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10 in the subspace W spanned by U1point) Find the closest point to yand U2The closest point is...

Question

10 in the subspace W spanned by U1point) Find the closest point to yand U2The closest point is

10 in the subspace W spanned by U1 point) Find the closest point to y and U2 The closest point is



Answers

In Exercises 11 and $12,$ find the closest point to $y$ in the subspace $W$ spanned by $\mathbf{v}_{1}$ and $\mathbf{v}_{2} .$
$$
\mathbf{y}=\left[\begin{array}{r}{3} \\ {-1} \\ {1} \\ {13}\end{array}\right], \mathbf{v}_{1}=\left[\begin{array}{r}{1} \\ {-2} \\ {-1} \\ {2}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}{-4} \\ {1} \\ {0} \\ {3}\end{array}\right]
$$

Thank you, man. What? He was six. So minimizing. No squad off the distance. Fixed sequins X minus one. Will Esquire? Yes. Why? A Squire. This is equal to X Esquire plus one minus two X. Yes, My squad. This is equal. Do. It's It's quite plus one minus two weeks. Plus excess far since X is equal to lane. It's the goes affects. It was to excess. Well minus X plus one. This is a fix. So have death. Sex becomes going to do X minus two kisses four X minus. Tool, this is Theo. Therefore doing was for Rex. What? We can see X equals. Uh huh. Since X equals Y, then four white. Oh, Sweet, Huh? No, we have one upon two on one of goingto Is that point? Um What was next? Just two. This is the answer.

What point on this plane is going to be closest to my given point? Well, to know how far apart they are, I need to use the distance formula to find the distance. I'm gonna take the square root of the difference of my exes squared, plus the difference in my wise squared, plus the difference in my Z's squared. Now, I would really like to have this in two variables instead of three. So I'm going to look at the equation for my plane, and I'm gonna rewrite that and solve for Z So Z equals two minus x plus. Why? And I can substitute that in for Z in my distance formula, and I get X minus one squared plus why minus one squared plus one minus x plus y squared. Now, before I take my partial derivatives, I'm going to simplify under that radical as much as I can multiply things out, add my common Ah, any common terms that I have and what I end up with is two x squared plus two y squared minus four x minus two x y plus three. Okay, Now, in order to find where those are the closest I am minimizing the distance. A minimum happens at a critical point. So to find my critical point, I need to find the derivative with respect to X and with respect to why. So let's do X first. I'm going to take that partial derivative, which is on the denominator two times my entire radical and my numerator will be four X minus four minus two y. If I factor what I can take a two out of the top in the bottom, I can simplify that just slightly. 22 X minus two minus y over. Two times my radical all the way across. Okay, Now I need to do the same thing in terms of why so my derivative, by partial derivative with respect to why has the exact same denominator. But my numerator will be in this case four y minus two X and Aiken Simplify that by pulling out a two from the top and the bottom. And what I end up with is I just realized I copied it to here where I shouldn't have. What I end up with is the square root of two X squared plus two y squared minus four x minus. two x y plus three in the denominator and my numerator is to why minus X and I want these to be equal to zero. Well, that means on the top two X minus two minus y equals zero or two x minus Y equals two or on the bottom. I have to. Why minus X equaling zero or X equals two. Why, by substitute that in two times. God, too. You make that clearer. So I'm not messing up is much there. I have to Times X, which is to why minus y equals two. Or why equals 2/3. If y equals 2/3 I go pin plug that back in and I get X equals 4/3 and back to my original equation showing the plane I get a value for Z also equal to 4/3. So that is the point that is closest to the given point. The point is 4/3 2 3rd and 4/3

Hey, it's clear, So immune right here. So the normal vector of the plane is also directional is, although the directional vector of the perpendicular line Sophie is equal to one. Make it a one on one. The equation off the line, passing through a pointy and parallel to the direction v. ISS, given by lt because P. Plus TV and tea is over. Real values. So this for us this is going to be equal to one plus two tea common to minus G. Come with three plus T. Then we're going to solve for tea X minus y. Let's see. Is it good for Plug it in, you get one plus two minus to my honesty. Waas three plus two u. C. Four Do you get to third and substituting in the equation? We get the corn off the point, so we're in it where X we get 5/3 or why we get 4/3 and foresee get

I think. Okay. So again, we're going to use the fact that the distance it was going to be, uh, x minus your ex, not square. All right. Plus y minus year. Why not? Which is one this case? Screw that Z ailing this year is you know, it squared. All right, so we want to minimize this, and we have the plane. Um X minus Joe, I plus three z is equal to six. All right, So you know this that if we minimize what's insight, so call f that what f b this with his in parentheses and ignore the square root. The square root will also be with the mice and so called g the following. And we also have eat condition. On top of that, the equal zero. So that's our boundary. Um, and, uh, f is this here? So let's just differentiate with respect. The x, y and Z together ingredients derivative X, with respect to X nursery experience to x right. Um, the derivative with respectable I will be too. Why minus one. Everything with respect is he will be. Is he minus one times two. These will all be equal to you. We um, the Times, whatever the ingredient is over here, So gxe happens to be one. Since it's just one G y happens to be negative TUNEL. So this is Ah, we end at times negative too. Um, Jeezy happens to be three. So the citizens live the times three All right, so we have the fact that two X equals lambda. So let's just point that in everywhere. So we'll have that if we divide through by two here we're just getting negative. Negative, Linda. Right? So that employees, that one minus y is equal to lived up, which is equal than two X So why is equal to why is equal Thio one minus two x here. If we plug into X, we're going to get to Z minus one is equal the six x divide by two on both sides and then sold for Z. So this is a three Alright, so Z is equal. Thio three x plus one Plug that back into the equation above and you will get X plus. No, it was minus two names one minus two x then plus three times three x plus one. It's equal to six. So you have plus four x plus nine x 13 plus once of 14 x negative too, Was three is going to give us one some X is he called the six. Alright, so subtract one divided by 14 you'll get 5/14. Right? Um so that is X. And then Z is equal to three times that 15 uh, plus one. So 16. I'm sorry. Hold on one moment. So three times that. 15 plus 1 16 little for, uh, 16 of her 14. Why is equal to go one minus two times X. So that is one minus 5/7, which is two or seven. Yeah, well, jeez, I mean a little mystique here. So we have 15/14 plus 14 or 14 until that is 29/14. So we have our 0.5 14/14 to over seven and 29 over 14 and, uh, yeah, we're done.


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