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WeBWork Math132_Sm19 HW2 12 webwork yeditepe edu trMAA MATHEMATICAL ASSOCIATION OF AMERICA Logged in as 20150703056. Log' Out 4webwork math132_sm19 hw2 12HW2: ...

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WeBWork Math132_Sm19 HW2 12 webwork yeditepe edu trMAA MATHEMATICAL ASSOCIATION OF AMERICA Logged in as 20150703056. Log' Out 4webwork math132_sm19 hw2 12HW2: Problem 12Previous ProblemProblem ListNext Problempoint) Find the point P where the line X = 1 +t,Y= 2t,2 = -3t intersects the plane x + Y - 2 = -3.P =Note: You can earn partial credit on this problem:

WeBWork Math132_Sm19 HW2 12 webwork yeditepe edu tr MAA MATHEMATICAL ASSOCIATION OF AMERICA Logged in as 20150703056. Log' Out 4 webwork math132_sm19 hw2 12 HW2: Problem 12 Previous Problem Problem List Next Problem point) Find the point P where the line X = 1 +t,Y= 2t,2 = -3t intersects the plane x + Y - 2 = -3. P = Note: You can earn partial credit on this problem:



Answers

Use the formula in Exercise 12.4.45 to find the distance from the point to the given line.

$ (4, 1, -2) ; x = 1 + t , y = 3 - 2t , z = 4 - 3t $

Questions were given the question on the line where we have X echo to chew. Why go to the three blasts Tuesday, the ego to the managed to minister today And we have the question of the plan where we have the sixth X plus three y minus forms three equal to minus trail. Now you find in the problem intersection all we need to do. We look in the X y Z into x y and say in the equation of the plan. So we have the six times two plus three times three plus two day and minus four times managed to manage to today equals two months trail. It will simply finding the throughout plus nine plus six, the plus eight plus 82 months trail. Now we see that from the tea here. When we get you go to the 14 day, then we equal Thio this week and, uh, 12 plus nine plus eight and bless true royal. So he could to minus 41. So it means that the ego equity minus 41 other 14 So it isn't the day here we put into the equation the line the equal to the minus 41/14. So we should get Now we have the point of the intersection now actually going to do on the time. So we have three and then minus 41 hour seven. So we get Nico to the minus 20/7. Here we have the man ist to plus 41 hour seven, so we get echo to the 27 hour seven.

Hello. Precaution is taken from vectors and geometry of the space. And the question is find the equation of lines and planes. Find the question of a find a point at which the line intersect a given page. So that line is X. Is equal to -2 T. Why is equal to tt That is equal to one thirsty. Okay and X plus two Y minors. That is equal to seven. So we need to find a point which which is passing. Uh We need to find a point at which line intersects the plane. So in order to evaluate that point let us execute the value of X. Y. Z out of the line. In the given train equation we get to minus two T. There's two into 20 minus of one plus T. Is equal to seven. If we fail to solve it we get to minus duty last 60 -1 -3 is equal to Sam And for those all we need we get five to minus duties. 20 Plus one is a tradition. Well that's all we need is equal to seven minus 26 of what we that is 2 50 is equal to two then X is equal to 2 -4. So that will be team two minus. To introduce minus to invite is your car too Treaty. So I will be going to so that will be equal to six and said this is equal to one thirsty. That is equal to one plus two which is equal to three. So the given point to which that line intersect with the plane is minus 263, which uses the required points, So Hope disclose her.

Person. We're trying to find the parametric equations foot I intended to the intersection Kurtz curves are ex script was to live with. Two. C is equal to four and why is equal to one in a sexual coin? Sorry, it's one comma one cover 1/2 And so let's just first to find f x y Z to the X squared plus two by plus two C minus four and g of x y c to be blind minus one Again, this tangent line must be orthogonal to both radiant of F ingredient ghee on This was explained on page 811 of the textbook. So if you don't know why, um, you can go there They get an explanation there. But essentially we know that the tangent line to any curve must be perpendicular to ingredient right, That's just by definition, the Grady must repair a perpendicular to the detention line. And so we know that Ah, the tension line must be perpendicular to both LF right and LG And so for that reason, we know that this LF prostate algae is the vector that defines the direction of this tension. And once we find the factor that defines the direction of the tension line and we know coordinate a point on the tension line. We can write a set of Parametric equations defining this line. And so the first order of business this determining this Del half of 11 1/2 crossed LG of 11 1/2. And so let's try to figure out So L f is equal to two acts 22 right, and then so so up with 11 1/2 must be too common to comment to you Just plug in X y Z s 11 of 1/2 respectively. Similarly, down with G is equal to zero comma one common zero and then therefore LG at 11 1/2 Also zero Commons here, one common zero. And so now I want to find the direction of this tangent line You simply across the two vectors and I'm going to just and that gives us negative to common zero comment to. And ah, if you don't trust me or if you're doing this problem by yourself, you should definitely work this out like manually to check that the cross product is indeed negative to common is you're going to? Well, I won't work through because everyone has different ways of doing the cross product right there. So many, like different schools, different teachers teach different methods of determining cross products. And so, uh, it's it'll be confusing if you're exposed to a lot of different ways. But just know that your way of determining crossword because Justus effective is line and you should be able to get negative two comments here, too. And so this is the vector that defines the direction of the tangent line. This is the point on the line so we can write. Ax equals one minus two t Why is equal to a one plus zero T, which is just one si is equal to 1/2 plus two t and that's our a set of Parametric leading. So again the key was just, you know, finding the cross product of the two ingredients. And then once he did that, we just used our knowledge of, of like of, like, parametric equations to plug everything in right because you strike access equal to the X coordinate of the point, which is one plus the ah x corner of the spectre times T So we get the finest to tea. And then similarly, for why we use the why components of but the point and the vector and similarly Percy. And then we're able to find her tangent line.

For this question we're getting to Curry X Plus Wife's were seen equals four and X equals one. And at that 10.1 comma, one common one. We're trying to find the Parametric equations of the line tangent to the intersection, the skirts. And so, uh so, first of all, let's right. It's created F of X y Z as X plus y squared plus C minus four and g of x y z b X minus one And so, essentially that if we think about the definition of the tangent line of these two curves, we know that the tangent line is perpendicular to both the Grady in of F and the Grady in a GI. That's how we that's how like the tangent line is formed, right? And so the tangent line must go in the direction. So if we have a line of the form, essentially the tangent line must be in the direction off. Radiant Death cross radiant G permit. And so we need to find greedy enough cross creating a G at the 0.1 comma one cover one. So, first of all, it's fine degree. Annabeth, Right. Sorry. The greeting of s is equal to 12 Why bottom and so the great enough as that 111 is equal to one common too common one. Likewise, the greening of she had it With respect. X Y Z is simply 100 And so the ingredient of G at one comma, one comma one is equal to 100 And so now we must try to find the cross product off great and F cross grading of G at one comma, one comma one. And so, if we do this and we work it out so I won't go, I'm just going to assume that we we know how to do cross products. It's not weaken. Go back earlier into the book and they do explain how to do cross products. But for the sake of time under, it's gonna, uh I'm just gonna sit to write to the cross border. Yet Syria too negative too. And if you are confused, ah, Or if you didn't get this, check your work. Yeah, and if not, just look back earlier in the book, they do talk about how to find cross buttocks. But essentially, once you find this, we know that essentially that the line will be of the Forum X is equal to x o plus the x component of this so plus zero t Why is equal toe? Why component the Y component of the given point plus two times T and then Z will be equal to the Z component, plus my plus negative to t. So since we know that at a given point is 111 you simply plug in and get the X is equal to one. Sorry, there's no there's no zero here. Why is equal to one plus two? T and Z is equal to one minus two t and that's our tangent line. So to recap, essentially, we're giving these two curves, right? We create these two functions and essentially what we do is we think OK, if we're trying to make a parametric equation of this function, we need to know the vector that defines the direction of this line. Right. If we're trying to be a parametric equations like a tangent line, we need to know like the direction off Mike essentially the vector that defines the direction in which this tangent line is moving. And so we think Okay, What direction will this doctor be in? Well, it has to be perpendicular toe the Grady int off the first function and stand in two. And it also has to be perpendicular to be Grady of the second function. That's tangent to. So that must mean that the vector itself must be in the direction of the cross product of these two Grady INTs. So once we realized that all that's left is for us to explicitly find this, um, after mentioned across product and I ah, since so we worked it out and we found that the cross product is this zero common too common Negative too. And Ah, yeah, You can check my work on the cross border. Essentially, After we do that, we just plug it into this equation, right? And then we get our desired parametric equations.


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