5

VI IFI 8ji LI'Lji I :1 1 1| 1 3 1 L V 1 8 j + Wt € 8 { 6 1 1 1 WW Ii 3 1 f IL 1 ! ! 1 2 1 8 { i 3 3 2 [ i 1 V W Ff L Eee EEi @E L 1 NL I 1 3 1 2 4 1 1 ...

Question

VI IFI 8ji LI'Lji I :1 1 1| 1 3 1 L V 1 8 j + Wt € 8 { 6 1 1 1 WW Ii 3 1 f IL 1 ! ! 1 2 1 8 { i 3 3 2 [ i 1 V W Ff L Eee EEi @E L 1 NL I 1 3 1 2 4 1 1 9 0 1 2 0 + 8 ! 3

VI IFI 8 ji LI'L ji I : 1 1 1| 1 3 1 L V 1 8 j + Wt € 8 { 6 1 1 1 WW Ii 3 1 f IL 1 ! ! 1 2 1 8 { i 3 3 2 [ i 1 V W Ff L Eee EEi @E L 1 NL I 1 3 1 2 4 1 1 9 0 1 2 0 + 8 ! 3



Answers

$\mathbf{v}_{1}=\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right), \quad \mathbf{v}_{2}=\left(\frac{1}{2}, \frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right)$ $\mathbf{v}_{3}=\left(\frac{1}{2},-\frac{1}{2}, \frac{1}{2},-\frac{1}{2}\right)$

In this problem. We asked our ass to find the center of mass of this two dimensional system with the masses. 2 8 to 1 and four in the positions. Negative three negative. 100 And negative one, too. In order to find the center of mass, we first need to calculate the center mass in the ex direction, which we will do by calculating the moment in the UAE direction divided by the total mass. Once you've done that, we will calculate the, um, center of mass in the wind direction. Just calculate the moment in the ex direction of over the center of mess. Okay, So calculating Expert, um, the moment of wise calculating by multiplying the masses by the X coordinates. So we have eight times negative. Three plus one time, zero plus four times negative. One all over. The total mass, which is given by eight plus one is nine plus four gives us 13 doing some arithmetic. We can find out that eight times negative three is negative. 24 0 minus forgives us negative. 28 over 13. Now, to find the center of mass in the UAE direction, we're going to take the moment of X, which is the masses times there. Why Coordinates? So we're going to have eight times negative. One plus one time, zero plus four times two all over the total mass, which is still 13 here. Only two are particularly. Get negative 80 plus h, which gives us zero over 13 for our center of mass. Explore why BAR is given by the coordinate point. Negative 28 over 13 from a zero.

