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8. Which of the following pairs of vectors are parallel? I.a = (3,2) and b = (-9,-6) II: PQand OR_ where P = (1, 1,0),@ = (3,5,2),R=(-1,-2,-1), and 0 is the origin....

Question

8. Which of the following pairs of vectors are parallel? I.a = (3,2) and b = (-9,-6) II: PQand OR_ where P = (1, 1,0),@ = (3,5,2),R=(-1,-2,-1), and 0 is the origin. III: & = (2,1) and PC , where P = (0, 1) and @= (2,1)only only II only III only I and II all of them

8. Which of the following pairs of vectors are parallel? I.a = (3,2) and b = (-9,-6) II: PQand OR_ where P = (1, 1,0),@ = (3,5,2),R=(-1,-2,-1), and 0 is the origin. III: & = (2,1) and PC , where P = (0, 1) and @= (2,1) only only II only III only I and II all of them



Answers

$19-20$ Determine whether the given vectors are orthogonal,
parallel, or neither.
$$
\begin{array}{l}{\text { (a) } \mathbf{a}=\langle- 5,3,7\rangle, \quad \mathbf{b}=\langle 6,-8,2\rangle} \\ {\text { (b) } \mathbf{a}=\langle 4,6\rangle, \quad \mathbf{b}=\langle- 3,2\rangle} \\ {\text { (c) } \mathbf{a}=-\mathbf{i}+ 2 \mathbf{j}+5 \mathbf{k}, \quad \mathbf{b}=3 \mathbf{i}+4 \mathbf{j}-\mathbf{k}} \\ {\text { (d) } \mathbf{a}=2 \mathbf{i}+6 \mathbf{j}-4 \mathbf{k}, \quad \mathbf{b}=-3 \mathbf{i}-9 \mathbf{j}+6 \mathbf{k}}\end{array}
$$

Okay. This question is asking us to deal with parallel vectors. So wants to know if each of these air parallel Savi. So to do this, I like to get one of the components for V to be a one or two or some easy to work with numbers so you can see the relation between the coordinates. So let's divide both components by three. So that means that V and all vectors paralleled Avi are in the form to calm a three where there's two of something in the X component and three of it on the white component. So let's check a 12 common 18. Well, to get from to to 12 you have to multiply by six. And if that holds for three, we know the vectors air parallel. So three times six is indeed 18. So this is true because two times V is what this factor is. So it's in the same direction, then three comma too well to get from 6 to 3, we have to divide by two. But if we divide nine by two, we do not get to. So these air not parallel than 23 Well, we already established that 23 is sort of our direction vector we're looking for here. So this is true cause this is just busy. Divided by three. So it's same direction. Negative six. Negative nine. Well, this is also parallel because this is just negative V, which means it points in the opposite direction than negative 24 Negative 27. Well, if we divide negative 24 by four, we get six. I apologize if we divide negative 24 into a vector. One word can we want to get it into six? So 24 divided by four is indeed six 27. Divided by four is not nine, so they're not parallel, but this time negative. 24 1926 If we divide by negative four, we get six comma nine and that is V. So this is it. Also parallel. It's just negative. Four times V. So its opposite direction. So again, just reiterate what we did here. For each of these vectors, we tried to convert it into the form of our vector V, and if both components matched their parallel and if both components aren't the same, then they're not parallel. And if there's a negative sign in front of the scaler, they point in the opposite direction.

Couldn't conduct the Juventus am be there but allow And only if the better I can be written down as a multiple of better be. And when it came to wind up uh, in the real number yeah, in this question from the part I now we're giving them. But be here you go to the 69 that is that we can take the tree outside Uganda 203 and in the part a were given the better trail 18 so it would take out Onda six Then we can get equal to the 23 So this one for that means that we by our innovative V and pointing the same direction here fun of p were given that three too. So it's not the same as the 23 solution will be not gonna allow to limit away for the sea were given the to three So their fathers and will be barrel out innovatively and same direction For the d, we have equal gender minus six months nine now Well, it can be written down as a minus three our side and then we have the two on the three And because of the minus here. So it means that will be better to the V. But opposite direction here. Fun of fun. Ah, a We have the Manus 24 1 27. So that would account. Um, a monastery outside. We have you would be enough eight and And I their foundation. Not perhaps a little bit of we on in the air. We have the minus 24 months. 36. It would take around. Uh, listen, you can take under six. Uh, and the trial one is through outside that we have to end it. Three because of the man is here. So the falls in a barrel A t v. But opposite direction here.

