volume-of-a-parallelepiped-three-vectors-mathbfu-mathbfv-and-mathbfw-are-given-a-find-their-scalar-6

Question

Volume of a Parallelepiped Three vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ are given. (a) Find their scalar triple product $\mathbf{u} \cdot(\mathbf{v} 	imes \mathbf{w}) .$ (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine.
$$
\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{v}=3 \mathbf{i}-\mathbf{j}-\mathbf{k}, \quad \mathbf{w}=6 \mathbf{i}
$$

Rishi Kavikondala · Accepted Answer

in this problem, we have to start off by finding the scaler triple product. Given the vector is UV and doubly so. Let&#x27;s start off by finding V Cross W The cross W which we confined using the formula listed here is V two times W three. So that&#x27;s negative. One time zero, which is just zero minus V three times W two. That&#x27;s gonna be negative. One time zero, which was just zero and second part V. Three times W one. The three is negative one. W one is six that makes this term negative. Six and then v one times W three Deli 30 Therefore, the second term is zero. And for the third part V one times W two w 20 So this can automatically be made zero minus V two times w one that&#x27;s native one times six. That&#x27;s just negative. Six. And now when we simplify this, we get an answer of zero common negative six comma six. Okay. And from here we can find the dot product of you and our cross product. I do. God, he crossed w The way we do this is by multiplying all the corresponding components together So starting from the beginning, we have zero times two, obviously just zero plus negative. Two times native six, which is just negative. 12. We&#x27;re sorry. 12 plus negative. Three times six, which is just negative. 18. So negative. Eat. That makes this answer equal to native six. Because we&#x27;re just doing 12 minus 18. So our answer right here is negative six. And because we did not get an answer of zero, this means that these vectors are not co plainer. In other words, they do not all exist in the same plane. We know that to find the volume. Therefore, all we have to do is take the absolute value over a scaler triple product. So the volume is just the absolute value of negative six, which is just six

Megan Mcfarland · Answer

in this problem, you&#x27;re given three vectors and asked if I&#x27;m the dog product between you and the vector from the cross product to be cross W. And so first, we&#x27;re going to convert all of these doctors into the normal vector formula used to seeing Yume is going to be too negative too negative. Three b is going to be three negative one negative warm and w is going to be six 00 Okay, now we&#x27;re going to take the cross W this is per day be cross w, and that is equal to them Matrix Where the first rose i j k the second round with the components of B which are three negative one negative one the third row with components of W 600 Now we&#x27;re going to do this and we find that for eyes component to find that we&#x27;re gonna cross up the first column and take the determinant of this two by two matrix and detonated 1 p.m. zero comes or minus negative one times a year. That serum in a zero that&#x27;s just zero. Then we always attracted jeez, component. And to get Dr Boehner, we&#x27;re gonna cross at the middle call and take the determines of this to buy shoe matrix and we get three times zero minus negative. One time. 60 minus negative. Six year post 626 And to get Caisse component been across at the third Column, take the determines of this two by two matrix and we get three times zero minus negative. One time. Six That zero minus negative. 60 plus six is six and we have found our vector V cross W which is zero negative. 66 Now we&#x27;re going to take the dot part of this with the vector you, which again is to negative to negative three. Let me do this. We get the following two times zero Yes, negative. Two times negative. Six plus negative. Three times six Do time. 00 negative. Two times in a 6 to 12 negative. Three times six is 18 me a 12 minus 18. That&#x27;s negative. Six. So instead of part is going to be negative six. And then for Part B. They first asked us if the vectors or co planner determine if they&#x27;re co plainer. We see if the doc products between you and the cross product be crossed. W&#x27;s equal to zero in part a. We see that is not equal to zero. So therefore they&#x27;re not co plainer and then to find the volume of the apparel pipe ed. It is actually quite simple. All we do is take the dot product we found between you and the cross product of across W, and we take the absolute value of it. And so we&#x27;re going to take the absolute value of negative six. Let me see that that is equal to six, and that is the volume of a parallel piper.

Megan Mcfarland · Answer

in this problem were given three vectors and asked to provide the dot product between you and the cross product of V Cross W. And so first up in this is going to be to find that cross product to be across W. And so to do that, we&#x27;re going to first draw Matrix with its first row. Was I Jane K. It&#x27;s second row as the components of be because we&#x27;re doing be cross W. And so if we rewrite B and W in their normal doctor forms, we&#x27;re going to see the B is equal to the following victor. There&#x27;s no I components. We get zero. The J coefficient is negative. One in the K coefficient is one, and then W is going to be equal to the vector 111 And so these components go in the second row, and W&#x27;s components go in the third row. And so when we do this, we get the following eyes component to get that everyone across that the first call on take they call the determinant of this two by two matrix. It&#x27;s going to be negative one times one minus one times one or negative one of minus one, which is negative. Two. That we always subtract the Jays component and to get that warm going across that the middle column Take the determines of this two by two matrix and get zero times one minus one times warm zero minus one negative one. The negatives. Cancel that, and to get Kay when across at the third calm. Take the determines of this two by two matrix we get zero times one minus negative, one times warm or zero minus negative, warm, dizzy or plus one which is just one. And so we found our vector for the cross product of E Cross. W is going to be negative, too. One word. Everyone who doesn&#x27;t collect that with you and if we rewrite you into are the normal doctor. Former use of scene is going to be one negative 11 And so when we do this stop product, we get the following. We&#x27;re going to get war times negative, too, plus negative warrant times one close. The one times one one times negative two is negative. Two close later, one times one that&#x27;s negative. Once we get negative. Two minus one. That&#x27;s negative. Three and two that one times more was just wondering a negative three plus one. That&#x27;s negative too. And so our answer to party is going to be too negative too. Now, in part B were asked if the vectors air co planing to determine this. We have to see if the dot problems between you in the new Doctor V Cross W is equal to zero. If it is equal to zero, the vectors are complainer. We can see in this case it is not equal to zero. Therefore, the vectors air, not complainer. And so now we have to find the volume of the pair a little pipe ed to do. This is actually quite simple. We&#x27;re going to take the dog product we just found in part A, and we want to take the absolute value of it. So we&#x27;re gonna take the absolute value of negative, too, and we get to and that right there is the volume of the parallel Piper

