in-exercises-7-10-let-h-be-the-hyperplane-through-the-listed-points-a-find-a-vector-mathbfn-that-is-

Question

In Exercises $7-10$ , let $H$ be the hyperplane through the listed points. (a) Find a vector $\mathbf{n}$ that is normal to the hyperplane. (b) Find a linear functional $f$ and a real number $d$ such that $H=[f : d]$
$$
\left[\begin{array}{l}{1} \ {1} \ {3}\end{array}ight],\left[\begin{array}{l}{2} \ {4} \ {1}\end{array}ight],\left[\begin{array}{r}{-1} \ {-2} \ {5}\end{array}ight]
$$

Anthony Ramos · Accepted Answer

in this exercise, we need toe find there is a new hyper plane. The age that that passed through three points. The one on the 13 we to is a point to. For one onda 0.3, find us minus one minus 25 So we got another on hyper playing that passed through these two points on. We need to find first the normal vector, the normal vector to this plane. So technically thes is, or hyper plane age on here. In this plane lies the 3.3123 So we need to find a vector that the normal vector to these hyper plane. Okay. And then after that, we need to find the linear functional such that this age is equals to the that linear, functional f on that number. Okay, so let&#x27;s start by considering the normal vector in this case on, we&#x27;re going to construct in affordable way. So these three points lies on the same on the same plane. Okay, so we to the three. So we can find vectors that lies also on this hyper plane by just taking the difference between these two points so we can define here a vector Alfa, and I&#x27;m going to call this victory beat on how to construct the normal vector. We can do this in two ways. The usual way is that it&#x27;s supposed that this normal vector should be perpendicular to these two vectors here. Alfa Um beater. So we just need to defined two equations and that product Alfa should be close to zero on end that product, Vita should because 20 And here we obtain our system off a linear linear system off equations on we can solve. However, there is another way to do this we can instead off considering thesis complicated could be complicated system We&#x27;re going to consider geometric property of the vectors. So let&#x27;s suppose that here is we one hears me to. Here is the three we find here are vectors Alfa on Vita on We know that the cross product off Alfa with Vita so we can define the normal vector Just as the cross product off these two vectors here could just a even simpler. So let&#x27;s compute these vectors Alfa on a vita. So here we just need to take three to minus We won and here the three minus we won from this we obtain. This is two for one minus 113 on Dhere. Yes, minus one minus 25 minus on 13 So, Alfa, the Alfa factor is 13 minus two on de Vita Vector is minus two minus three to Okay, so we got these two vectors, and now we can compute the cross product. These two were We can use the determinant way. So that means that you were going to put the component I J k. So here is 13 minus two, minus two, minus three to on. From this we obtain that is cross product or the normal vector is 0 to 3. Okay, so this is a normal vector that we try to find for the part A. Now, now that we have defined it in normal vector, we can define the functional. So how to define functional? Just remember that the equation off hyper plane is the normal vector times a vector X equals to be We&#x27;re X is on our end, depending on the dimension off the hyper plane. Okay, so this is similar. This is going to be or functional on Duh. So let&#x27;s Let&#x27;s compute this so the functional in this case is defined. X one x two x three Will be the normal vector times x one x two x three But that is equal to say 0 to 3 The product here x one x two x three So or functional if 6123 is two x two plus three x three Okay, now we need to find the value off the on that it seemed the is obtained by evaluating dysfunctional on some off the point B one B two or B three that lies on the hyper plane so we can choose any of this in particular. I&#x27;m going to choose the first one, the big one. So let&#x27;s evaluate this. A to point B one on this is equals to 11 three, so f 11 three is equals to to times one plus three times three on this is equals to 11, so the value off the is illa so at the end, or Hypercar hyper plane is defined as a function of F D, where f is equals to two x two plus three x three. Andi is equals to 11