Okay, we're given three different vectors and were asked to do things with their magnitudes. So notice the double lines mean magnitude. So for a and let's look at A. M. Because they are very similar but very different also. So notice for a we have to first add our vector U. And V. And then we find its magnitude. But for be we're gonna be finding the magnitude for each and then summing them. So for a let's go ahead and first add our vector you. So I'm just writing it and then I'm going to write my vector V underneath it. And then I can sum them together by adding my columns. So I get a to I minus two J plus two K. So now to find my magnitude I will take the square root of the square of all of the pieces. So we have a two squared a negative two squared, which would be just like a two squared. So in your calculator you can just throw that in as a two squared. And so really what we have is the square root of 12 and that's our magnitude. We can also bring out um you know four times three for 12. And so we can do a square four and bring out a two. So two square root of three. Okay, so now for part B we're going to actually find the magnitudes separately. And so for are you vector, we would be doing the square root of one squared, negative three square, but the same as three squared. And for my calculator, I don't want to get confused in with those negatives. They're not apprentices. And so we end up um um Then also doing the same thing with R. V. And it just has the two components one square plus one square. Okay, so now we have to add the two of them together. Well, our first was the square root of 14, and the second was the square root of two. No go back and get that correct. That is the square root. Okay. Okay, so let's go on to see. So first we already found our um our value for our vector U. Which was the square root of 14. So now whether the negative negative two is multiplied in at the beginning or at the end it's a property Of um are vectors that scaler piece can be pulled out front, so it's kind of like factored out of each piece. Um however the magnitude is just going to be too, because we're losing our direction and so we can take our two multiplied by our square root of 14. And then we're going to be adding that to two times are other vector, which was the square root of two. Okay, so for um d we are going to hold off and do that, we need some more space. But ian f aren't so bad. So let's go ahead and do e. So what it's asking us to do is telling us to take all of our vector W and divide each term by the magnitude. Now what this does is it produces a unit vector. So a unit vector really means that here and we'll see it. It will just show us our direction, but not our magnitude. So let's go ahead. We have all of our pieces. And then our magnitude is going to be that two square plus two squared plus negative four squared. And so that places on the bottom of square root of 24, and we can consider that's a two square root of six. So now we divide each term by two square root of six or two's cancel out our four divided by two is still just a two. And we get what's called a unit vector in a unit vector Really has the magnitude of one and they're really for comparing directions. Okay, now F looks a little bit um more different. Um But it just has the magnitude on the outside. Well we called it a unit vector because it has a length of one. And so its magnitude is one. So just by definition the magnitude of the unit vector is one. Okay, so we skipped over D. So let's now go back and look at um D. So we're gonna have to take three times are vector U. We're going to have to take not negative five times our vector V. So notice I'm just distributing these things and then we are also taking one of our vector W. And then I'm just adding up all my eyes, which gives me zero I all my jays, which is a negative 12 J. And then all my K. Is which is positive two K. Now it's asking for magnitude. So it's kind of like I have a zero squared plus a negative 12 squared plus a two squared. So in the end, that gives me the square root of 148. And if I factor out of four, I get the get Almost wrote 36 but knew this was not 12 squared, which would be 144. So it ends up that it's two times the square root of 37.

Okay, we're asking to find the norm of the. Now the norm of V is the magnitude of the So it's really, if you consider that these are components, um they would form triangles. And we're really looking for our high pot news because that is going to be the distance Between 00 and the point that we are going to. Okay, so for a we will be finding the magnitude of the by taking the square root of one squared plus negative one squared. And obviously it doesn't matter if we bring our negative in because it is going to get squared. Um but that will end up being squared too. So when I write my problems out, I will put the negatives in there. Um But often on my calculator, I don't, if you forget apprentices, you're taking a negative of a one squared, which can be an issue. So just um, I would say in your calculator because you know, you're squaring it, just put the positive in there and so you don't make any um silly mistakes. So this guy is the square root of negative one squared plus seven squared, which is square at a 50 50 is made of 25 2. So we can take the square root of 25 put a five in front five square too. Now notice our next um vector has three components, and if you have a hypothesis of two of them and you square it and you add it to the third squared and take that square root, um that also works. So what happens is you can again just take the some of your squares, it works fine. So if you're finding the magnitude, we can just do a negative one squared plus a two squared plus a forest squared, and then take the square root of that. So let's say we have 14 16, So that is going to be 21, so square root of 21 and our last one also have the three components. So we will do a three squared a two squared in a one squared, so that's nine plus four plus one. So that will be the square root of 14.

