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12.5.35Find the distance from the point to the line:P(3 8 3)" X= 3+6t, y =8 + 5t 2 =3The distance is units (Simplify your answer )...

Question

12.5.35Find the distance from the point to the line:P(3 8 3)" X= 3+6t, y =8 + 5t 2 =3The distance is units (Simplify your answer )

12.5.35 Find the distance from the point to the line: P(3 8 3)" X= 3+6t, y =8 + 5t 2 =3 The distance is units (Simplify your answer )



Answers

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

$ (0, 1, 3) ; x = 2t , y = 6 - 2t , z = 3 + t $

We're looking for the distance from the point to negative 34 to the line given by X equals four minus t y equals three plus two tea and Z equals negative five plus three t It would be worthwhile in most of these problems involving vectors. Thio, draw a diagram So we know where we are in space. This point is that too negative? Three, four So approximately we are here. This line is translated from the origin for three and down five. So here's our point. You're here. That's this position and then we have a direction Vector that takes us minus one in X. Huh? Up to in why and up three You. I see. So this specter this portion here that I'll call you is the direction vector of the line and this vector from he down to this starting point of the line, I will call Vector Rip Specter V and Vector. You is the direction vector that is minus 12 and three and vector V is going from P down to this starting point. So that is for three minus five, minus two negative three or that gives us a point 14 um, and negative nine. That is this vector. The what we're looking for is the distance between here and here's who we want Perpendicular two, this vector here and we'll call that d if we extend this this south parallel to you. And we stand this out parallel to the to see that you get this parallelogram and the area of a parallelogram is the base times the height. And in this case, the base is the magnitude of this vector. You and the height is this distance D. The area of a parallelogram is also the cross product off the two sides. So in this case, you cross the and that leaves us with the distance that we're trying to find being the cross product. Oh, you and V provided by the magnitude of you because this is a distance, this is going to be thing gonna be an absolute value. So in order to find this distance, we need to find this cross product the cross product of you. The is the determinant. I j K Yes. One, 23 one For negative nine, which is I times the determine 234 negative nine minus j determine of minus 131 minus nine. Us. Okay, that's the determinant. Minus 1214 And solving this, huh? Gives us, um negative. 18 minus 12. Negative. 30 i nine minus three or six. So minus six J and minus four minus. Tube is minus six. Okay. And so the magnitude of you cross V will be the square root of negative 30 squared. Plus, I get it. Six squared. Plus get six squared, divided by the square root of negative one squared. Plus two squared. Well, it's pretty squared. And if we do this, uh, we come out with the square root of 972 over the square root of 14 which will reduce nine square roots of 42 over seven. So the distance that we have determined between this point and this line is nine square roots of 42 divided by seven

You need to find a distance between the point which is you go to three minus one far and alive in head of equation X Ecco CI farm honesty y equals do tree Best duty and Zico Dumanis five us through tea they conduct distant former between upon to the line will be d equals Didn't know be asked cost effective V over the nominal bit of we hear the B will be the one given in the question. The answer will be the bone on the line here. So it would choose the Seeker Chisel the answer acreage. Oof. Far too. Um honest five and event of the we can get any calls through the minus 12 and three. So the next thing we need to find a better be asked. So we have a better be asked if we could do. We had to ask him. I minus the police again. The one far on Ben, uh, minus nine. And now we need to cross with the V here. Suddenly recover being again here. Therefore, we cost the be asked. Cross with three. Get me go to drought. The first carbon and we could trial, plus Ah, 18. So they were eco just off the And then the second component will be three miners. Nice psycho Teoh Uh, minus six units. AFL emphasizing could becomes a six and then the last component will beat you. Plus for would be six. Um, Therefore, we have everything now and then We should be able to computer Liston now with equal to the square root. I'm the 30 square blocks six square that's six square, inviting by normal of being a B minus one squared, plus three plus two square plus three square go to on the top. We have a 30 square plastic Scrabulous six square. You could just go with the nice seven job running by the the nominee taken one left from last night because you have 14 on, then it would deny 72 divided by four thing. We don't have the nice answer here. So which is equal to the square of the far right is next divided by seven, and it is the least damage in the pond to the line

This question Will even the pond every communist e the big tree minus one far. And we have that lie am here X incurred uniform honesty. Why echoed you That three pressed duty The ICO jet a minus five plus three day. Now with the question of the line, we can fire another pawn being we will entity ecology zero. If I don't being, we could, you know, for three minus five and we can find a better be They were in code, you know, hair will be one for on my list and I And now we can also find a darshan better v and we ico June the minus 12 and three and recording a distant legend upon aged a lie. And we could get a better i V because with a better be taken the gnome running by the nominal but to obey. And here we get equal to the cross border. Um uh, job them Go ICO too. Hey, I would be on throughout. This will be a trail. Plus you never know and will be 12 on then. Uh, plus 18 Sick agenda 30 day and you have three minus, you know. So it would be minus 12 and the last one handed to All right to bless foreign coaches six and dividing by the number but a ve coach. You know, one blessed for Preston, I So in which in, if I get echoed you the time again. Uh, 30 square plus drought square for six square. So you gonna want our right there. Only wedding by the square Root on faulting the venom in front in. And then we can be going to no five. So far, there are seven square root.

Okay, This question wants us to find the distance between these two vectors. So to do this, we need to notice that the shortest path between any two vectors is the perpendicular distance between them. So what we're looking for is the magnitude of the perpendicular component. So to do this will first find the parallel component. And I'll say that the parallel of why that's in the same direction as you is just the projection on to you of why, Which would be Why don't you, over the magnitude of you squared times you so filling in the relevant numbers here, Why don't you would be eight times three is 24 plus one time six, which is six. And then we divide by the magnitude of you squared, which is eight squared plus six squared. And then we distribute that into our vector. You, which is still 86 then simplifying. We get 30 over 100 times 86 or why parallel would be equal to 24/10 and then 18/10. And we can simplify that even more into 12 5th and 9/5. So then we confined the perpendicular component just by subtracting off the part that's parallel, so we'd get 31 minus 12 59 5th or what we would get here would be, while three is 15 5th. So 15 minus 12 we get 3/5 on top and then in the bottom. One is 5/5 and we'd get a four here with the negative sign. Then all we need to dio is take the magnitude of this perpendicular component, which would be the square root of 3/5 plus 4/5. Both of them squared, and that would just give us the square root of 1/25 times. Three squared plus four squared. And that's just a 345 triangle, so we just get square root 25 or 25 or one.


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