James Chok · Answer

okay. In this question, you want to find the normal better that is normal to the high percentage covenant by B one, B two, B three and the linear functional, if such that all the author of this whole thing. So first off, who perform actually things again. So v to once we won you go to four minus one gives you three minus two minus minus two gives you 03 mice. One cause you to the three months being one is seven minus one is his sixth minus four minus minus two minus 24 months. One give you re so then your normal vector. You know better. What dick that&#x27;s about being one win is being so It was 6 to 0 and you know more vector about the three month be one most possible victims there surviving these two out fully. You have three and one plus zero times in to Jesus Syria. So, plus to end to June +33 physical to zero and six n one my spine two and two plus three and 3 to 0. Just looking for anyone you think too negative to end three other three. So that&#x27;s choose n 3223 and end one easy to shoot. So the reason why would choose minus three is just so I can catch it on the street. There&#x27;s no other reason why you could do the same thing for in 321 you get the same results in putting it back into this equation. You have six times to give you 12 minus to end. Two plus three times minus treaties. You mice line is it? This is your then upload Modest line gives you three are moving out onto the other side. You have negative too. And Chu is attitude 93 So end too easy with you. Sure. So our mom a vector. It&#x27;s just negative. 33 over two on two. So if this is a normal vector to get rid of the half, we can save this set All normal vector to be equal to Thio End to 2000 original, which is minus six, three and four. I don&#x27;t know, I have these ones. This is true and this is minus three. So this is 23 and mine, right? Okay, this is two, 32 and 93. So then this is equal to four, three and minus six. So the reason why we can do this right he had to be scaling is if you think back on two dimensions. If you&#x27;re no more, Victor was like this. Then if you&#x27;re playing your planes problem Dick Lecter, this normal vector in this plane, this plane will this line will still be perpendicular to two times Inspector. So that&#x27;s why you can just simply scale it to be this. So I can&#x27;t We know that a little victor is full three and my six. Well, then a linear functional, as you expected, is X factor in Vector. So that&#x27;s four x one plus three x two minus six x three. So then, to find the real number d or you need to do is simply substitute X. Better is XlV one or maybe two? We&#x27;ll be three. Doesn&#x27;t really matter. In this case. I&#x27;ll just prove you won. So be one is equal to are think one Maya&#x27;s shoe and one transfers So off the one vector What Time&#x27;s one gives you for three times I minus two gives you minus six minus six mile tied by minus one gives you a negative six. So this is four minus minus four minus six is minus two minus 60 minus. It&#x27;s so this is the

James Chok · Answer

okay. In this question, we want to find the hydroplane governed by these four vectors. So, of course, the person that we do is we shift the vectors. So that&#x27;s to be two months to be. One jean minus one gives you 12 months to give. Zero minus one minus zero. Gives a minus one monastery. Monastery gives your minus three. Now we three models we won one minus one gives you 03 months to give one to minus 32 710 G seven and finally the full. What does he want? So three months, one gives you two tumors. Tickety zero minus one minus zero, minus one and minus one minus zero. You&#x27;re minus one. Now let&#x27;s end be our normal Veta. So then we have to solve and Victor Harvey choose money to be won, which is equal to in one month&#x27;s by and three. My spy. Three in four. This is equal to zero because because n must have been taken to this letter, right? Yeah, it didn&#x27;t. We do the same thing for the other two victors. So in vector, about be three miles be one is equal to and two. That&#x27;s What? Judy in three plus by seven in four is equal to zero finally and Victor before my sweet one. You think too, due in one minus phi in three months by end, pool is equal to zero. Now we&#x27;ll take two of these and subtract it this way. So to minus two days, right, negative one minus minus two gives you plus one. So we end three and minus one minus. Why, minus three times two to that gets you. That&#x27;s bye bye in or is equal to zero. So now we let end for be equal to one. Well, you got one. So then, in three, easy go to minus five in four, which is effective, modest by So to get into the substitute in one and informed and three into this equation. So you have and two plus by two times minus five years. You minus 10 plus by seven times once That&#x27;s just plus seven. Is it Caesar? And two minus 10 past seven. Q T minus three is equal to zero. So en tu is equal to three now getting in one. We use anyone informed and three in this equation. So anyone minus five minus five. So that&#x27;s plus fire minus three times what? Kissing minus three. Is it good teaser? So N one plus two was exert. So n one easy between minus two. So your normal vector is then minus two three minus five and one. So this is just simply taking and one is it to monitor and 2 to 3 In three months, I&#x27;ve been 40 to 1. Yeah, so then a linear functional is in f functional. Expect is n, but thanks, Victor. That&#x27;s just minus to x one plus by three x two months by five x three plus by X Oh, now it&#x27;s a final de. All you need to do is the substitute f As a function of this state, V one doesn&#x27;t really matter what people you could be Veach. You&#x27;ll be free before doesn&#x27;t matter. So we have the one in the equation be one equal to 1 to 0 and if it&#x27;s a function of the one is minus two times. But why? Using minus two plus three times by two Getting plus six months part time +00 plus four times your plus zero disease There that this is just cool. So your linear functional is given by this with D executive for