So in this question, were given vectors you the interview and were asked to do certain operations. So we'll be doing is question one by one. The first question we were asked to find the length of director You place feed. First of all, we need to find what is you place we and then we'll have to find the magnitude of the length of that vector. So when I am minus three, shapeless to case you the is I pledge city When adding those two victors, we'll be getting it as to I minus two j plus Tookie. So the magnitude is the Squire off the route off the square of the oceans so well before place for bless Forward left the summit. So we'll be getting Route 12. So by simply hang really be getting it as to route three. So this is the first answer. So moving on to the next question in the next question, you're asked to find the some off the magnitudes or the length of directors U and V. That is normal view plus normal week. So we know toe find the norm of the magnitude of any vector is the Squire of route off the Squires of the sun off their corresponding corruption. So one square plus minus three squared plus two square is the magnitude root. It'll be the magnitude of you. And before we it will be ruder one squared plus one square. Starting all these things will be getting Route 14 39 food plus one X p 14 plus two. So taking root to outside would be simplifying it to the form a road to in the road seven plus one. So that's the second answer, which is a simplified form coming on to the next question. So let me raise the A part so we can write it here. So the next question is pretty, pretty much seen as B. You'll understand it. So it would be see is actually my length, or of the magnitude of minus two times you plus two times magnitude of the length of week. So just remember this the magnitude off any vector K obvious models off K into magnitude of factor V. She'll be using this property here so that we can he pretty much easily do this problem so it will be magnitude or models of minus two in tow magnitude of U plus two times magnitude of week. So models of minus two is nothing but to, and we'll be taking to common from both these things. So we'll be getting modelers off magnitude off you, plus magnitude of we. We already know the some off length of magnitude off you. Plus we, because we have calculated didn't be it will be just equating that we'll be getting the answer pretty much easily. So with the answer for this is the route to into seven Route seven plus one. So the final answer would be to root two into Route seven plus one. So that's the answer for option. The sub question. See, moving on. Let me erase option. The question be so the next question is de in here were given the conditions. It's that three U minus five. Replace W on. We have to find the length off that particular vector. So we have to do these operations, which is three U minus five plus w. And then we have to find the model is or the length of that particular vector. So three you is three times I'm sorry. It is three times I minus three g close to key, minus five times I plus d Bless Toe I plus two J minus walking. So that would be three and five, minus 90 plus 60 minus five I minus five G plus two I plus two G minus 40. So it's two g. So I those two j minus four. So we'll have to do simplify this thing So we know three eyes here, minus five is here to eyes, ears. So three plus 25 I minus five ever be zero I So the I common and gets canceled out. So minus nine is dead minus nine. G minus. Vijay would give us 14 g plus. Tuesday would give us minus two allergy, so it will be minus 12 G plus six K minus. Focal give us two K. So the magnitude off minus 12 plus two K would be route off 12 Squire minus 12 square plus two square. That would be 1 44 plus four that would give us Route 1 48. And Route 1 48 is nothing but four into 37. So would be getting it as two or 37. Sorry, that would be two times Route 37. There's a final answer for option D simplified and find lands of options. So let me raise all these things so that we can move onto Option E. Okay, so the option e yes, one by model s off W in two w. So when we look at this, this is actually a unit vector in the direction off. W we know if we divide director by its model s or the length would get a unit vector in the direction of that particular vector. So they're actually trying to find the unit director in the direction of public. So here we know the bluest y plus to the minus four. Okay, so do I. Plus Tuesday minus four k. We have to divide it by the model is of the length of the EC terribly, which is actually to Squire plus two square minus four square on the route off that practical value. But be getting it has to I plus two j minus four k divided by Rudolf 16 plus forward plus four. That would be four plus four plus 16 which would be Route 24 which can be written us. I told to Route six. So? So I buy to Route six. That's two G Buy two Road six minus for Sorry, that's actually to load six. Let me raise it so it can be written last two g buy 26 minus four K by two Road six So we can canceled that divided by two properly. So we'll be getting I've I wrote six plus Jay by Route six, minus two by two by six. This would be the answer for this particular option. So we're going toe the last support off this question, which is F it is. We were asked to find the magnitude off Option E, which is one by models off the in tow Masari models of W into W. So we already know the the option E is a unit director, so the magnitude off Amy unit director would be one. So we pretty much know that particular condition the magnitude of the unit director is always one. So this answer it straight, or we can evaluate or check this particular condition. Bye. One by road six. The whole square less one by road six the whole square plus minus 206 The whole square and the root off. This complete thing should give us one because 106 106 and minus 206 Other corruption off the unit director. So we're just trying to find the magnitude off the unit director, which will always get it as one because let's find it out. One plus one plus 11 plus one plus food by Road six would give B six by six what is one. So in both manner, we got the same answer, which is one. So that's the final answer. I just want


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