So we're dealing with vectors. Specifically, we're dealing with the dot product of vectors, and we want to know if it's thes vectors, orthogonal, parallel or neither. So to do that, the dot product is really useful. We're going to dot the a vector with the B vector and what we end up getting as this is for part A. We would get a negative five times six plus three times negative, eight plus seven times. Two. When we do that, we end up getting a negative 40. And that doesn't equal zero. So because it doesn't equal zero, we know without a doubt that the vectors aren't orthogonal. And then what we want to dio is find the angle between the given vectors. So we are going to take the magnitude of the A vector. We get this squared of 83 take the magnitude of the B vector, and we get the square root of 104. So according to corollary six, we know that data is going to equal the Arcos Einar aside. Like to write it? We could write it as the inverse co sign, um, of negative 40 over the square to 83 Arizona four. Keep in mind this came from all the values that we got. So we have the negative 40 represented here, the square to 83 screwed of one of four. What we end up getting as a result is 115.5 degrees and this is not, um, equal to a multiple of pie, so we know that they are not parallel. So we know that they're not orthogonal either. So they are going to be neither. Then for part B, we're just gonna follow the same process today is neither be with the given vectors that we have. We take their dot product. We see that it's four times negative three plus six times to that's negative 12 plus 12, which is just zero. So we know that they are in fact, orthogonal. Then for part C, we take the dot product again. We get negative one times three plus two times four plus five times a negative one. That will be a negative five plus eight minus three. That will give us zero. So we know once again that these vectors A and B are orthogonal. And then lastly for part D, we take the dot product and we see that they are not equal to zero eso. Now we want to look for to see if they're parallel. So we take the magnitude of the A vector. We get 56 take the magnitude of the B vector, get 1 26. Then we'll find data which is going to be Thean verse co sign of negative 84. 0, and keep in mind that the reason why we get negative 84 is because a dot B was equal to negative 84 which was not zero eso. With that, we get negative 84 over the square to 56 times the square root of 1 26. This is going to end up giving us through the inverse coastline of negative one which we know to be pie. So because that equals a multiple of pie, we can say with assurance that these two vectors are in fact, parallel

Okay, so this one has four parts, so? Well, go ahead to start with a here. Um, we need to figure out what are these actors orthogonal Kyle? Or neither. So for party, we have vector a being equal to negative five, three and seven and a vector B is equal to six. Negative eight and two. Okay. So to determine if they are perpendicular, we want to find their dot product. So a dot B is going to be equal to negative five times six plus three times negative, eight plus seven times two. And this is negative 40. And this does not equal zero. So therefore they are not perpendicular now. If they were parallel, then there would be some sort of scaler. Such that is equal to some scaler time, Inspector B. So to term in this, I think what would six need to be multiplied by to be equal to negative five, right, So six times negative 5/6 would give me negative five. So what happens if I multiply a negative eight by the same thing? If I were to take negative eight and multiply it by native 56? Well, that does not equal three. Right? So therefore, that's killer does not exist. Okay, so these are not parallel? Yes. Is the answer here then is going to be neither Neither perpendicular or parallel. So moving on to part B this time were given Vector A is equal to 46 and vector B is going to be equal to negative three two. So start by taking the dot product, and that is equal to four times negative. Three plus six times two plus zero time zero. So that's gonna give us negative. 12 plus 12 0 Awesome. So that means that they are perpendicular, okay? And we want to take a look and see if we can find some scaler such that a is equal to a scale air times be. And so what would we need to multiply? Negative three by to get four. So if I were taking be would need to be four divided by a negative three. Right? Would give me four. So if I multiply to buy this same four over negative three No, this this is gonna give me native thirds, which does not equal six s. So therefore, there is no no scaler that that does that on. So that means it's not parallel. So yes, perpendicular, but not Kayla. Okay, so we move on to part. See here Vector is equal to negative one, two and five, and the vector B is equal to 34 negative one. So to determine if they're perpendicular will take a look at the dot product that's gonna be equal to negative one times three plus two times four plus five times negative one that's gonna give us negative three plus eight minus five, and that is equal to zero. Okay, so that means that they are indeed perpendicular. And then we want to see if to be fair, once you know they're perpendicular, you know that they are not parallel. Um, but to be thorough, you know, Ah, doing the process of determining whether or not they are. We can go ahead and see if there's a scaler such that is equal to a scale or times be. And to do that we say what we we multiply times three to get negative one, and so he would take negative three. I'm sorry. Just take positive three, and we would multiply that fire and negative 1/3 So then we'll take this negative 1/3 and multiply it by four. And we would get negative 4/3 which does not equal to. And so it's not parallel. Yes. So for this one, we have again just a perpendicular hair vectors and on to Part D. The final installation. Here we have Vector a being equal to two six, negative four and vector be being equal to negative three, negative nine and six And so first to determine whether or not they are perpendicular we take the dot product and that would be equal to two times negative. Three plus six times negative, nine plus negative for times six so that it's gonna be negative. Six minus 54 minus 24. So negative 84. Okay, so then we know it is not perpendicular. So now we'll take a look and see if we can find a scale where a is equal to some scaler times be So what will we multiply by? Negative three to get to Well, that would be a negative 2/3. Okay, So what happens if we take negative nine and multiply that by a negative 2/3? Then we get six Okay, well, that equals the second element. A. So that's promising. So let's take six and multiply that by a negative 2/3. And what does that get? Well, that gives us a negative four. Okay, that minds Oppa's well. So the scaler is negative 2/3 and that means that he's won. Thes vectors are indeed parallel.


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