James Chok · Answer

Okay, so in this question, we want to find the hyper plane. Given that the hyper plane is equal to the roast base would be so in question. If you&#x27;re, um six in section 6.1, we know that the roast base of B is equal to is perpendicular to the last message. Be so all we need to do to find the 70 club vector off be We just have to find the hectic denounced recipe because Director in big particular is in the right space. So to find a victor in the null space, all we need to solve is B X is equal to zero. So we get X one just by four x two miles five x three Is it easier? So this first line comes from this vector times by extra extra. So be one times x one for X two months, five x tree And in this one come from the next row. So we minus two x two plus by eight x three is equal to zero. So solving for X two, we have ex too easy to four x three. Well, so were the x three b one X to be full extra careful and something for next one we have. X one is a excellent plus all time by forgetting 16. Marcus Fox had 1 55 because he&#x27;s right. So x one plus 11 is it to zero? So x one is even too modest level. So Okay, we know that the Victor Morris 11 four and one is inside is an element in the null space O b. Now, because this is in a small space, it be this vector must be perpendicular to the right place to be. So to fight for a lenient functional, we can simply take our n vector to be distracted right here at 11 for one. And then if x box is this simply n but X. But so my 11 x one trust by four x two plus by x three. So to find the to find hate, we can just set Deacon, scissor user, And we&#x27;re done

James Chok · Answer

Okay, so we want to find the hybrid thing described by when is given by this measure is right here. So then the Vigna functional if functional ex Victor, it&#x27;s just a s. So this is just to x one us five x two Mars by three x three plus by six Next five Now this is effectively the same thing as Ryan x one x two x three x four, which is either to to x one plus by five x two Minds by three x three Suspects six x full. So the biggest difference between these two is that here using X fire and here, using X four now in describing hybrid friends, this wouldn&#x27;t really matter so much because he just switcher to access is so It doesn&#x27;t really matter too much now, since F since hedges and null space off, eh? It follows that this f should be equal to zero. So your linear functional is given by this thing with D is equal to zero

James Chok · Answer

okay. In this question, we want to find the l line l that prostitutes pimps be wanting to be true. The pistol rule. Find door translated like I don&#x27;t know which positive. Very be to be one. Performing calculations minus two minus one, which is minus three minus one minus four is just minus five. So the point is, in albums, it must be perfectly clear to this line out here. So if you spend a nickel in the Bumble Bee to mind what our victim, this should give you so in one. And Chief Well, I thought that was minus three and Mars by which is even too negative. Three and one and two. Today&#x27;s long for anyone. We just have this need to name two live and chewed over. So then we just choose a model for into cancer satisfaction. So this is choose end to minus three at an and one is equal cheered. So then you have and it&#x27;s even true. Five and wise three. So then we&#x27;re functioned x one x two is even to you and X, which is to be five x one minus three x tooth to this function. Come dysfunction represents all lines with which is parallel to this. Inspected by open ridiculous Inspector. Right here. So to get back online, Ellis on what we need to do except servitude. One of these points in anticipation. So we can either pull minus two miles. One no one in for closer this case. It was wonderful. A 14 executive flats, as one wishes you find minus three times What was she 12. This is my son Thio Online l is along. L is Why x one? Twice three x two 97 Oh, by x one one&#x27;s three x to just seven is equal to zero.

Problem

In Exercises $7-10$ , let $H$ be the hyperplane t…

Problem 7 Easy Difficulty

Answer

a) $\vec{n}=\left[\begin{array}{l}{0} \\ {2} \\ {3}\end{array}\right]$
b) $f\left(x_{1}, x_{2}, x_{3}\right)=2 x_{2}+3 x_{3}$
$d=11$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Catherine R.

Kayleah T.

Caleb E.

Samuel H.

Vectors Intro

Vector Basics - Example 1

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a) $\vec{n}=\left[\begin{array}{l}{0} \\ {2} \\ {3}\end{array}\right]$b) $f\left(x_{1}, x_{2}, x_{3}\right)=2 x_{2}+3 x_{3}$$d=11$

Linear Algebra and Its Applications

Catherine R.

Kayleah T.

Caleb E.

Samuel H.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Catherine R.

Kayleah T.

Caleb E.

Samuel H.

a) $\vec{n}=\left[\begin{array}{l}{0} \\ {2} \\ {3}\end{array}\right]$
b) $f\left(x_{1}, x_{2}, x_{3}\right)=2 x_{2}+3 x_{3}$
$d=11$

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications