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X 28d" 5 Sp Sp Sp 2 Sp Sp'21) 212 1 ) 1 1 1 1nil liwav A i...

Question

X 28d" 5 Sp Sp Sp 2 Sp Sp'21) 212 1 ) 1 1 1 1nil liwav A i

X 28 d" 5 Sp Sp Sp 2 Sp Sp' 21) 212 1 ) 1 1 1 1 nil liwav A i



Answers

(a) $P_{1}(3,5), P_{2}(2,8)$ (b) $P_{1}(7,-2), P_{2}(0,0)$ (c) $P_{1}(5,-2,1), P_{2}(2,4,2)$ (a) $P_{1}(-6,-2), P_{2}(-4,-1)$ (b) $P_{1}(0,0,0), P_{2}(-1,6,1)$ (c) $P_{1}(4,1,-3), \quad P_{2}(9,1,-3)$

Hello there. So uh anticipation we have this matrix A. And we need to do, we need to check, you need to do two things. The first thing is to verify the common space of a is an orthogonal set and second we need to find an Ortho normal basis for that space. So we know that become space of A. Is generated by the columns of the matrix. So that means that we need to check that the space Amiga in this case for my B one, B two, B three, he's our throne. Okay. So each of the columns will correspond to one of the factors that generates this space. So the meaning that this omega set that generates actually so basically become space of a will be this one. Yeah. Amina. So this omega said is or or a phono means that the inner product of all the possible combinations of the of the components are equals to zero. So we want in there 2. 1 would be three odd, the two would be three. So we need to verify that each of these products are equal to zero. So in the case of The one would be to that is equal to 1/10 times. So we have here is one. So first I take as Common factor from V21 health and then from the one 1/5. So that's why obtain here 1/10 As a common factor. And then we multiply the elements of the factors that means -1 was one and as it goes to zero For everyone would be three. We have that Here is a forgotten -2. So there's that goes to 1/15. one was 1 -2. That again physicals to zero And finally be too would be three. That is 1/6 of -1. What's going equals to zero? So suspected. The general the generators of the common space of a uh foreman or normal or tonal set. Okay, so each of these vectors, we want Be one between the three. Alright, so no, with each other, that means that easily. Now we need to extend this to an Ortho normal basis for that. We need to just normalize each of the Each of these vectors. We want two and 3. So remember that realization means that you can define vectors you want us? We one divided the norm. You too. It's going to be too divided two. And the same for you three. So all of these victories are going to be unitary. Um the corresponding in their problems. So in this case the correspondent Norms of factors are the following. So the norm of B one is equal to The Square Root of three. Life For two equals 2. S squared of 2/2, four. B three. These norm is equals two this quarter to over three. Mhm. So we have here all the norms and then we can normalize the vector. So doing that, we obtain that You won is because to one over the square root of three, 11 one For you to the z equals two. one over the square root of two of minus one. Yeah, zero. And you three As equals to one over squared of six times of one, one minus two. And this set, so the set Formed by U one U 2 and U three corresponds to an orphan normal basis. Or did come space of A.

Well, everyone, this is Ricky and they were working on a problem. Five from a model test number six. And so this is a question asking us about sphere that's inscribed in a cube. And what the ratio of the falling sphere to the booth. All right, So to start this question, let's write down the formulas for both of our severe and for acute first fear, 4/3 choir cute. And for the volume of the Cube, a heart issues this will rather Cuba's the length aside. Yes, threes as cute. And then, if we translate this where the length of side ISS equal to the radius spear, get or rather the diameter of the spear, get what you are. That's a cute Wallace to paint way Now if we look at the ratio votes spear. Q. Start with Why are huge? Huge. You see that? Ready? I drop out. And for my what a fine two. Everything's fine Pi over 63.14 over six, which is about 0.52 I hope this video is helpful. Seeing that

Okay, so this question we just want to find the oh image off given sense. First sets minus two minus 10 12 and three. And we have Fo X is equal to floor excluding my life. Okay, so we just have to perform this computation about everything. So we have ever minus two. He's gonna be floor maestro sweat. That's four divided by three, which is 1.3 repeating. This is just one if a modest one is going to be one squints. Thus when they won by three, which is just 0.3 repeating that's just zero fo zero is zero squared zero divided by three. So that's a serious side. This just for zero. This is zero. If all one is for one square to that, just one divided by three. That's his full. Is there a 0.3. Repeating was just zero f old chu for two spaces. That's 41 by three, which is one for three paintings. Us one off three is in full off three squid. That gives you nine. What? By three. That's a 43 So then our limit our image all the say s Is this going 013 to be careful, you don't put twice. Um, you don't put the same number twice more than twice into the set. Because, strictly speaking, sanitation needs to be ridden as unique numbers. So even though the 7001 is equivalent to 011 you don't write like this. You're right. In terms of organ, you numbers. So like this. No, uh, B s is equal to 012345 Wherefore s where x is well exclaimed, divide everything. Okay, so F or zero is just going to be zero squared, divided by three semesters. Zero because zero, if one is you 21 switch, that's one divided by three. This is your f off to visit would choose where that's four divided by three. This is one. If both three is then three sweat, which is nine. Divide by three. So that's his three. Divided by three. So that's also 95 by 3 53 four gives year floor 16. So for about four square is 16 divided by three. So that's gonna be, uh, 15 like that. We 5.3 repeating. So this is five. And finally fo five Easy to 25 divided by three Which should give you that. So then our sets is in 013 live in eights. Okay. See S is equal to 15 7 11 So f or F or X is easier for exploded. Three. It's one is just gonna be one squared divided by three So that's just zero f 055 squared gives 25 divided by three So that's still don't eight f 07 So seven squared gives you 49 divided by three. To that excuses, 16 f o 11 gives you 4 11 square So that's 121. What about three? So that gives you four. So then your image is then 08 16 and 40 And finally we have the s is equal to 26 10 and 14. 14 f f o x is equal to well, exclaimed the wildlife Now F 02 is in full. Also to squeak is or divided by three. So that's just one F off six is It was a 6 20 to 36. You want about three? That's just 12. Every 10 tense. What gives you 100 divided by three seconds. You 33 ever 14 is 14 square, which I am not very sure. Also start putting spread you all right. Three, which is just 65. So then you set your image effort s busy one, 12 33 and cc. Also one notes that I want to say as well is that you don't always have to write This sets in strictly ascending order or in some sort of order. You can just very easily right this whole thing as 1 33 12 65 It doesn't really matter or not so much. What matters more is a uniqueness off, but that's just a quick next.

It will have X here We have exciting one word of the why therefore, yes. And he's a function is, uh, sure.


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