times not J but times one again us. And then we ignore the third column negative one times zero minus two times negative one will be zero minus negative too, Which is 0-plus two times four. And since we don't have any eyes jay's or k's this is not a factor but just a number six times one minus negative three, That's plus three times 1 Plus two times 4. And so we're looking at six plus three plus eight is 17. The other thing we need from our formula remember is the magnitude of a cross B. And so we can get that by calculating the cross product of A and B. Once again we're looking at six minus zero, I minus negative three minus zero jay plus zero minus negative too. Okay, Giving us the vector six three two. If you want to calculate the magnitude of a Crosby, that'll just be the magnitude of 63 two. Which is the square root of six squared Plus three squared plus two squared or The square root of 36 plus nine plus four. And that's the square root of 49 or just seven. Since we determined that the distance is the ratio of the two numbers that we just found, we get the distance is let's go back to that cross product Or the triple product with 17 over seven. I said earlier, this wasn't the only way we could have found this. Since we already know the cross product a Crosby, you could have the magnitude of that and then found the magnitude of C. Got a cross B. Just using the vector C. And a dot product. That would avoid having to do the cross product twice. Thanks for watching.

For this problem we are asked to find the shortest distance from the origins of the plain. X plus two, Y plus three. Z equals 12. So to begin, we know that the square of the distance from the origin to any point, it's going to be d squared equals x squared plus y squared plus z squared. So we want to minimize the square of the distance because if we minimize the square of the distance, we minimize the distance as well. So what we really want to do here is minimize F. Of X Y Z X, Y. Z. Where X plus two, Y plus three equals 12. Which we can rearrange and get easily. That X then needs to equal 12 minus two y minus three zed. Which then we can substitute that into our method X Y Z function to get a function of just two variables app of Hawaiian said. So we'll get that F of y Z is going to be 12 minus two, Y minus three. Zed squared plus y squared plus Z squared. The next thing that we want to do is expand that out and simplify which I'll do off screen. So the function is then going to become 10 0 squared plus. Uh Let's see here, tens, x squared minus 70 Tuesday plus 12, wizard Plus five, Y Square -48, Y Plus 1 44. Now to minimize this, we start by taking our partial derivatives with respect to y and Z. So the partial derivative with respect to why It's going to be 12 said plus 10 y -48. We want this to equal zero. In addition, we have the partial derivative with respect to said It's going to be 20, said minus 72 Plus, Excuse Me, 12 Y. And that's it. We just want that to equal zero then. So we can now rearrange and solve to get Z and Y, which I'll do off screen. So we should get a final result here of Y equals 12/7. Yeah, Z equals 18/7 And then substituting that into the expression for X. They have 12 -20 for over seven minus 18 times 3/7. So we get that X with, then needs to equal 6/7.


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