Ryan Mcalister · Answer

this question wants us to find the scaler triple product of a of three vectors vectors that it gives our vector A which is equal to to I minus two J minus three k. I&#x27;m gonna rewrite this as just one vector. Right now, it&#x27;s a what&#x27;s called a linear combination of three different unit vectors, all scaled and added together. I&#x27;m gonna rewrite This is just one vector, and that&#x27;s going to be just a simple s too negative to negative three. Because as we know, I is 100 j 010 and k is 001 when we scale them and added together, this is what we get. I&#x27;ll do the same thing for the other vectors. B, which is, um three. I minus j minus K is just equal to three negative one negative one and see, which is just, Ah six. I is just going to be 600 So these are our vectors and were asked to find a scaler triple product. The scale a triple product is, of course, from your book a dotted with Be cross. See? No, let&#x27;s start with finding be cross e b crusty. We&#x27;re gonna find using our regular method of the determinant of a three by three Matrix will have the first row. B i j K are unit vectors. Our second and third row will be B and C. So three negative one negative one and of course, zero or excuse me six 00 When we find the determinant of this will do a co factor, decomposition along the first row will get the determinant of negative one negative 100 times I minus. Remember, we subtract when we&#x27;re doing the J term here. 36 negative 10 times j And then we add the determinant of 36 negative 10 times k. Now we know how to find determines of to a two. Major sees it. This is just going to be zero I, uh, minus six J because we have a positive six as their determinant and we are subtracting it and then we&#x27;re adding six case. Let me rewrite this and a little bit prettier of fashion is just gonna be negative. Six j plus six k. And in other words, that is zero negative. Six positive six. So we have our cross product now all that&#x27;s left to do is dot it with our a vector, remember, Are a vector was too negative to negative three So too negative to negative. Three dotted with zero negative. Six negative positive. Six. And what we do here is we take our, uh we add up the products of the corresponding, uh, components. So are I. Components multiplied together is to time zero, which is zero r j components multiple together. Negative two times. Think of six is positive 12 and R K components negative. Three times six isn&#x27;t negative. 18. So what that is 12 miners 18 is, of course, negative six. And that is our scaler triple product a dotted with the across C. Now the next part of this question asks us if this is co plainer. And if not, it wants us to find the volume of a shape that these three, uh, factors make what we know right away that since the scale a triple product is negative six. It&#x27;s not complainer. Any skeletal product that&#x27;s not zero is from three factors that are not co plainer. So a and B and C or not complaining, which means they are not all in the same plane, so they formed a parallel, a pipette apparent little pipe ed. If I can try to draw it on a two dimensional screen, will look something like this. Let&#x27;s say this is B and C is somewhere behind it. So let&#x27;s imagine this is ah, a little bit behind B. And there&#x27;s an angle here and this is in one plane. Of course, any two vectors, um, can always be put in tow. One plane and now a we know is not in this plan, so it&#x27;ll be sticking out a little bit. Let&#x27;s say it&#x27;s up here. The parallel of hype Ed, this defines is something like this. It has a base of B and C that will look like this, let&#x27;s say and then height defined by a like this, and we can connect this all here, and it&#x27;s not very well drawn, but you can imagine what it really would look like. So it&#x27;s essentially a parallelogram prism, but then you slanted a little bit with a This is Specter A So how are we gonna find the volume of this? Well, there&#x27;s a trick, but I want to show you the long way first, and we&#x27;ll get, um we&#x27;ll meet up with the trick at the very end, and it&#x27;s pretty exciting. So, uh, buckle in first, we want to find for any prism. We&#x27;re always gonna find the volume as base times height. So let&#x27;s find the volume of the base. First, we&#x27;re gonna treat B and C as the base. So here is vector B, and this is now the two dimensional, like looking straight at the base from above. And here, see? And we&#x27;ve got, like, something like this and something like this. So when we&#x27;re looking at B and C, let&#x27;s say this is data DiVall In the area of a parallelogram like this is the base times the height perpendicular to that base of this right here. Well, we know what see is and we know if it is. So this height right here is simply see sign Seda, Remember, since we&#x27;re finding the opposite side of this right triangle here that&#x27;s gonna be see Scient. Ada. So the height or excuse me? The area of this parallelogram is the magnitude of B times, the magnitude of C times sign of data. And now this is where it gets exciting because you probably already recognize this as the magnitude of be cross. See? So the magnitude to be cross see is what we&#x27;re looking for as the base. Well, we had we be cross. He was part of this scale, a triple product. So let&#x27;s see how this will play into it. We have the base and we want the height. While the height again is not gonna be a it&#x27;s going to be the, uh, the portion of a the component of a that is perpendicular to our base or orthogonal from our base. What we know that be cross see, is orthogonal to this Plan B is B cross. He is perpendicular to be and to see. So if there&#x27;s an angle fader between be cross C and A, we want to find the component of A that is alongside this, uh, this value here alongside Be cross, See? Well, that&#x27;s gonna be a times this co sign of data. The blue part is eight times the co sign of Fada. So we want to multiply the base times this height. The base we know is the magnitude of be cross. See, I&#x27;m gonna write this in. Ah, let me to read here. The base is a magnitude of be crossed, see? And the height is the manager of a times the co sign of Fada. And now this is where it gets really, really exciting. Because this is exactly the formula for a dotted with be cross. See? Well, hold on. We just figured out what a daughter be cross. He was. We said it was, um we said it was negative. Six. So we&#x27;re finding the magnitude of negative six. Well, magnitude just means absolute value. The absolute value of six is, of course, positive. Six. So we found that the volume of this scale a triple proud of this peril pipette is equal to the absolute value of the scale. A triple product, which is six. And that&#x27;s gonna be true for any three non co plainer, uh, vectors. The area of the volume of this parallel pipe. It will be the absolute value of the scale. A triple product. So this is the really exciting part. Um, I hope you enjoyed. That&#x27;s your final answer.

Megan Mcfarland · Answer

in this problem, you&#x27;re given a three vectors and asked to find that you don&#x27;t that in parentheses v across the W. And so to do that first we&#x27;re going to determine what be cross W. Is. And to do that, we&#x27;re going to create a matrix with its first row as I, J and K. It&#x27;s second row with the components of the because we&#x27;re doing it be cross W. And so that&#x27;s going to be negative. Three, two and one. And the third row is going to have the components of W zero eight and 10. No, you&#x27;re the cross product. And to get eyes component first, we&#x27;re going to a cross out the first column and take the determinants of this two by two matrix just going to be two times 10 minus one times eight. That is 20 minus eight that is 12. Then we&#x27;re going to subtract OJ&#x27;s component, as we always do. Then we went across at the middle column and take the determines of this two by two Matrix is going to be noted three times 10 minus one times zero. That&#x27;s negative. 30 minus zero. That&#x27;s negative. Three negatives. Cancel out to create a positive and then to get K&#x27;s component. We&#x27;re going to cross out the third comma and we want to take the determinants of this two by two matrix and we want to get negative three times eight minus two times zero. That&#x27;s negative. 24 minus zero. That&#x27;s just negative. 24. Okay, And now we have found our doctor for V Cross W, which is 12 30 Negative 24. And we&#x27;re going to dock product that with you, which is 123 Okay. And when we do that, we&#x27;re going to multiply one times 12 plus two times 30 Those three times negative 24 one times 12 is 12 plus two times 30 12 plus 60 72 24 times three is also 72. So we get 70 to minus 72 that is equal to zero. So our answer for part A is going to be zero now. Part B then asks about whether the vectors are co plainer and for the doctors to be co plainer. You dot the cross w has to be equal to zero. Sure enough, that&#x27;s what we found. And so we found the directors are indeed co plainer. And because they&#x27;re co planner, we don&#x27;t need to find the volume of the parallel pipe. And so that is our answer to party.

Rishi Kavikondala · Answer

in this problem, we&#x27;re given three vectors U V and W. And the first part is to find the scaler triple product, which we can do by using this formula listed at the top. So for part A, let&#x27;s start off by finding the cross product the cross w so ve crossed w is equal to Okay, Well, starting with the two times w three, so be, too is one W three is zero. That&#x27;s gonna be zero minus the three times W two be three is one W two is four. So that&#x27;s gonna be for and then, for these second number, we have V three times w one That&#x27;s gonna be one time seven, which is just seven minus V one times w three w three is zero. So it&#x27;s just going to be for the second part. And then for the third part of the cross product, we have V one times w two v one is one w to us four. That gives us four minus V two times w won&#x27;t be too is one Billy one is seven. So that gives us seven. And when we simplify this, we get that the cross product is equal to negative for comma seven comma native three. Now we have to find the dot product of this vector and you so you you got be cross. W is equal to these sums of the products of the corresponding terms in these two vectors. So starting off with the first terms in each of the two vectors, we have negative for times three. That&#x27;s gonna be native 12th plus zero times seven. That&#x27;s going to be zero and then plus negative four times native three. That&#x27;s just 12. And when we find the sum of these three terms, we get zero. And obviously the magnitude of zero is just zero. So therefore, our scaler triple product is equal to zero. Now for part B, we&#x27;re asked to determine if these three vectors are co plainer. We know that they are complainer because the scaler triple A product is equal to zero. And we also know that if the volume is zero, then these three vectors are co player. The scaler triple product is equivalent to the volume, which means that since both are equal to zero, these three vectors are co plainer and the volume is zero. So our final answer is that they are co plainer and the volume is zero

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(a)-6
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